Introduction to Schemes

Introduction to Schemes

Geir Ellingsrud and John Christian Ottem

University of OsloVersion 2.2

18th May 2022

Contents

Contents 2

Chapter 1: Sheaves 14

Sheaves and presheaves 14

Morphisms between (pre)sheaves 19

Stalks 22

Interlude: A primer on limits 24

A family of examples – Godement sheaves 29

Sheafification 31

Kernels, images and quotients 36

Direct and inverse images 42

Sheaves defined on a basis 45

Chapter 2: The Prime Spectrum 49

The spectrum of a ring 49

Affine spaces 54

Distinguished open sets 55

Irreducible closed subsets 57

Morphisms between prime spectra 63

Chapter 3: Schemes 70

The structure sheaf on the spectrum of a ring 70

Locally ringed spaces 75

Affine schemes 79

Schemes in general 81

Properties of the scheme structure 86

Chapter 4: Gluing and first results on schemes 92

Gluing maps of sheaves 92

Gluing sheaves 93

Gluing schemes 95

Gluing morphisms of schemes 97

Universal properties of maps into affine schemes 98

Brave new varieties 101

Chapter 5: Examples constructed by gluing 103

Gluing two schemes together 103

A scheme that is not affine 104

The projective line 105

The affine line with a doubled origin 109

Semi-local rings 110

The blow-up of the affine plane 111

Projective spaces 114

Line bundles on P1116

Hyperelliptic curves 120

Double covers of PnA 121

Hirzebruch surfaces 122

Chapter 6: Geometric properties of schemes 123

Noetherian schemes 123

The dimension of a scheme 126

Normal schemes and normalization 131

Chapter 7: Fibre products 136

Introduction 136

Fibre products of schemes 139

Examples 145

Base change 148

Scheme theoretic fibres 150

Chapter 8: Separated schemes 155

The diagonal 155

Separated schemes 157

Chapter 9: Projective schemes 166

Motivation 166

Basic remarks on graded rings 168

The Proj construction 170

Functoriality 178

Projective schemes 181

The Veronese embedding 182

More intricate examples 183

Chapter 10: Sheaves of modules 187

Sheaves of modules 188

Quasi-coherent sheaves 195

Coherent sheaves 206

Categorical and Functorial properties 208

Closed immersions and closed subschemes 213

Chapter 11: Locally free sheaves 218

Locally free sheaves and projective modules 219

Invertible sheaves and the Picard group 222

Extending sections of invertible sheaves 224

Operations on locally free sheaves 226

Chapter 12: Sheaves on projective schemes 228

The graded tilde-functor 229

Serre’s twisting sheaf O(1) 233

The associated graded module 236

Quasi-coherent sheaves on Proj R 238

Closed subschemes of projective space 240

The Segre embedding 242

Locally free sheaves on P1243

Two important exact sequences 243

Two examples of locally free sheaves 244

Appendix: Graded modules 247

Chapter 13: First steps in sheaf cohomology 249

Some homological algebra 250

Cech cohomology of a covering 252

Cech cohomology of a sheaf 257

Chapter 14: Computations with cohomology 261

Cohomology of sheaves on affine schemes 261

Cohomology and dimension 265

Cohomology of sheaves on projective space 266

Extended example: Plane curves 269

Extended example: The twisted cubic in P3270

Extended example: Non-split locally free sheaves 271

Extended example: Hyperelliptic curves 272

Extended example: Bezout’s theorem 274

Cech cohomology and the Picard group 274

Chapter 15: Divisors and linear systems 277

Weil divisors 279

Cartier divisors 288

Effective divisors and linear systems 293

Appendix 295

Quadrics 296

Extended example: Hirzebruch surfaces 300

Chapter 16: Maps to projective space 303

Globally generated sheaves 304

Morphisms to projective space 305

Application: Automorphisms of Pn310

Projective space as a functor 310

Projective embeddings* 313

Ample invertible sheaves and Serre’s theorems 315

Chapter 17: Differentials 319

Zariski tangent spaces 321

Derivations and Kähler differentials 324

Properties of Kähler differentials 328

The sheaf of differentials 334

The Euler sequence and differentials of PnA 337

Relation with the Zariski tangent space 339

The conormal sheaf 343

The quadric surface 344

Chapter 18: Properties of morphisms 346

Finite morphisms 346

Flat morphisms 349

Smooth morphisms 349

Etale morphisms 349

Proper morphisms 349

Finite and quasi-finite morphisms 358

The valuative criterion* 361

Chapter 19: Curves 363

Curves 363

The genus of a curve 363

Divisors on Curves 364

Morphisms of curves 365

Hyperelliptic curves 367

Chapter 20: The Riemann–Roch theorem 370

Chapter 21: The Serre duality theorem 373

Proof of Serre duality for X = P1374

Two cohomological lemmas 374

Schemes obtained by gluing two affines 375

The dualizing sheaf 376

The dualizing sheaf equals the canonical sheaf 377

Chapter 22: Applications of the Riemann–Roch theorem 379

Very ampleness criteria 379

Curves on P1 ˆ P1380

Curves of genus 0 382




Curves of Genus 4 389

Chapter 23: More on vector bundles 391

The vector bundle associated to a locally free sheaf* 391

Vector bundles in general* 392

Extended example: The tautological bundle on Pnk * 394

Chapter 24: Further constructions and examples 399

Grassmannians 399

Some explicit blow-ups 402

Resolution of some surface singularities 407

A scheme without closed points 412

Every finite partially ordered set is a spectrum 415

A most peculiar scheme 419

A: Some results from Commutative Algebra 425

Discrete valuation rings 425

Unique factorization domains 426

Hartog’s extension theorem 427

Projective modules 427

Dimension theory 427

B: More on sheaf cohomology 428

Flabby sheaves 429

The Godement resolution 431

Sheaf cohomology 432

Cech vs sheaf cohomology 436

Godement vs. Cech 441

C: Solutions 443

Bibliography 471

Acknowledgements

Thanks to Georges Elencwajg, Frank Gounelas, Johannes Nicaise, Dan Petersen,Kristian Ranestad, Stefan Schreieder for comments and suggestions. Also,thanks to Anne Brugård, Edvard Aksnes, Søren Gammelgaard, Simen WestbyeMoe, Torger Olson, Nikolai Thode Opdan, Gabriel Ribeiro, Magnus Vodrup andQi Zhu for numerous corrections to the text.

Any further comments or corrections are welcome:https://docs.google.com/document/d/1T7R9ROah2RyR6mXMesEHZgk2de4cSOC-GMXgRSGbd34/

edit?usp=sharing

https://docs.google.com/document/d/1T7R9ROah2RyR6mXMesEHZgk2de4cSOC-GMXgRSGbd34/edit?usp=sharing

https://docs.google.com/document/d/1T7R9ROah2RyR6mXMesEHZgk2de4cSOC-GMXgRSGbd34/edit?usp=sharing

Introduction

Introduction to the schemes formalism

If X is an affine variety over an algebraically closed field k, then it has anaffine coordinate ring A(X) of regular functions X Ñ k, and A(X) completelycharacterises X up to isomorphism. The ring A(X) is a particularly nice ring;it is finitely generated as a k-algebra over an algebraically closed field, andit has no zero-divisors (two non-zero regular functions can not have a zeroproduct). The correspondence X ÞÑ A(X) sets up a bijective correspondencebetween algebra and geometry: affine varieties correspond exactly to the k-algebrasof the above type, and morphisms between affine varieties correspond to k-algebra homomorphisms between the coordinate rings. The modern formulationof algebraic geometry, due to Grothendieck, vastly generalizes this picture,replacing the k-algebras with the more general category of commutative rings.

AlexanderGrothendieck(1928 – 2014)

$

’

&

’

%

finitely generated k-algebrasover an algebraically closed field,with no zero-divisors

,

/

.

/

-

Ă

!

rings)

ˇ

ˇ

ˇ

ˇ

ˇ

ˇ

!

affine varieties)

Ă

!

affine schemes)

.

The affine varieties are the building blocks of general varieties; a variety is aspace which comes with an open covering made out of affine varieties (it is‘glued’ together by affine varieties). A scheme is similarly a space which is gluedtogether by affine schemes.

Thus schemes provide a vast generalization of algebraic varieties. Themain advantage with schemes is that many of the geometric arguments andintuition carry over from the classical setting of varieties, and algebro-geometricarguments can be used to study more general problems in other branches ofmathematics (e.g., number theory).

11

We should emphasise that the main goal of algebraic geometry is still tounderstand algebraic varieties over a field. However, enlarging our category toschemes offers tremendous benefits. Not only do schemes provide the ‘correct’way of setting up the theory, but there are many examples where scheme theoryprovides more flexibility and conceptual clarity (e.g., being able to switch baserings).

General ringsThere are of course many reasons why going from this rather restrictive classof rings to general ring is desirable. The most obvious example comes fromnumber theory, where one often studies solutions to equations with integercoefficients (‘Diophantine equations’) over non-algebraically closed fields (suchas Q) or more general rings (such as Z, Zp or C(t)).

For instance, Fermat’s last theorem implies that there are no solutions to theequation x3 + y3 = z3 except the ‘trivial solutions’ where one of the variablesis 0. In geometric terms, there are no rational points on the projective curveV(x3 + y3 ´ z3) in P2, besides the three points (0 : 1 : 1), (1 : 0 : 1) and(1 : ´1 : 0).

To make sense of this statement, we need to be able to speak of plane curvesdefined over the field Q. There is a naive fix, to just go to the algebraic closure,but this misses the main points: The curves x3 + y3 = z3 and x3 + y3 = 9z3 areisomorphic over Q The coordinate change

z1 = 3?9z is allowedover Q, but not overQ.

, but the first has only a few trivial points over Q and thesecond has infinitely many.

Prime ideals, rather than maximal ideals.Points of affine varieties can be studied in terms of maximal ideals. If X Ă An

is an affine variety, with coordinate ring A(X) = k[x1, . . . , xn]/I, then points ofX are in bijection with the maximal ideals in A(X), which are all of the form(x1 ´ a1, . . . , xn ´ an). Moreover, morphisms of affine varieties φ : X Ñ Y are inbijective correspondence with their coordinate rings φ˚ : A(Y)Ñ A(X).

However, when going to general rings, we will see that it is in fact muchmore natural to include all the prime ideals, not just the maximal ones. Oneof the first definitions in scheme theory is the affine scheme Spec A for a ringA; this is a topological space whose set of points in the set of prime ideals inA. Thus any ring gives rise to a topological space, and ring maps φ : A Ñ Binduce maps φ˚ : Spec B Ñ Spec A, by sending a prime ideal p P Spec A toφ´1(p) P Spec B.

The transition from maximal to prime ideals is something that then is almostforced on us by functoriality: Consider the inclusion k[x] ãÑ k(x): the maximalideal (0) of k(x) pulls back to the prime ideal (0) of k[x] which is not maximal.Including the set of all prime ideals, we obtain a very nice category which

12

behaves essentially just like the category of commutative rings. In fact, one ofthe main basic theorems of scheme theory is that the category of affine schemesis equivalent to the category of rings.

Beside of these categorical advantages, there are also a number of placesin more classical settings where thinking of points defined by prime idealsprovides an extra conceptual clarity.

For instance, in the Fermat example, we consider the affine scheme

X = Spec Z[x, y, z]/(x3 + y3 ´ z3)

(where now X denotes set of prime ideals in the ring Z[x, y, z]/(x3 + y3 ´ z3)).For number theoretic-purposes it turns out to be fruitful to look for morphismsSpec Z Ñ X; as we will see, these are in bijection with ring maps going theother way, i.e. ring maps

Z[x, y, z]/(x3 + y3 ´ z3)Ñ Z.

or equivalently, since such maps are determined by where the variables x, y, z aresent — integers a, b, c such that a3 + b3 = c3. Likewise, morphisms Spec Q Ñ Xcorrespond to solutions of a3 + b3 = c3 over the rational numbers.

Nilpotents and zero-divisorsNilpotents and zero-divisors appear already in Bezout’s theorem, when we need‘multiplicities’ to get a satisfactory intersection theory. For instance, the conicC = Z(y´ x2) intersects a general line in two points, but the line y = 0 only atthe origin (with multiplicity two). Looking at the ideals, it is intuitive that theintersection should correspond to the ideal (y´ x2, y) = (x2, y) which gives usthe intersection number (’multiplicity 2’), but that ideal does not correspond toan algebraic subvariety of C, as it is not radical.

Similar phenomena appear naturally when you take ‘limits’ (or ‘deforma-tions’) of algebraic varieties. As a basic example, consider the family of curvesin A2 given by Vt = Z(y2 ´ tx). Then for t ‰ 0, Vt is curve of degree 2 (a conic),whereas for t = 0 it is a ‘double line’. The limit as t Ñ 0 is problematic; if youinsist that it should correspond to an affine variety, it would have to be theclosed set Z(y), which has degree 1, and then the degree of Vt does not dependcontinuously on t.

These problems vanish when passing to schemes: it is completely unproblem-atic to work with affine schemes associated to the coordinate rings k[x, y]/(x2, y)or k[x, y]/(y2) respectively. Note however that we have to adjust our viewpointabout regular functions: For instance, in the second example the element ‘y’ isnon-zero in the ring k[x, y]/(y2) - but it takes the value 0 at each of the closedpoints of the underlying topological space (the x-axis).

13

GluingA variety is by definition glued together by affine varieties. There are howeveritems in the definition that restrictions on how this can be done: one allows onlyfinitely many affines, and the end product has to be irreducible, and satisfy the‘Hausdorff axiom’ (that the diagonal is closed). In analogy with this, a ‘scheme’is defined to be a space which is glued out of affine schemes. But here there areno restrictions. An important feature of the catefory of schemes is that it allowsus to glue unconditionally; for an arbitrary collection of schemes, together withsome ‘gluing data’, one can produce a new scheme.

The perspective of viewing a space as glued out of simpler spaces can befound everywhere in mathematics. For instance, differential manifolds are bydefinition spaces constructed by gluing together copies of Rn. Another exampleis the projective space Pn, which is commonly studied using its covering by(n + 1) affine spaces An.

The perspective stands in contrast to, describing a projective variety X by itsequations in some projective space Pn. The first viewpoint is more intrinsic, andmakes definitions more natural and computations simpler. For instance, we candefine the ‘normalization’ of a variety by normalizing each chart (but it wouldbe unthinkable to try to study this in terms of explicit equations). Perhaps evenmore strikingly, even when gluing together affine varieties we may obtain newvarieties which may fail to be embeddable in any An or Pn. So to study thevariety, we have no choice but to use its local charts.

Chapter 1

Sheaves

The concept of a sheaf was conceived in the German camp for prisoners of warcalled Oflag XVII where French officers taken captive during the fighting inFrance in the spring 1940 were imprisoned. Among them was the mathematicianand lieutenant Jean Leray. In the camp he gave a course in algebraic topology(!!)during which he introduced some version of the theory of sheaves. In modernterms, Leray was aiming to compute the cohomology of a total space of afibration in terms of invariants of the base and the fibres (and the fibration itself).To achieve this, in addition to the concept of sheaves, he also invented ’spectralsequences’.

Jean Leray(1906 – 1998)

After the war, Henri Cartan and Jean-Pierre Serre among others developedthe theory further, and finally the theory was brought to the state as we know ittoday by Alexander Grothendieck.

1.1 Sheaves and presheaves

A common theme in mathematics is to study spaces by describing them in termsof their local properties. A manifold is a space which looks locally like Euclideanspace; a complex manifold is a space which looks locally like Cn; an algebraicvariety is a space that looks locally like the zero set of a set of polynomials.Here it is clear that point set topology alone is not enough to fully capture thesethree notions. In each case, the space comes equipped with a distinguishedset of functions that capture the essence of what the space is: C8-functions,holomorphic functions, and polynomials respectively.

Sheaves provide the general framework for discussing such functions; theyare objects that satisfy basic axioms satisfied in each of the examples above.To explain what these axioms are, let us consider the primary example of asheaf: the sheaf of continuous maps on a topological space X. By definition, Xcomes with a collection of ‘open sets’, and these encode what it means for amap f : X Ñ Y to another topological space Y to be continuous: for every open

sheaves and presheaves 15

U Ď Y, the set f´1(U) should be open in X. For two topological spaces X andY, we can define for each open U Ď X, a set of continuous maps

C(U, Y) = t f : U Ñ Y | f is continuous u.

Note that if VĎU is open, then the restriction f |V of a continuous function f toV is again continuous, so we obtain a map

ρUV : C(U, Y)Ñ C(V, Y)

f ÞÑ f |V .

Moreover, note that if WĎVĎU, we can restrict to W by first restricting to V,and so ρUW = ρVW ˝ ρUV . The collection of the sets C(U, Y) together with theirrestriction maps ρUV constitutes the sheaf of continuous maps from X to Y.

An essential feature of continuity is that it is a local property; f is continuousif and only if it is continuous in a neighbourhood of every point, and of course,two continuous maps that are equal in a neighbourhood of every point, have tobe equal everywhere. Moreover, continuous functions can be glued: Given anopen covering tUiuiPI of an open set U, and continuous functions fi P C(Ui, Y),so that for each x P X the value fi(x) does not depend on i, we can patch themaps fi together to form a continuous map f : U Ñ Y which satisfies f |Ui = fi

for each i; we just define f (x) = fi(x) for any i such that x P Ui.Essentially, a sheaf on a topological space is a structure that encodes these

properties. In each of the examples above, there is a corresponding sheaf ofC8-functions, holomorphic functions, and regular functions respectively. Onemay think of a sheaf as a distinguished set of functions, but they can also bemuch more general mathematical objects which in a certain sense behave as setsof functions. The main aspect is that we want the distinguished properties to bepreserved under restrictions to open sets, and that the objects are determinedfrom their local properties.

PresheavesThe concept of a sheaf may be defined for any topological space, and the theoryis best studied at this level of generality. We begin with the definition of apresheaf.


Definition 1.1 Let X be a topological space. A presheaf of abelian groupsF on X consists of the following two sets of data:

i) for each open UĎX, an abelian group F (U);ii) for each pair of nested opens VĎU a group homomorphism (called

restriction maps)

ρUV : F (U)Ñ F (V).

The restriction maps must furthermore satisfy the following two conditions:

i) for any open UĎX, we have ρUU = idF (U);ii) for any three nested open subsets WĎVĎU, one has ρUW = ρVW ˝

ρUV .

We will mostly write s|V for ρUV(s) when s P F (U). The elements of F (U)

are usually called Sections of a presheaf(seksjoner av etpreknippe)

sections (or sections over U). The notation Γ(U,F ) for thegroup F (U) is also common usage; here Γ is the ‘global sections’-functor (it isfunctorial in both U and F ).

The notion of a presheaf is not confined to presheaves of abelian groups.One may speak about presheaves of sets, rings, vector spaces or whatever youwant: indeed, for any category C one may define presheaves with values in C.The definition goes just like for abelian groups, the only difference being thatone requires the gadgets F (U) to be objects from the category C, and of course,the restriction maps are all required to be morphisms in C.

We are certainly going to meet sheaves with a lot more structure than themere structure of abelian groups, e.g. sheaves of rings, but they will usuallyhave an underlying structure of abelian group, so we start with these. Thatbeing said, sheaves of sets play a great role in mathematics, and in algebraicgeometry, so we should not completely wipe them under the rug. Most resultswe establish for sheaves of abelian groups can be proved mutatis mutandis forsheaves of sets as well, as long as they can be formulated in terms of sets.

SheavesWe are now ready to give the main definition of this chapter:


Definition 1.2 A presheaf F is a sheaf if it satisfies the two conditions:

i) (Locality axiom) Given an open subset UĎX with an open coveringU = tUiuiPI and a section s P F (U). If s|Ui = 0 for all i, thens = 0 P F (U).

ii) (Gluing axiom) If U and U are as in (i), and if si P F (Ui) is acollection of sections matching on the overlaps; that is, they satisfy

si|UiXUj = sj|UiXUj

for all i, j P I, then there exists a section s P F (U) so that s|Ui = si

for all i.

Note that the Locality axiom says that sections are uniquely determined fromtheir restrictions to smaller open sets. The Gluing axiom says that you areallowed to patch together local sections to a global one, provided they agree onoverlaps, just like what was the case in the first example of continuous functions.

There is a nice concise way of formulating the two sheaf axioms at once. Foreach open cover U = tUiu of an open set UĎX there is a sequence

0 F (U)ś

i F (Ui)ś

i,j F (Ui XUj)α β

(1.1)

where the maps α and β are defined by the two assignments α(s) = (s|Ui)i, andβ(si) = (si|UiXUj ´ sj|UiXUj)i,j. Then F is a sheaf if and only if these sequencesare exact. Indeed, exactness at F (U) means that α is injective, i.e., that s|Ui = 0implies that s = 0 (the Locality axiom). Exactness in the middle means thatKer β = Im α, i.e., elements si satisfying si ´ sj = 0 on Ui XUj must come froman element s P F (U) (the Gluing axiom).

This reformulation is sometimes handy when proving that a given presheafis a sheaf. Moreover, since F (U) = Ker β, we can often use it to compute F (U)

if F (Ui) and F (Ui XUj) are known.Example 1.3 The empty set. There is a subtle point about taking U to be theempty set in the definition of a sheaf. If F is a sheaf, we are forced to defineF (H) = 0. Indeed, the empty set is covered by the empty open covering,and since the empty product equals 0, the sheaf sequence (1.1) looks like0 Ñ F (H)Ñ 0 Ñ 0. K

Examples(1.4) Continuous functions. Take X = Rn and let C(X, R) be the sheaf whosesections over an open set U is the ring of continuous real valued functions onU, and the restriction maps ρUV are just the good old restriction of functions.Then C(X, R) is a sheaf of rings (functions can be added and multiplied), andboth the sheaf axioms are satisfied. Indeed, any function f : X Ñ R, which


restricts to zero on an open covering of X is the zero function. Also, givencontinuous functions fi : Ui Ñ R agreeing on the overlaps Ui XUj, we can formthe continuous function f : U Ñ R by setting f (x) = fi(x) for any i such thatx P Ui.

In fact, the argument from the beginning of this chapter shows that for anytwo topological spaces X and Y, the presheaf F (U) = C(U, Y) of continuousmaps f : U Ñ Y forms a sheaf (they are sheaves of sets here, because we cannotin general add or multiply maps).(1.5) Holomorphic functions. For a second familiar example, let X Ď C be anopen set. On X one has the sheaf OX of holomorphic functions. That is, for anyopen U Ď X the sections OX(U) is the ring of complex differentiable functionson U.

One can relax the condition of holomorphy to get the larger sheaf KX ofmeromorphic functions on X. This sheaf contains OX as a subsheaf, and thesections over an open U are the meromorphic functions on U. Meromorphic means:

holomorphic on all ofU except for a set ofisolated points, whichare poles of thefunction.

In a similar way,one can get smaller sheaves contained in OX by imposing vanishing conditionson the functions. For example if p P X is any point, one has the sheaf denotedmp of holomorphic functions vanishing at p. Convince yourself that this indeedis a subsheaf of ideals of OX.(1.6) A presheaf which is not a sheaf. Let us continue the set-up in Example 3 toexhibit an example of a presheaf which is not a sheaf. Let X = Czt0u, and letOX denote the sheaf of holomorphic functions. OX contains the subpresheafgiven by

F (U) = t f P OX(U) | f = g2 for some g P OX(U) u.

This is not a sheaf, because the Gluing axiom fails: the function f (z) = z isholomorphic, and has a holomorphic square root near any point x P X, but itis not possible to glue these together to a global square root function

?z on all

of X. Note however, that the Locality axiom holds as F is a subpresheaf of thesheaf OX.(1.7) Constant presheaf. For any space X and any abelian group A, one has the

Constant (pre)sheaves(konstante(pre)knipper)

constant presheaf whose group of sections over any nonempty open set U equalsA and equals 0 if U = H. This is not a sheaf whenever A is not the trivial group.For instance, if U = U1YU2 is a disjoint union, any choice of elements a1, a2 P Awill give sections over U1 and U2 respectively, and they automatically match onthe intersection, which is empty. But if a1 ‰ a2, they cannot be glued. In fact, the constant

presheaf is a sheaf ifand only if any twonon-empty opensubsets of X havenon-emptyintersection. Algebraicvarieties with theZariski topology areexamples of suchspaces.

There is a quick fix for this. We can define the following sheaf AX by letting

AX(U) = t f : U Ñ A | f is continuous u

where we give A the discrete topology. For a connected open set U we thenhave AX(U) = A. More generally, since f must be constant on each connected

morphisms between (pre)sheaves 19

component of U, it holds true that

AX(U) »ź

π0(U)

A, (1.2)

where π0(U) denotes the set of connected components of U. As before, we alsomust put AX(H) = 0.

The new presheaf AX is now a sheaf, the constant sheaf on X with value A.That being said, the sheaf AX is not quite worthy of its name as it is not quiteconstant.(1.8) Skyscraper sheaf. Let A be a group. For x P X define a presheaf A(x) by

A(x)(U) =

#

A if x P U,

0 otherwise.

It is straighforward to check that this is a sheaf, usually called a Skyscraper sheaves(skyskraperknippe)

skyscrapersheaf skyscraper sheaf.Our main interest in this book will still be the following:(1.9) Algebraic varieties. Let X be an algebraic variety (e.g. an algebraic set inAn

k or Pnk ) with the Zariski topology. For each open U Ď X, define

OX(U) =

#

f : U Ñ k

ˇ

ˇ

ˇ

ˇ

ˇ

for each point x P U, f can be represented asa quotient of polynomials g/h where h(x) ‰ 0

+

This is indeed a sheaf: locality holds, because if f : U Ñ k restricts to the zerofunction on an open covering, it is the zero function. If we are given regularfunctions fi : Ui Ñ k on an open overing Ui of U, that agree on the overlaps,they certainly glue to a continuous function f : U Ñ k Again: define

f : U Ñ k byf (x) = fi(x)whenever x P Ui .

. The function f is alsoregular, because it restricts to fi on Ui, and these are locally expressible as g/hthere.

K

1.2 Morphisms between (pre)sheaves

A Morphism ofpresheaves (morfi avpreknipper)

morphism (or simply a map) φ : F Ñ G of (pre)sheaves on a space X is a collectionof group homomorphisms φU : F (U) Ñ G(U) indexed by the open sets in Xand compatible with the restriction maps. In other words, the following diagramcommutes for each inclusion VĎU:

F (U) G(U)

F (V) G(V).

φU

ρUV ρUV

φV

In this way, the sheaves of abelian groups on X form a category AbShX whoseobjects are the sheaves and the morphisms the maps between them. The


composition of two maps of sheaves is defined in the obvious way as thecomposition of the maps on sections. Likewise, we have the category AbPrShX

with the presheaves of abelian groups as objects and morphisms the mapsbetween them. For short, one frequently uses the names Abelian (pre)sheaves

(abelske (pre)knipper)abelian sheavesabelian

sheaves and abelian presheaves respectively.As usual, a map φ between two (pre)sheaves F and G is an isomorphism

if it has a two-sided inverse, i.e. a map ψ : G Ñ F such that φ ˝ ψ = idG andψ ˝ φ = idF .

If F is a presheaf on X, a Sub(pre)sheaf(under(pre)knippe)

subpresheaf G is a presheaf such that G(U) Ď F (U)

for every open U Ď X, and such that the restriction maps of G are induced bythose of F . If F and G are sheaves, of course G is called a subsheaf.Example 1.10 Let X = R and let let Cr(X, R) be the sheaf of functions f :U Ñ R which are r times continuously differentiable (note that this is indeeda sheaf ). The differential operator D = d/dx defines a morphism of sheavesD : Cr(X)Ñ Cr´1(X). K

Example 1.11 The sheaf of homomorphisms. Given two abelian presheaves F andG on a space X, we may form a presheaf HomX(F ,G) by letting the sectionsover an open U be given by

HomX(F , )G(U) = Hom(F |U ,G|U),

and letting the restriction maps, well, be the restrictions: if VĎU is anotheropen sets and φ : F |U Ñ G|U is a map, the restriction of φ to V is simply therestriction φ|V . Notably, HomX(F ,G) will in fact be a sheaf when G is a sheaf.

Let us verify the two sheaf axioms, and we begin with the Locality axiom.Assume that φ : F |U Ñ G|U is a section and that φ|Ui = 0 for all members froman open cover tUiuiPI of U. For every open VĎU and every section s P F (V) itthen holds that φ(s)|VXUi = 0 for all i, and hence φ(s) = 0 by the Locality axiomfor G.

Then to the Gluing axiom: we are given a bunch of maps φi : FUi Ñ GUi

which coincides on the overlaps Uij = Ui XUj, and we are to define a mapφ : F |U Ñ G|U restricting to each φi. This amounts to giving appropriate mapsfrom FV to GV for each open VĎU, but replacing each Ui by V XUi we maywell assume that V = U. The Gluing axiom for G permits us to construct φU :pick a section s P F (U), and form si = φ(s|Ui). On the overlap Uij one has

si|Uij = φi(s|Ui)|Uij = φi|Uij(s|Uij) = φj|Uij(s|Uij) = φj(s|Us)|Uij = sj|Uij ,

and the si’s may be glued together to give the section φU(s) in G(U). Therequired properties, that the φU’s are compatible with restrictions and each isadditive, follow readily, but industrious students are recommended to check it.

K


Exercises(1.1) Let X be the set with two elements with the discrete topology. Find aˇ

presheaf on X which is not a sheaf.(1.2) In the notation of Example 3, the differential operator gives a map ofˇ

sheaves D : OX Ñ OX, where as previously X Ď C is an open set. Show that theassignment

A (U) = t f P OX(U) | D f = 0 u

defines a subsheaf A of OX. Show that if U is a connected open subset of X,one has A (U) = C. In general for a not necessarily connected set U, show thatA (U) =

ś

π0(U) C where the product is taken over the set π0(U) of connectedcomponents of U.

The next exercise suggests another way of thinking about sheaves: they are‘sections’ of a map Y Ñ X from some space into X:(1.3) A Riemann surface. Let X = Czt0u and let Y denote the complex ’parabola’ˇ

Y = t (x, y) | y2 = x u Ă CˆC

Let π : Y Ñ X be the projection onto the first factor. Consider the presheafon X given by

G(U) = t f : U Ñ Y | f is holomorphic, and π ˝ f = idU u.

Show that G is a sheaf.(1.4) The Möbius strip. Consider the Möbius strip M with its projection π : M Ñ

S1 to S1. This is an example of a fiber bundle; any fiber of π is homeomorphic tothe same space, the unit interval [0, 1]. The sheaf F of sections of π is definedby

F (U) = t s : U Ñ M | s is continuous and π ˝ s = idU u.

Show that F is a sheaf on S1.(1.5) Let X Ď C be an open set, and assume that a1, . . . , ar are distinct points inX and n1, . . . , nr natural numbers. Define F (U) to be the set of those functionsmeromorphic in U, holomorphic away from the ai’s and having a pole orderbounded by ni at ai. Show that F is a sheaf of abelian groups. Is it a sheaf ofrings?(1.6) The definition of a presheaf may be phrased purely in categorical termsin the following way. Consider the category openX, whose objects are the opensets U in X and whose morphisms are the inclusion maps U Ă V between opensets. Show that a presheaf with values in the category C is the same thing as acontravariant functor

F : openX Ñ C.

M

stalks 22

1.3 Stalks

Suppose we are given a presheaf F of abelian groups on X. With every pointx P X there is an associated abelian group Fx called the Stalk of a (pre)sheaf at

a point (stilken til et(pre)knippe i et punkt)

stalk of F at x. Onethinks of Fx as a sort of ‘limit’ of the groups F (U) as we zoom in on x usingsmaller and smaller neighbourhoods containing x. The elements of Fx arecalled

Germs of sections(kimen til en seksjon)germs of sectionsnear x; they are essentially the sections of F defined

in some sufficiently small neighbourhood of x. The definition of Fx goes asfollows: We begin with the disjoint union

š

xPU F (U) whose elements we indexas pairs (s, U) where U is any open neighbourhood of x and s is a section ofF (U). We want to identify sections that coincide near x; that is, we declare(s, U) and (s1, U1) to be equivalent, and write (s, U) „ (s1, U1), if there is an openV Ď U XU1 with x P V such that s and s1 coincide on V; that is, if one has

s|V = s1|V .

This is clearly a reflexive and symmetric relation. And it is transitive as well:if (s, U) „ (s1, U1) and (s1, U1) „ (s2, U2), one may find open neighbourhoodsVĎU XU1 and V1 Ď U1 XU2 of x over which s and s1, respectively s1 and s2,coincide. Clearly s and s2 then coincide over the intersection V1 XV. Thus therelation „ is an equivalence relation.

Definition 1.12 The stalk Fx at x P X is defined as the set of equivalenceclasses

Fx =ž

xPU

F (U)/ „ .

In case F is a sheaf of abelian groups, the stalks Fx are all abelian groups.This is not a priori obvious, because sections over different open sets can not beadded. However if (s, U) and (s1, U1) are given, the restrictions s|V and s1|V toany open VĎU XU1 can be added, and this suffices to define an abelian groupstructure on the stalks.

The germ of a sectionFor any neighbourhood U of x P X, there is a natural map F (U)Ñ Fx sendinga section s to the equivalence class where the pair (s, U) belongs. This classis called the Germ of a section

(Kimen til en seksjon)germ of s at x, and a common notation for it is sx. The map is a

homomorphism of abelian groups (rings, modules,..) as one easily verifies. Onehas sx = (s|V)x for any other open neighbourhood V of x contained in U, orexpressed in the lingo of diagrams, the following diagram commutes:

F (U) Fx

F (V).

ρUV (1.3)

stalks 23

When working with sheaves and stalks, it is important to remember the threefollowing properties; the two first follow right away from the definition, and thethird one is easily deduced from the two first.

o The germ sx of s vanishes if and only if s vanishes on some neighbourhoodof x, i.e. there is an open neighbourhood U of x with s|U = 0.

o All elements of the stalk Fx are germs, i.e. every one is of the shape sx forsome section s over an open neighbourhood of x.

o The abelian sheaf F is the zero sheaf if and only if all stalks are zero, i.e.Fx = 0 for all x P X.

Example 1.13 Let X = C, and let OX be the sheaf of holomorphic functionsin X. What is the stalk OX,x at a point x? If f and g are two sections of OX

over a neigbourhood U of the point x having the same germ at p, the fact thatf and g admit Taylor series expansions around x, implies that f = g in theconnected component containing x of the set where they both are defined. Thusthe stalk OX,x is naturally identified with the ring of power series converging ina neighbourhood of x. K

Morphisms of (pre)sheaves induce maps of stalksA map φ : F Ñ G between two presheaves F and G induces for every pointx P X a map

φx : Fx Ñ Gx

between the stalks. Indeed, one may send a pair (s, U) to the pair (φU(s), U), andsince φ behaves well with respect to restrictions, this assignment is compatiblewith the equivalence relations; if (s, U) and (s1, U1) are equivalent and s and s1

coincide on an open set V Ď U XU1, the diagram (1.3) gives

F (U) G(U)

Fx Gx

F (V) G(V)

φU(s)|V = φV(s|V) = φV(s1|V) = φU1(s1)|V .

Here we have (φ ˝ψ)x = φx ˝ψx and (idF )x = idFx , so the assignments F ÞÑ Fx

and φ ÞÑ φx define a functor from the category of abelian sheaves to the categoryof abelian groups.

The support of a sheafFor a sheaf F on the topological space X we define the Support of a sheaf

(støtten til et knippe)supportof F , denoted by

Supp(F ), bySupp(F ) = tx P X |Fx ‰ 0u.

In a similar way, for a section s P F (U) we define the Support of a section(støtten til en seksjon)

support of s, denoted bySupp(s), as the set of points x P U such that the image sx P Fx of s is not zero.

interlude: a primer on limits 24

Observe that if s P F (X) is a section and x is a point such that sx = 0 in Fx,then there is an open neighbourhood V Ď X containing x such that sy = 0 forall y P V. It follows that the support of s is a closed subset of X.

In contrast, the support of a sheaf is in general not closed (see Exercise 1.16.)However, for sheaves of rings, the support is always closed: this is because aring equals the zero ring 0 if and only if 1 = 0, and the support of a sheaf ofrings therefore equals the support of the section 1, which is closed.

1.4 Interlude: A primer on limits

Another notation for the stalk of F at x is

Fx = limÝÑxPU

F (U).

This is the direct limit (also called the colimit or the inductive limit) of all theF (U)’s when U runs through the partially ordered set of open sets containingx. The direct limit is a general construction in mathematics, so let us give afew more details here for future reference. To fix the ideas, we will work withmodules over a ring A, which is enough for our purposes in this book. There is adual concept named inverse limits which also is important in algebraic geometryand will appear later in the text .

Direct limitsRecall that a preordered set is a set endowed with a relation i ď j which issymmetric; that is, i ď i for all i, and transitive; that is, if i ď j and j ď k,then i ď k. A preordered set resembles a partially ordered set, but lacks theanti-symmetry property: it might be that i ď j and j ď i without i and j beingequal. We say that a preordered set I is Directed orders

(direkte ordninger)directedif the following condition holds:

for any two indices i and j there is a k P I such that k ě i and k ě j.A Directed systems of

modules (direktemodulsystemer)

directed system of modules (Mi, φij) is a collection tMiuiPI of A-modules,indexed by an directed index set I, and a collection of A-linear maps φij : Mj Ñ

Mi, one for each pair (i, j) so that j ď i, satisfying the two conditionsMk Mj Miφjk

φik

φijo φij ˝ φjk = φik whenever k ď j ď i;

o φii = idMi .

The Direct limits (direktegrenser)

direct limit of the system (Mi, φij) is an A-module limÝÑ

Mi together with acollection of A-linear maps

φi : Mi Ñ limÝÑ

Mi

which satisfy φi ˝ φij = φj, and which are universal with respect to this property.That is, for any A-module N and any given system of A-linear maps

ψi : Mi Ñ N


such that ψi ˝ φij = ψj, there is a unique map η : limÝÑ

Mi Ñ N satisfying ψi =

η ˝ φi.

Mj

limÝÑ

Mi

Mi

φj

φij

φi

Mi limÝÑ

Mi

N

φi

ψiη

1.14 The definition of the direct limit may be formulated in any category: justreplace the words ‘A-module’ with ‘object’ and A-linear by ‘arrow’. In generalcategories it may easily happen that direct limits do not exist. However, thecategory of modules over a ring is a well behaved category, and here all limitsexist:

Proposition 1.15 Let A be any ring. Every directed system (Mi, φij) of mod-ules over A has a direct limit, which is unique up to a unique isomorphism.

Proof: We begin with introducing an equivalence relation on the disjointunion

š

i Mi. Essentially, two elements are to be equivalent if they become equalsomewhere far out in the hierarchy of the Mi’s. In precise terms, x P Mi andy P Mj are to be equivalent when there is an index k larger than both i and j suchthat x and y map to the same element in Mk; that is, φki(x) = φkj(y). We writex „ y to indicate that x and y are equivalent; the first point to verify is that this isan equivalence realtion.

Mn

Ml Mm

Mi Mj MkObviously the relation is symmetric, since φii = idMi

it is reflexive, and it being transitive follows from the system being directed:Assume that x „ y and y „ z, with x, y and z sitting in respectively Mi, Mj andMk. This means that there are indices l dominating i and j, and m dominating jand k so that the two equalities φli(x) = φl j(y) and φmj(y) = φmk(z) hold true.Because the system is directed, there is an index n larger than both l and m, andby the first requirement above, we find

φni(x) = φnl(φli(x)) = φnl(φl j(y)) = φnm(φmj(y)) = φnm(φmk(z)) = φnk(z),

and so x „ z. The underlying set of the A-module limÝÑ

Mi is the quotientš

i Mi/ „, and the maps φi are the ones induced by the inclusions of the Mi’sin the disjoint union.

The rest of the proof consists of putting an A-module structure on limÝÑ

Mi

and checking the universal property. To this end, the salient observation isthat any two elements [x] and [y] in the limit may be represented by elementsx and y from the same Mk: Indeed, if x P Mi and y P Mj, choose a k thatdominates both i and j and replace x and y by their images in Mk. Forminglinear combinations is then possible by the formula a[x] + b[y] = [ax + by]where the last combination is formed in any Mk where both x and y live; thisis independent of the particular k used (the system is directed, and the φij’sare A-linear). The module axioms follow since any equality involving a finitenumber of elements from the limit may be checked in an Mk where all involvedelements have representatives.

Finally, checking the universal property is straightforward: The obvious mapfrom the disjoint union

š

i Mi into N induced by the ψi’s is compatible with


the equivalence relation and hence passes to the quotient; that is, it gives thedesired map η : lim

ÝÑMi Ñ N. o

Apart from the universal property, there are two ‘working principles’, reflect-ing the working principles for stalks, one should bear in mind when computingwith direct limits:

o Every element in limÝÑ

Mi is of the form φj(x) for some j and some x P Mj.

o An element x P Mj maps to zero in limÝÑ

Mi if and only if φij(x) = 0 forsome i ě j.

Example 1.16 Union as a direct limit. If each Mi are submodules of some A-module M, and the maps Mj Ñ Mi are given by inclusions Mj Ă Mi, then thedirect limit is simply the union:

limÝÑ

iMi =

ď

i

Mi.

K

Example 1.17 Stalks as a direct limit. Let X be a topological space, and considerthe directed set I of open neighbourhoods U of a point x P X ordered by inclu-sion. If F is a presheaf on X, then setting MU = F (U), the above constructionof the direct limit lim

ÝÑU MU is exactly the same as the previous definition of thestalk Fx. K

Example 1.18 Localization as a direct limit. Let A be a ring and S a multiplicativesubset. We put a preorder on S by declaring s ď t when t = us for some u P S,and this makes S a directed set. Next, for s ď t with t = us, there exists a ringhomomorphism fts : As Ñ At, which is defined by fts(asń) = auntń. In thisway the family of rings tAsusPS forms a directed system of rings — one easilychecks that the properties required of a directed family hold.

For each s P S, there is a localization map As Ñ S´1A, so from the universalproperty of the direct limit, we obtain a canonical A-linear map

φ : limÝÑsPS

A f Ñ S´1A.

We contend this is an isomorphism. The map φ is surjective: any element inS´1 A is of the form as´1 with s P S; this element lies in As and hence in theimage of φ. The map φ is injective: if asń P A f is mapped to 0 in S´1A, thenfor some t P S it holds that ta = 0, hence asń = 0 P Ast, and φ is injective. K

Inverse limitsThe dual concept of a direct limit are the Inverse limits (inverse

grenser)inverse limit (also called the projective

limit or just the limit) of anInverse systems(inverse systemer)

inverse system tMiuiPI . These systems and their limitsare defined similarly to the direct systems, just with the arrows reversed. In fact,


an inverse system indexed by I is nothing but a direct system indexed by theopposit ordered set Iop, though the limits will have rather different properties.

To be precise, the staging is as follows: we are given a collection of A-modules Mi, indexed by a directed preordered set I and for every pair i, j fromI with i ď j, we are given an A-linear map φji : Mj Ñ Mi (note that they go‘backwards’) which satisfy the compatibility conditions:

o φki ˝ φjk = φji when i ď k ď j;

o φii = idMi .

Mj

limÐÝ

Mi

Mi

N limÝÑ

Mi

Mi

φji

φj

φi

η

ψiφi

The inverse limit of the system is a module limÐÝiPI Mi together with a collection

of universal maps φi : limÐÝiPI Mi Ñ Mi satisfying φi = φji ˝ φj. That is, for any

other module N together with maps ψi : N Ñ Mi such that ψi = φji ˝ ψj there isa unique A-linear map η : N Ñ lim

ÐÝiPI Mi satisfying ψi = φi ˝ η.

Proposition 1.19 Every directed inverse system of modules has a limit.

Proof: Consider the productś

i Mi and define a submodule by

L =

(xi)ˇ

ˇ xi = φji(xj) for all pairs i, j with i ď j

. (1.4)

The projections induce maps φi : L Ñ Mi, and we claim that L together with thesemaps constitute the inverse limit of the system. A family of maps ψi : N Ñ Mi

gives rise to a map η : N Ñś

i Mi by the assignment x ÞÑ (ψi(x)), and it takesvalues in L when the ψi’s satisfy the compatibility constraints ψi = φji ˝ ψj. Thismap is clearly unique, and hence we get the desired universal property. o

Example 1.20 Inverse limits and intersections. If all the Mi are submodules ofsome fixed module M, and the maps Mj Ñ Mi are the inclusion, then the inverselimit is simply the intersection:

limÐÝiPI

Mi =č

iPI

Mi Ă M.

K

Example 1.21 p-adic integers. An important application of inverse limits isto form so-called ‘completions of rings’. The primary example is the p-adicnumbers. Let p be a prime number and consider the modules Mi = Z/piZ.They form an inverse system indexed by N0 with φij being just the canonicalreduction map Z/pjZ Ñ Z/piZ that for j ě i sends a class [n]pj mod pj to theclass [n]pi mod pi. The system may be visualized by the sequence

. . . Z/pi+1Z Z/piZ . . . Z/p2Z Z/pZ.

The inverse limit is denoted by Zp and is called the ring of p-adic integers. K

a family of examples – godement sheaves 28

Exercises(1.7) Let A be a ring and a P A an element. Let a direct system indexed by N

be given by Gi = A for all i and fij(x) = ajíx for i ď j. Determine the directlimit lim

ÝÑiPI Gi.(1.8) Let A be a ring. Show that the inverse limit of the inverse system

. . . A[x]/mi+1 A[x]/mi . . . A[x]/m2 A[x]/m

where m = (x), and the maps are the canonical reduction maps, is isomorphicto the ring of formal power series A[[x]].(1.9) Show that the map Z Ñ Zp sending n to ([n]pi)i is an injective ringhomomorphism. Show that the assignment x ÞÑ p defines an isomorphismZ[[x]]/(x´ p)Ñ Zp.(1.10) Assume that tGiuiPI is a directed (resp. inverse) system with transitionsmorphisms fij in a category C. Let JĎ I be a subset which is directed whenendowed with the ordering induced from I. Then tGjujPJ is a directed (resp.inverse) system.

a) Show that there is a canonical morphisms limÝÑjPJ Gj Ñ lim

ÝÑiPI Gi, respectivelya canoncal morphism lim

ÐÝiPI Gi Ñ limÐÝjPJ Gj, whenever the involved limits

exist.

b) The subset J is said to be Cofinal subsets(kofinaleundermengder)

cofinal if there for every i P I is a j P J with j ě i.Show that the morphisms defined in a) are isomorphisms whenever J iscofinal in I.

(1.11) Let tGiuiPI be a directed (respectively inverse) system of abelian groups.Assume that I is discrete, that is no two elements are comparable; that is, i ď jonly when i = j. Show that lim

ÝÑiPI Gi =À

i Gi, respectively limÐÝiPI Gi =

ś

i Gi,provided the sum respectively the product exists in C.(1.12) Assume that I is a directed set in which every element is dominatedby a maximal element. Let tGiuiPI be a direct (respectively inverse) system ofabelian groups indexed by I. Show that lim

ÝÑiPI Gi is isomorphic to the direct sumÀ

Gj, respectively limÐÝiPI Gi is isomorphic to the product

ś

Gj, where the sum,respectively the product, extends over all maximal elements in I.(1.13) Show that arbitrary direct and inverse limits exist in the category Sets

and Rings of sets, respectively of rings. Hint: Adapt the proofs above.(1.14) Exhibit a directed system in the category sets of finite sets that does notˇ

have a direct limit in sets.M

1.5 A family of examples – Godement sheaves

We will consider a class of rather peculiar sheaves, called Godement sheaves(Godement knipper)

Godement sheaves, todemonstrate the versatility and the generality of the notion of sheaves. They


will also be important later, when we define the sheafification of a presheaf, andthey open the path to a convenient definition of the cohomology of a space withcoefficients in a sheaf.

Let X be a topological space. Suppose that we are given, for each point x P X,an abelian group Ax. The groups Ax can be chosen in a completely arbitraryway, at random if you will. The choice of these groups gives rise to a sheaf A onX whose sections over an open set U Ď X are given as

A (U) =ź

xPU

Ax,

and whose restriction maps are defined as the natural projections

ρUV :ź

xPU

Ax Ñź

xPV

Ax,

where V Ď U is any pair of open subsets of X. The restriction map just ‘throwsaway’ the components at points in U not lying in V.

Roger Godement(1921–2016)

Proposition 1.22 A is a sheaf.

Proof: The Locality condition holds, because if the family tUiuiPI of open setscovers U, any point x0 P U lies in some Ui0 , so if s = (ax)xPU P A (U) is a section,the component ax0 survives in the projection onto A (Ui) =

ś

xPUiAx. Hence if

s|Ui = 0 for all i, it follows that s = 0.The Gluing condition holds: Assume we are given an open cover tUiuiPI of

U and sections si = (aix)xPUi P

ś

xPUiAx ai

x is an element of Axover Ui matching on the intersectionsUi XUj. The matching conditions imply that the component of si at a point x isthe same whatever i is as long as x P Ui. Hence we get a well-defined sections P A (U) by using this common component as the component of s at x. It isclear that s|Ui = si. o

Definition 1.23 The sheaf A is called the Godement sheaf of the collectiontAxu.

The construction isnot confined to abeliangroups, but works forany category wheregeneral products exist(like sets, rings, etc).

So what is the stalk of A at a point x? It is tempting to think that it shouldbe Ax, but in fact the group can be extremely complicated. Of course there is amap Ax Ñ Ax from the stalk to Ax, but that is in general the best you can say.For example, suppose Ay = Z/2 for all y P X, and that there is a sequence xn

of points in X converging to x P X, with xn = xm for all n = m. Then Ax mapsonto the (infinite) set of tails of sequences of 0’s and 1’s.* ˚Two sequences are

said to define the sametail if they merelydiffer at finitely manyplaces. This anequivalence relationwith uncountablymany equivalenceclasses

Namely, every openneighbourhood of x contains almost all of the xn.

On the other hand, if every neighbourhood of x contains a point y such thatAy = 0, then Ax = 0.


Skyscraper sheavesWe met this kind of sheaves already in Example 6. They are very specialinstances of Godement sheaves where the abelian groups Ax are zero for all xexcept for one, where it takes the value A. The sections are described by

Γ(U, A(x)) =

#

A if x P U,

0 otherwise.

If x P X is a closed point, the stalks are easy to describe: they are zero everywhereexcept at x, where the stalk equals A. Indeed, if y ‰ x, then y lies in the openset Xztxu over which all sections of A(x) vanish.

However, if txu is not a closed point (i.e., Xztxu is not open), one still hasthe Godement sheaf A(x), but the argument above does not work, and thedescription of the stalks is more complicated.Exercise 1.15 Assume that x is not closed and let Z = txu be the closure ofthe singleton txu. Show that the stalks of A(x) are (A(x))y = 0 if y R Z and(A(x))y = A for points y belonging to Z M

Exercise 1.16 Find examples of sheaves F (e.g. Godement sheaves), where thesupport is not closed. M

Slightly generalizing the construction of a skyscraper sheaf, one may formthe Godement sheaf A defined by a finite set of distinct closed points x1, . . . , xr

and corresponding abelian groups A1, . . . , Ar. Then one sees, bearing in mindthat an empty direct sum is zero, that the sections of A over an open set U isgiven as Γ(U, A ) =

À

xiPU Ai. The stalks of A are

Ax =

#

0 when x ‰ xi for all i,

Ai when x = xi.

A barcode sheaf ?The Godement sheaf associated with a presheafLet F be an abelian presheaf on X. The stalks Fx of F give a collection ofabelian groups indexed by points in X, as good as any other collection, and wemay form the corresponding Godement sheaf, which we denote by Π(F ). Thesections of Π(F ) are given by

Π(F )(U) =ź

xPU

Fx, (1.5)

and the restriction maps are the projections as for any Godement sheaf. This sheaf is sometimescalled the sheaf ofdiscontinuoussections of F .

There is an obvious and canonical map

κF : F Ñ Π(F )

sheafification 31

that sends a section s P F (U) to the element (sx)xPU of the product in (1.5).This map is functorial in F , for if φ : F Ñ G is a map of sheaves, one has thestalkwise maps φx : Fx Ñ Gx, and by taking appropriate products of these, oneobtains a map Π(φ) : Π(F )Ñ Π(G). Over an open set U, it holds that

Π(φ)((sx)xPU

)=(φx(sx)

)xPU ,

and there is thus a commutative diagram of sheaves

F Π(F )

G Π(G).

κF

φ Π(φ)

κG

(1.6)

It is not hard to check that Π(idF ) = idΠ(F ) and that Π(ψ ˝ φ) = Π(ψ) ˝Π(φ)

for two composable morphisms between abelian presheaves on X, so that Πdefines a functor from the category of abelian presheaves on X to the categoryof abelian sheaves on X.

1.6 Sheafification

Given an abelian presheaf F on X there is a canonical way of defining an abeliansheaf F+ that in some sense is the sheaf that best approximates F . The mainproperties of F+ are summarized in the following proposition.

Proposition 1.24 Given an abelian presheaf F on X. Then there is an abeliansheaf F+ and a natural map κF : F Ñ F+ such that:

i) κF is functorial in F ; φ : F Ñ G inducesφ+ : F+ Ñ G+

ii) κF enjoys the universal property that any map of abelian presheavesF Ñ G where G is a sheaf, factors through F+ in a unique way. Thisproperty characterizes F+ up to a unique isomorphism;

iii) If G is a sheaf, there is a natural bijection

HomAbPrShX (F ,G) = HomAbShX (F+,G) (1.7)

where on the left hand side, G is considered as a presheaf;iv) κF induces an isomorphism on stalks: Fx » F+

x for every x P X.

Of course it is nice to have seen how to construct F+ and κF from F byhand, but in fact, once having seen that it is possible, we will never need theexplicit construction again. All of the arguments using F+ in this book use onlythe four properties in the proposition, which uniquely characterize the pair F+

and κF (up to unique isomorphism). Compare this to thedefinition of the tensorproduct: we never usethe explicitconstruction in termsof generators andrelations; all we everuse is its formalproperties.

Exercise 1.17 Prove that, if it exists, the sheafification is unique up to a uniqueisomorphism. M

sheafification 32

When F is a subpresheaf of a sheafWhat prevents the presheaf F from being a sheaf is of course the failure of oneor both of the sheaf axioms. To remedy this, one must factor out all sectionsof F whose germs are everywhere zero, and one has to enrich F by addingenough new sections so that the Gluing axiom holds.

The simplest case of this is when F comes embedded as a subpresheaf of asheaf G. Then the Locality axiom already holds for F , because it holds for G.In this special case, we have an explicit description of the sheafification of F ;it will be equal to the Saturation (metning)saturation of F in G, which is essentially as the smallestsubsheaf of G which contains F .

To define it, let us say that a section s P G(U) locally lies in F if for someopen covering tUiuiPI of U one has s|Ui P F (Ui) for each i.

Definition 1.25 We define the sheaf saturation F of F in G by

F (U) = t s P F (U) | s locally lies in F u.

The sheaf saturation F is again a subpresheaf of G (with restriction mapsbeing the ones induced from G). In fact, F is, almost by definition, a sheaf. Thesections of F (U) lie in G(U), so the Locality axiom holds because it holds for G.Furthermore, given a set of gluing data for F ; that is, an open covering tUiuiPI

of an open set U and sections si P F (Ui) matching on intersections, the si’s canbe glued together in G, since G is a sheaf, and since they are born to locally liein F , the glued section s lies locally in F as well and hence is a section of G(U).So also the Gluing axiom holds for F .

Note that the saturation F contains F as a subsheaf. If F is already a sheaf,we don’t get anything new, so that F = F .

Let us check that the saturation F indeed satisfies the universal property ofsheafification: Let φ : F Ñ G be a morphism of presheaves, where G a sheaf.If U Ă X is open, and s P F (U), then there is an open covering Ui of U suchthat s|Ui P F (Ui) for each i. Then φUi(s|Ui) are elements of G(Ui) which agreeon the overlaps, so they glue to a section t P G(U), and we define the mapF (U)Ñ G(U) by sending s to t.Exercise 1.18 Prove the uniqueness part for the universal map of sheavesF Ñ G. M

Example 1.26 Bounded continuous functions. Consider the sheaf C(X, R) fromExample 2 and the subpresheaf F , defined by setting F (U) = Cb(U, R), thegroup of bounded continuous functions. Then Cb(X, R) is not a sheaf, becausethe Gluing axiom fails: the open sets Ui = (í, i) form a covering of X = R, andthe function fi(x) = x|Ui defines a well defined element Cb(Ui) for each i. Thefi’s agree with the function f ( f ) = x on each overlap Ui XUj, but they do notglue to an element in Cb(R), because the function x is not bounded.

sheafification 33

In this example, the saturation of the presheaf Cb(X, R) is in fact all ofC(X, R), because any continuous function is locally bounded. K

The general constructionIf F is a general presheaf, there is a nice and canonical way to map it into asheaf using the Godement sheaf Π(F ) associated to F . Recall the canonical mapκ : F Ñ Π(F ) that sends a section s of F over an open U to the sequence ofgerms (sx)xPU P

ś

xPU Fx = Γ(U, Π(F )). This map certainly kills the ‘doomed’sections, i.e. those whose germs all vanish. And we can get an actual sheaf bytaking the image of κ in Π(F ):

Definition 1.27 For an abelian presheaf F on X, we define its sheafificationF+ as the image sheaf Im κ in Π(F ). In other words, F+ is the saturation ofthe subpresheaf U ÞÑ Im κU in Π(F ).

For simplicity, we also write κF or simply κ, for the canonical map F Ñ F+.It might help to unravel this definition. Over an open set U Ď X the sections

of F+ are given by

F+(U) =

#

(sx) Pź

xPU

Fx

ˇ

ˇ

ˇ(sx) locally lies in Im κ

+

=

$

’

&

’

%

(sx) Pź

xPU

Fx

ˇ

ˇ

ˇ

for each x P U, there exists an open V Ď Ucontaining x and t P F (V), such thatfor all y P V we have sy = ty in Fy

,

/

.

/

-

Thus elements of F+(U) can be thought about as sequences (sx)xPX of germs ofF , but only those sequences that arise from local sections of F are allowed.

Lemma 1.28 The sheafification F+ depends functorially on F . Moreover, if F is asheaf, κ : F Ñ F+ is an isomorphism, so that F and F+ are canonically isomorphic.

Proof: Suppose that φ : F Ñ G is a map between two presheaves. Let s be asection of Π(F ) over some open set U so that s locally lies in F . In other words,there is a covering tUiu of U and sections si of F over Ui with s|Ui = κF (si).Hence by (1.6) one has

Π(φ)(s|Ui) = Π(φ)(κF (si)) = κG(φ(si)).

This means that Π(φ)(s) lies locally in G, and hence Π(φ) takes F+ into G+.Moreover, there is a commutative diagram

F F+ Π(F )

G G+Π(G).

κF

φ φ+Π(φ)

κG

sheafification 34

In case F is a sheaf, the map κF maps F injectively into Π(F ), and F = Im κFis its own saturation, hence κF is an isomorphism. o

Lemma 1.29 Given an abelian presheaf F on X. Then the sheaf F+ and the naturalmap κF : F Ñ F+ enjoys the universal property that any map of abelian presheavesF Ñ G where G is a sheaf, factors through F+ in a unique way. This propertycharacterizes the pair κF and F+ up to a unique isomorphism.

F F+

G

κ

Proof: If G in the commutative diagram above is a sheaf, the map κG : G ÑG+ is an isomorphism and φ+ ˝ κ´1

G provides the wanted factorization. Theuniqueness statement follows formally: given two abelian sheaves F+ and F 1satisfying the above, we get by the universal properties two maps F+ Ñ F 1 andF 1 Ñ F+, whose compositions are the identity by uniqueness. o

This also gives item iii) in Proposition 1.9: for a presheaf F and a sheaf Gthere is a natural bijection

HomAbPrShX (F ,G) = HomAbShX (F+,G), (1.8)

where on the left G is considered as a presheaf. A fancy way of restating thisis to say that the sheafification functor F ÞÑ F+ is an adjoint to the forgetfulfunctor ι : AbShX Ñ AbPrSh from sheaves to presheaves.

We now turn to point iv) of Proposition 1.9:

Lemma 1.30 Sheafification preserves stalks: Fx = (F+)x via κx.

Proof: The map κx : Fx Ñ (F+)x is injective, because Fx Ñ (Π(F ))x isinjective. To show it is surjective, suppose that s P (F+)x. We can find anopen neighbourhood U of x such that s is the equivalence class of (s, U) withs P F+(U). By definition, this means there exists an open neighbourhood V Ď Uof x and a section t P F (V) such that s|V is the image of t in Π(F )(V). Clearlythe class of (t, V) defines an element of Fx mapping to s. o

The next lemma tells us that the sheafification is easy to compute in the case Fis already a subpresheaf of a sheaf. This will be the case for many of the sheavesin this book.

Lemma 1.31 If F is a presheaf which is a subpresheaf of a sheaf G, then F+ is naturallyisomorphic to the saturation of F in G.

Here is another example where one can describe the sheafification explicitly.Example 1.32 Constant sheaves. Recall Example 5 in which we showed that theconstant presheaf given by F (U) = A is usually not a sheaf (where A is anabelian group). In this case, the sheafification F+ is exactly the constant sheafAX defined by

AX(U) = t f : U Ñ A | f continuous u. (1.9)

sheafification 35

where A is given the discrete topology. To prove this, consider the map ι :AX Ñ F+ which over an open set U Ă X, sends f : U Ñ A to the element( f (x))xPU P

ś

xPU A (note that this element lies in F+, since f is locally constant).Furthermore, on stalks this is simply the identity map ιx : A Ñ A. Since ι is amap of sheaves which is an isomorphism on stalks, it is an isomorphism. See Proposition 1.20.K

Example 1.33 Regular functions and saturation. Consider again the sheaf ofregular functions on an algebraic variety, as defined in Example 7. There isa subtlety in the definition of a regular function; the function is only locallyexpressible as a quotient g/h; we do not require a global representative. This isrelated to the concept of saturation.

To see the issue, consider the case where X is an affine variety over k withcoordinate ring A(X), and consider the ‘naive sheaf of regular functions’ givenby

oX(U) =

#

f : U Ñ k

ˇ

ˇ

ˇ

ˇ

ˇ

f = g/h for polynomials g, hwith h(x) ‰ 0 for every x P U

+

This is not a sheaf in general. For instance, consider X = Z(xy´ zw) Ă A4.Then x/z defines a section of oX over U = tz ‰ 0u and w/y defines a sectionover V = ty ‰ 0u. Using the defining relation xy = zw we see that these twosections agree on the overlap U X V. However, they do not glue together toan element of oX(U YV). Indeed, there is no quotient g/h of polynomials inx, y, z, w representing the corresponding regular function f : U YV Ñ k. See also [EO, Example

3.5].The presheaf oX sits naturally in the sheaf C(X, k) of continuous maps U Ñ k,

and the saturation is exactly the usual structure sheaf OX on X. Therefore, whilethe two expressions x/z and w/y can not be glued together to an elementof oX(U YV), they can be glued together to an element in OX(U YV). Thussaturation explains the subtlety of the definition of a regular function: a functionf : U Ñ k is only locally expressible as a fraction of polynomials; there is noglobal representative. K

Example 1.34 Inverse limits and sections. Whereas direct limits gives us stalks,inverse limits give a way to compute sections. In the context of sheaves, theslogan is: ‘Direct limits have a localizing effect, while inverse limits effectuateglobalizations.’

Consider an open set U of the topological space X and a sheaf F on X.Assume given an open covering U = tUiuiPI of U which is directed underinclusion; i.e. the intersection of two members from U contains a third, then therestriction maps induce an isomorphism F (U) » lim

ÐÝiPI F (Ui). Indeed, the mapsρUUi : F (U)Ñ F (Ui) comply with the compatibility request ρUUi = ρUjUi ˝ ρUUj

for UiĎUj, and they thus give a canonical map F (U)Ñ limÐÝiPI F (Ui).

In view of the description (1.4) this is an isomorphism: that s maps to zero,means that ρUUi(s) = s|Ui = 0 for each i, which by the Locality axiom entailsthat s = 0. Furthermore, sections si P F (Ui) so that sj|Ui = si for each inclusion

kernels, images and quotients 36

UiĎUj may, by the Gluing axiom, be glued together to give a section of F overU, and the map is surjective.

In fact, with slightly more care one establish that if F is a presheaf, thesections of the sheafification F+ is given as the inverse limit

F+(U) » limÐÝiPI

F (Ui). (1.10)

K

Exercise 1.19 Convince yourself that (1.10) holds true. M

1.7 Kernels, images and quotients

When working with abelian sheaves on a space the toolbox include some tradi-tional tools, well-known from abelian groups: the possibility to form kernels,cokernels and images of maps, which entails the notion of exact sequences. Wemay also form arbitrary direct sums and products of sheaves. In short, thecategory AbShX of abelian sheaves on X is an abelian category with arbitraryproducts and direct sums.

Kernels and imagesLet φ : F Ñ G be a map between two sheaves on X.

Definition 1.35 The kernel Ker φ of φ is the subsheaf of F defined by

(Ker φ)(U) = Ker φU .

In other words, (Ker φ)(U) consists of the sections in F (U) that map to zerounder φU : F (U)Ñ G(U).

The requirement in the definition is compatible with the restriction mapssince φV(s|V) = φU(s)|V for any section s over the open set U and any openV Ď U. Thus we have defined a subpresheaf of F . This is indeed a subsheaf:the Locality axiom holds for Ker φ because F is a sheaf. Moreover, if tUiu is anopen covering of U and tsiu a family of sections with si belonging to Ker φ(Ui)

that agree on overlaps, one may glue together the si’s to a section s of F overU. One has φ(s)|Ui = φ(s|Ui) = φ(si) = 0, and from the Locality axiom for G itfollows that φ(s) = 0. We leave it to the reader to verify that the stalk (Ker φ)x

of Ker φ at x equals Ker φx. This establishes the following:

Lemma 1.36 Let φ : F Ñ G be a map of abelian sheaves. The kernel Ker φ is a subsheafof F . It has the following two properties:

i) Forming the kernel commutes with taking sections: Γ(U, Ker φ) = Ker φU ;ii) Forming the kernel commutes with forming stalks: (Ker φ)x = Ker φx.


The map φ : F Ñ G is said to be injective if Ker φ = 0, a condition equivalent toφU being injective for each open U. In light of the previous lemma, this is alsoequivalent to the condition that Ker φx = 0 for all x; i.e. that all stalk-maps φx

are injective. One often expresses this in a slightly imprecise manner by sayingthat φ is injective on stalks.

When it comes to images the situation is not as nice as for kernels. Onedefines the Image presheaf

(bilde(pre)knippe)image presheaf contained in G by letting (Im φ)(U) = Im φU . However,

this is not necessarily a sheaf: if si = φUi(ti) are gluing data for the imagepresheaf with respect to a cover tUiu of U, there is no reason for the ti’s to matchon the intersections Uij = Ui XUj, even if the si’s do. In fact, we will see severalexamples where this fails later.

To remedy the situation we simply make the following definition:

Definition 1.37 For a sheaf map φ : F Ñ G we define the sheaf Im φ to be thesaturation of the image presheaf U ÞÑ Im φU . Im φ is the smallest

subsheaf of Gcontaining the imagesof φ.Forming the image of a map of sheaves does not always commute with

taking sections, but as we shall verify in the upcoming lemma, forming imagescommutes with forming stalks:

Lemma 1.38 Let φ : F Ñ G be a map of abelian sheaves. The image Im φ is a subsheafof G which satisfies:

i) For all open subsets U of X one has Im φU Ď Γ(U, Im φ);ii) For all x P X one has (Im φ)x = Im φx.

Proof: i): An element of t = φU(s) of Im φU is an element of G which clearlylocally lies in Im φ, so t P Γ(U, Im φ).

ii): Let tx P Im φx and pick an sx P Fx with φx(sx) = tx. We may extend theseelements to sections s and t over some open neighbourhood V, so that φV(s) = t,and t is a section of Im φ over V. This shows that Im φxĎ (Im φ)x. Conversely,if t is a section of G over an open U containing x locally lying in the imagepresheaf, the restriction t Ď V lies in Im φV for some smaller neighbourhood Vof x, hence the germ tx lies in Im φx. o

The map φ : F Ñ G is said to be surjective if the image sheaf Im φ = G. This isequivalent to all the stalk maps φx being surjective (one says φ is surjective onstalks). However, we underline that this condition does not imply that the mapsφU are surjective for every open U. (This is a cardinal observation; one might betented to say being the root of non-trivial mathematics)

However, for the map φ to be an isomorphism, one has the following


Proposition 1.39 Let φ : F Ñ G be a map of abelian sheaves. Then thefollowing four conditions are equivalent

i) The map φ is an isomorphism;ii) For every x P X the map on stalks φx : Fx Ñ Gx is an isomorphism;

iii) One has Ker φ = 0 and Im φ = G;iv) For all open subsets U Ď X the map on sections φU : F (U)Ñ G(U)

is an isomorphism.

Proof: Most of the implications are straightforward from what we have doneso far and are left to the reader. We comment just on the two main salient points.

Firstly, assume that all the stalk maps φx : Fx Ñ Gx are isomorphism, and letus deduce that all maps φU : F (U)Ñ G(U) on sections are isomorphisms. It isclear that φU is injective, since forming kernels commute with taking sections asin Lemma 1.17 on page 37.

For surjectivity, take an element t P G(U); we need to show that t = φU(s) forsome s P F (U). For each x P U there is a germ sx induced by a section s(x) of Fover some open neighbourhood Ux of x that satisfies φx(sx) = tx. Let t(x) = t|Ux .After shrinking the neighbourhood Ux, we may assume that φUx(s(x)) = t(x).Note that the t(x)’s match on the intersections Ux XUx1 , all being restrictions ofthe section t, and therefore the s(x)’s match as well because φUx is injective (aswe just observed above). Hence, the sections s(x) patch together to a section s ofF over U that must map to t since it does so locally.

Secondly, if all the φU’s are isomorphisms, we have all the inverse maps φ´1U

at our disposal. They commute with restrictions since the maps φU do. Indeed,from φV ˝ ρUV = ρUV ˝ φU one obtains ρUV ˝ φ´1

U = φ´1V ˝ ρUV . Thus the φ´1

U ’sdefine a map φ´1 : G Ñ F of sheaves, which of course, is inverse to φ. Weconclude that φ is an isomorphism. o

Exercise 1.20 Check that (Ker φ)x = Ker(φx) for a morphism of abelian sheavesφ : F Ñ G. M

Exact sequences of sheavesA complex of sheaves is a sequence

. . . Fi´1 Fi Fi+1 . . .φi´2 φi´1 φi φi+1

of maps of abelian sheaves where the composition of any two consecutive mapsequals zero, i.e. φj´1 ˝ φj = 0 for all j. We say that the sequence is exact at Fi

if Ker φi = Im φi´1. The short exact sequences are the ones one most frequentlyencounters. They are the sequences of the form

0 F 1 F F2 0φ ψ

(1.11)


that are exact at each stage. This is just another and very convenient way ofsimultaneously saying that φ is injective; that ψ is surjective; and that Im φ =

Ker ψ.

Lemma 1.40 A three term complex like the one in (1.11) is exact if and only if for eachpoint x P X, the induced sequence

0 F 1x Fx F2x 0φ ψ

of stalks is exact.

Proof: Bearing in mind that both forming kernels and images commute withforming stalks, this follows readily since a sheaf equals zero precisely when allstalks vanish. o

Proposition 1.41 For a short exact sequence 0 Ñ F 1 φÝÑ F ψ

ÝÑ F2 Ñ 0 andan open subset U, the following induced exact sequence is exact:

0 ÝÑ F 1(U)φUÝÑ F (U)

ψUÝÑ F2(U). (1.12)

Proof: By Lemma 1.17 the map φ is injective as a map of sheaves, henceinjective on all open sets U, so the sequence above is exact at F 1(U). To see itis exact in the middle as well, we need to show that Ker(ψU) = Im (φU). (Theimage presheaf is then given by the kernel of a morphism of sheaves, which isindeed a sheaf.)

That Im (φU)Ď Ker(ψU) is a consequence of taking sections being functorial:since ψ ˝ φ = 0, it follows that ψU ˝ φU = (ψ ˝ φ)U = 0, so everything in Im φU

lies in the kernel of ψU .For the opposite inclusion, Ker(ψU)Ď Im (φU), it may be helpful to look at

the following diagram for x P U:

0 F 1(U) F (U) F2(U)

0 F 1x Fx F2x 0

φU ψU

φx ψx

Note that the bottom row is exact since the sheaf sequence is exact.Let t P F (U) be so that ψU(t) = 0. Then for all x P U we have that

ψx(tx) = (ψU(t))x = 0, and the germ tx is an element in Ker(ψx) = Im (φx)

(here we use exactness at the stalks). This means that for every x P U there is anelement s1x P F 1x, say represented by (s1(x), V(x)) for some open neighbourhoodV(x) Ď U of x and section s1(x) P F 1(V(x)), such that φx(s1x) = tx. For any twopoints x, y P U we then have

φV(x)XV(y)(s1(x)|V(x)XV(y)) = t|V(x)XV(y) = φV(x)XV(y)(s

1(y)|V(x)XV(y)),


so that by the injectivity of φV(x)XV(y) (which we have already proved), therequired gluing condition

s1(x)|V(x)XV(y) = s1(y)|V(x)XV(y)

is satisfied, and the s(x)’s patch together to give a section s P F (U) with theproperty that for all x P U

s|V(x) = s1(x).

We then conclude that for every x P U

(φU(s))x = φx(sx) = φx(s1x) = tx,

since sx = s1x. This gives φU(s) = t as desired. o

One way of phrasing Proposition 1.22 is to say that taking sections overan open set U is a left exact functor; that is, the functor Γ(U,´) is left exact.This functor, however, is not right exact in general. The defect of this lackingsurjectivity is a fundamental problem in every part of mathematics where sheaftheory is used, and to cope with it one has developed cohomology.

ExamplesLet us give a few examples where the surjectivity on the right fails:(1.42) Differential operators. Let X = C and recall the sheaf OX of holomorphicfunctions and the map D : OX Ñ OX sending f (z) to the derivative f 1(z). Thereis an exact sequence

0 CX OX OX 0.D

This hinges on the two following facts. Firstly, a function whose derivativevanishes identically is locally constant; hence the kernel Ker D equals the con-stant sheaf CX. Secondly, in small open disks any holomorphic function is aderivative: e.g. if f (z) =

ř

ně0 an(z´ a)n in a small disk around a, the functiong(z) =

ř

ně0 an(n + 1)´1(z ´ a)n+1 has f (z) as derivative. However, takingsections over open sets U we merely obtain the sequence

0 Γ(U, CX) Γ(U, OX) Γ(U, OX).DU

Whether DU is surjective or not, depends on the topology of U. If U is simplyconnected, one deduces from Cauchy’s integral theorem that every holomorphicfunction in U is a derivative, so in that case DU is surjective. On the otherhand, if U is not simply connected, DU is not surjective; e.g. if U = C´ t0u, thefunction z´1 is not a derivative in U (there is no globally defined logarithm onU).(1.43) The exponential sequence. Let X = C´t0u. The non-vanishing holomorphicfunctions in an open set U Ď X form a multiplicative group, and there is a sheaf


O˚X with these groups as spaces of sections. For any f holomorphic in U theexponential exp f (z) is a section of O˚X. Hence there is an exact sequence

0 ZX OX O˚X 0,exp

where the first map sends 1 to 2πi. The rightmost map exp is surjective as a mapof sheaves, because non-vanishing functions locally have logarithms. However,over the open set U = X, the map is not surjective: the non-vanishing functionf (z) = z is not the exponential of a global holomorphic function.(1.44) Consider an algebraic variety X with the sheaf OX of regular functionson X. For any point p P X let k(p) denote the skyscraper sheaf See Example 6whose only non-zero stalk is the field k located at p. There is a map of sheaves evp : OX Ñ k(p)sending a function that is regular in a neighbourhood of p to the value it takesat p. This map sits in the exact sequence of sheaves

0 mp OX k(p) 0,evp

where mp by definition is the kernel of evp (the sections of mp are the functionsvanishing at p). Taking two distinct points p and q in X, we find a similar exactsequence

0 Ip,q OX k(p)‘ k(q) 0,evp,q

where Ip,q is the sheaf of functions vanishing on p and q, and evp,q is theevaluation at the two points.

If for example X = P1C (or any other projective variety), there are no global

regular functions on X other than the constants, and hence Γ(X,OX) = k. Butof course, Γ(P1, k(p)‘ k(q)) = k‘ k, so the map evp,q can not be surjective onglobal sections.

K

Quotient sheaves and cokernelsOne of the main applications of the sheafification process is to be able to definequotient sheaves. So assume that G ĎF is an inclusion of abelian sheavesand define a presheaf whose sections over U is the quotient F (U)/G(U). Therestriction maps of F and G respect the inclusions G(U)ĎF (U), hence passingto the quotients F (U)/G(U), and we may use these maps as restriction maps forthe quotient presheaf. The Quotient sheaf

(Kvotsientknippe)quotient sheaf F/G is by definition the sheafification

of this quotient presheaf. It sits in the exact sequence

0 G F F/G 0.

The cokernel Coker φ of a map φ : F Ñ G of abelian sheaves is then defined asthe quotient sheaf G/Im φ; it fits in the exact sequence

0 Ker φ F G Coker φ 0.φ

direct and inverse images 42

Example 1.45 To see why we have to sheafify in these constructions, con-sider again the exponential map from Example 21. The naive presheaf U ÞÑ

Coker exp(U) is not a sheaf: the class of the function f (z) = z restricts to zero inCoker exp on sufficiently small open sets, but it is itself not zero (since otherwisewe would be able to define a global logarithm on Cz0). K

1.8 Direct and inverse images

So far we have been interested in various constructions of sheaves on a fixedspace X. These constructions become more interesting if we involve continuousmaps between spaces and try to transfer sheaves from one space to the other.There are two constructs of this kind. One, the direct image, or pushforward, iscovariant and sends a sheaf on the source to a sheaf on the target; the other one,the inverse image, is contravariant; one may pull back a sheaf on that target to asheaf on the source.

The direct image of a sheafLet X and Y be two topological spaces and let f : X Ñ Y be a continuous mapbetween them. Assume that F is an abelian sheaf on X. This allows us to definean abelian sheaf f˚F on Y by specifying the sections of f˚F over the open setU Ď Y to be

( f˚F )(U) = F ( f´1U),

and letting the restriction maps F ( f´1U)Ñ F ( f´1V) be the ones from F .

Definition 1.46 The sheaf f˚F is called the pushforward sheaf or the directimage of F .

It is straightforward to see that f˚F is a sheaf and not merely a presheaf.Indeed, if tUiui is an open covering of U, then t f´1(Ui)u is an open covering off´1(U). A set of gluing data for f˚F and the given covering consists of sectionssi P Γ(Ui, f˚F ) = Γ( f´1Ui,F ) matching on the intersections, which means thatthey coincide in Γ(Ui XUj, f˚F ) = Γ( f´1Ui X f´1Uj,F ). These may thereforebe glued together to a section in Γ( f´1U,F ) = Γ(U, f˚F ), as F is a sheaf. TheLocality axiom follows for f˚F because it holds for F .Example 1.47 Let ι : txu Ñ X be the inclusion of a closed point in X. If A isthe constant sheaf of a group A on txu, then ι˚A is the skyscraper sheaf A(x)from Example 6 on page 19. More generally, when txu is not necessarily closed,


the pushforward sheaf ι˚A will be a Godement sheaf with ι˚(A)y = A wheny P txu and ι˚Ay = 0 otherwise. K

Example 1.48 Consider an affine variety X Ď An and let i : X Ñ An be theinclusion. For each open U Ď An define

IX(U) = t f P OAN (U) | f (x) = 0 for all x P Xu.

Then IX is a sheaf (of ideals) and we have an exact sequence

0 Ñ IX Ñ OAn Ñ i˚OX Ñ 0

K

For each morphism of sheaves φ : F Ñ G on X, we have for each opensubset U of Y an induced map

φ f´1(U) : F ( f´1U)Ñ G( f´1V)

which is compatible with the restrictions, and so we get a map f˚φ : f˚F Ñ f˚G.Trivially f˚(φ ˝ φ) = f˚φ ˝ f˚ψ for composable maps φ and ψ. This means thatf˚ defines a functor AbShX Ñ AbShY. As such it is left exact:

Lemma 1.49 The functor f˚ is left exact. That is, given an exact sequence of sheaveson X

0 F 1 F F2 0

then the following sequence is exact

0 f˚F 1 f˚F f˚F2.

Proof: Sections of f˚F being sections of F over inverse images of opens in Y,the lemma follows readily from the exactness of the sequence (1.12) on page 39

above. o

The pushforward also depends functorially on the map f :

Lemma 1.50 If g : X Ñ Y and f : Y Ñ Z are continuous maps between topologicalspaces, and F is a sheaf on X, one has

( f ˝ g)˚F = f˚(g˚F ).

(This is indeed an equality, not merely an isomorphism.)

Exercise 1.21 Prove Lemma 1.25. M

Exercise 1.22 Denote by t˚u a one point set. Let f : X Ñ t˚u be the one andonly map. Show that f˚F = Γ(X,F ) (where strictly speaking Γ(X,F ) standsfor the constant sheaf on t˚u with value Γ(X,F )). M


The inverse image sheafIf F is an abelian sheaf on X and U Ď X is an open subset, then F defines asheaf on U by restriction of sections. For an arbitrary subset Z Ď X, this naiverestriction does not directly give a sheaf on Z, because an open subset V Ď Z isusually not open in X. However, as in the definition of stalks, we can take limitsof F (U) as U runs over smaller and smaller open subsets containing V. Thisgives the following definition:

Definition 1.51 If i : Z Ñ X is the inclusion of a subspace Z Ď X, we definethe restriction ι´1(F ) as the sheafification of the following presheaf:

V ÞÑ limÝÑ

VĎUF (U).

When Z reduces to a singleton txu, we recognize the definition of the stalk,so that if ι : txu Ñ X denotes the inclusion, it holds that ι´1F = Fx.

We can extend this idea to any continuous map f : X Ñ Y and any abeliansheaf G on Y. This gives rise to the Inverse images of

sheaves (inverse bilderav knipper)

inverse image or the

Pullback of sheaves(tilbaketreknigen avknipper)

pullback of G, denoted byf´1G, which we shall now define. As above, we know the values of G on opensubsets V on Y, but we want a sheaf on X. The image f (U) of an open set U inX need not be open in Y, but we can look at “germ-like” equivalence classes ofsections of G(V) as V gets closer and closer to f (U). That is, we start out withthe presheaf f´1

p G on X defined by

Γ(U, f´1p (G)) = lim

ÝÑf (U)ĎV

G(V), (1.13)

where the indexing set consists of all open set V in Y containing f (U). Everysection of f´1

p G over U is induced by a section of G over some V containing f (U),and two such sections s and s1 over V and V1 induce the same section of f´1

p G(U)

if they agree over some open V2 with f (U)ĎV2ĎV XV1. The restriction mapscome from the universal property of the direct limit, since if U1ĎU, the set ofopens containing f (U) is contained in the set of those containing f (U1). Andthis leads us to the following:

Definition 1.52 Let f : X Ñ Y be a continuous map and G an abelian presheafon Y. The inverse image f´1G is the sheafification of the presheaf (1.13).

Note that while G may only be a presheaf, the resulting inverse imagef´1G is always a sheaf (sheafification is the special case when f is the identitymorphism).

We also note that the construction gives a functor f´1 : AbPrShY Ñ AbShX,indeed a map of sheaves φ : G Ñ H gives maps φV : G(V)Ñ H(V) for open setsV containing f (U) which are compatible with restrictions, and hence they passto the direct limit and induce a map f´1φ : f´1G Ñ f´1H.

sheaves defined on a basis 45

Note in particular that the stalk of f´1G at a point x P X is isomorphic toG f (x). Indeed, it suffices to verify this on the level of presheaves:

( f´1p G)x = lim

ÝÑxPU

f´1p G(U) = lim

ÝÑxPU

limÝÑ

f (U)ĎV

G(V) = limÝÑ

f (x)PV

G(V) = G f (x).

Example 1.53 The presheaf defined by (1.13) is not in general a sheaf. Forinstance, if f : X Ñ Y is the constant map with image y, the inductive limitin (1.13) will be the stalk Gy whatever the open UĎX is. So (1.13) defines theconstant presheaf with value Gy, and as we observed in Example 5 on page 18,this is not always a sheaf. K

The adjoint propertyThe definition of f´1G is natural, but a little bit hard to work with for actualcomputations, as it involves both taking a direct limit over open sets and asheafification. What’s much more important is what this sheaf does: It is theadjoint of the functor f˚ as a functor from AbShX to AbPrShY (and the latter is afunctor we understand well). The precise meaning behind that statement is thefollowing:

Theorem 1.54 Let f : X Ñ Y be a morphism, let F be an abelian sheaf on Xand let G be an abelian presheaf on Y. Then there is a natural bijection

HomAbPrShY(G, f˚F ) » HomAbShX ( f´1G,F ),

which is functorial in F and G.

This shows that the sheaf f´1G constructed before satisfies a very naturaluniversal property: sheaf morphisms φ : G Ñ f˚F correspond bijectively tomaps f´1G Ñ F . In particular, applying this to F = f´1G, and G = f˚F withthe identity maps, we see that there is a canonical map of sheaves

η : G Ñ f˚ f´1G,

which is functorial in G, and

ν : f´1 f˚F Ñ F ,

which is functorial in F .Exercise 1.23 Prove Theorem 1.28. Mˇ

1.9 Sheaves defined on a basis

Recall that a basis for a topology on X is a collection of open subsets B suchthat any open set of X can be written as a union of members of B. In many


situations it turns out to be convenient to define a sheaf by saying what it shouldbe on a specific basis for the topology on X. In this section we will see that thereexists a unique way to produce a sheaf, given sections that glue together overopen subsets from B.

Let us first make the following definition:

Definition 1.55 A B-presheaf F consists of the following data:

i) For each U P B, an abelian group F (U);ii) For all U Ď V, with U, V P B, a restriction map ρUV : F (U) Ñ

F (V).

As before, these are required to satisfy the relations ρUU = idF (U) and ρWU =

ρVU ˝ ρWV . A B-sheaf is a B-presheaf satisfying the Locality and Gluingaxioms for open sets in B.

Since the intersections V XV1 of two sets V, V1 P B need not lie in B, weneed to clarify what we mean in the Gluing axiom. Given a covering of U P B

by subsets Ui P B. If si P F (Ui) are sections such that si|V = sj|V for every i, jand every V Ă Ui XUj such that V P B, then the si should glue together to anelement in s P F (U).

The whole point with the notion of B-sheaves is expressed in the followingproposition. We will use this

construction when wedefine the structuresheaf in Chapter 3Proposition 1.56 Let X be a topological space and let B be a basis for the

topology on X. Then the following three statements are true:

i) Every B-sheaf F0 extends uniquely to a sheaf F on X;ii) If φ0 : F0 Ñ G0 is a morphism of B-sheaves, then φ0 extends uniquely

to a morphism φ : F Ñ G between the corresponding sheaves;iii) The stalk of the extended sheaf F at a point x equals

Fx = limÝÑ

xPU, UPBF0(U).

Proof: For clarity, let us denote ρ0VV1 : F0(U)Ñ F0(V) the restriction map over

subsets V, V1 P B with V1ĎV.Recall Example 19 on page 35 stating that sections of a sheaf over some

open set U may be expressed as an inverse limit of sections over members ofa basis contained in U. Bearing this in mind, one is compelled to define F (U)

as the inverse limit of the F0(V), where V runs over the basis of open subsets


contained in the open set U:

F (U) = limÐÝ

WĎU,WPBF0(W)

=

(sW) Pź

WĎU,WPB

F0(W) | ρ0WW1(sW) = s1W for all W1ĎW, W1 P B

(

.

Note that when VĎU and V P B, there is a canonical map

πUV : F (U)Ñ F0(V)

induced by the projectionś

WĂU, WPB F0(W) Ñ F0(V) onto the ‘V-th’ factor.These maps commute with the restriction maps ρ0

VV1 , in the sense that ρ0VV1 ˝

πUV = πUV1 for all V1 Ă V Ă U with V, V1 P B. In particular, we see thatthere is a canonical isomorphism πVV : F (V) Ñ F0(V) for every V P B (well,if V P B, it will be largest in the set of members of B contained in V, (byExercise 1.12 on page 28).

Furthermore, for each pair of open subsets U1ĎU, the projectionsź

WĂU,WPB

F0(W)Ñź

WĂU1,WPB

F0(W)

induce a mapρUU1 : F (U)Ñ F (U1).

By construction this map has the property that ρUU2 = ρU1U2 ˝ ρUU1 for eachchain of opens U2ĎU1ĎU, which means that F is a presheaf on X whichextends F0. It ensues from Example 19 on page 35 that F is a sheaf; indeed, ifF+ denotes the sheafification of F , that example yields the identity

F+(U) = limÐÝ

WĎU,WPBF (W) = lim

ÐÝWĎU,WPB

F0(W),

and this establishes i).We note that iii) follows immediately, since we may compute the inverse

limit that gives the stalk at a given point using subsets of B, which are cofinalin the set of open neighbourhoods of the point.

Proof of ii): saying φ0 : F0 Ñ G0 is a map of B-sheaves amounts to sayingthat the following diagram commutes for each pair V1ĎV of members of B:

F (V) G(V)

F (V1) G(V1).

(φ0)V

(φ0)V1

Taking the inverse limit over all open subsets V from B contained in U, weobtain a natural map

F (U) = limÐÝ

WPB,WĂUF0(W)Ñ lim

ÐÝWPB,WĂU

G0(W) = G(U),


which extends φ0. Again this must be unique, as it is completely determined byφ0 on stalks. o

Chapter 2

The Prime Spectrum

Affine schemes are the building blocks in the theory of schemes. Every schemeis locally an affine scheme; i.e. every point has a neighbourhood which isisomorphic to an affine scheme. This is similar to the notion of a manifold, wherea topological space is locally homeomorphic to an open set in Rn. However,affine schemes are infinitely more complicated than open sets in Rn and canhave extremely bad singularities.

The affine schemes are basically just a geometric embodiment of rings.affine schemesØ ringsThere is a one-to-one correspondence between rings and affine schemes, and

furthermore, given two rings, the ring homomorphisms between them are in aone-to-one correspondence with the scheme maps between the correspondingschemes (but the maps change direction, so the correspondence is what onecalls ‘contravariant’). This situation is in perfect analogy with what happens foraffine varieties, which are determined by their coordinate rings. However, inthat case there are heavy constraints on the rings involved; they are required tobe integral domains of finite type over an algebraically closed field.

Following the Grothendieck maxim of always working at the maximal levelof generality, we start out with a commutative ring A with a unit element In these notes the term

‘ring’ always refers to acommutative ring witha unit element ‘1’.Any map of ringsmust send 1 to 1.

. Weare going to define an affine scheme, written Spec A, called the prime spectrum ofA or simply the spectrum of A for short. As for any scheme, the structure ofSpec A will have two layers: there is an underlying topological space on whichthere lives a sheaf of rings. In this chapter we shall describe the underlyingtopological space and establish some of its fundamental properties, the structuresheaf of Spec A will be for the next chapter.

2.1 The spectrum of a ring

To motivate the definition, recall the situation for affine varieties and assume fora moment that A is a finitely generated k-algebra over an algebraically closedfield k which is an integral domain. In other words, A = A(X) is the coordinate

the spectrum of a ring 50

ring of a variety X, and the elements of A are the regular functions on X withvalues in k.

We know by Hilbert’s Nullstellensatz that points of X correspond to themaximal ideals in A: a point x correspond to the ideal mx of regular functionsvanishing at x, and conversely, every maximal ideal is the vanishing ideal of apoint.

The variety X has the Zariski topology in which the closed sets are the zerosets of ideals a in A; i.e. they have the form V(a) = t x P X | f (x) = 0 all f P a u.Or bearing the correspondence of points and maximal ideals in mind this maybe rephrased as V(a) = tm | m Ă A maximal and m Ě a u.

There is a natural way of generalizing this to all rings, and this involvesreplacing maximal ideals with prime ideals.

Definition 2.1 For a ring A we define its spectrum as Recall: prime idealsare proper ideals

Spec A = t p | p Ă A is a prime ideal u.

To distinguish between points in Spec A and ideals in A, we sometimes writepxĎ A for the point corresponding to x P Spec A in accordance with the practicefor varieties.

When A is a general ring, we cannot think of elements of A as functions intosome fixed field k. However, there still is an analogy between the elements fof A and some sort of functions on Spec A. If x is a point in Spec A and p = px,the localization Ap is a local ring with maximal ideal pAp, and one has the fieldk(p) = Ap/(pAp) (which also will be denoted by k(x)). The element f reducedmodulo p gives an element f (x) P k(p), which may be considered as the ‘value’of f at x; clearly f (x) = 0 if and only if f P p.

Definition 2.2 The field k(p) is called the residue field of Spec A at p.

Note in particular that for each f P A we may speak of the zero set V( f ) =tx P Spec A | f (x) = 0u in Spec A of the element f . It is important to note thatthe ‘values’ of an element f P A lie in different fields which might vary with thepoint. Thus we tweak our notion of a ‘regular function’ on X: they are not mapsinto some fixed field, but rather maps into the disjoint union

š

xPX k(x).We can put a topology on Spec A which generalizes the Zariski topology on

a variety and which will also be called the The Zariski topology(Zariski topologien)

Zariski topology. The definitions arevery similar: the closed sets in Spec A are defined to be those of the form

V(a) = t x P Spec A | f (x) = 0 for all f P a u

= t p P Spec A | p Ě a u,

where a is any ideal in A. Of course, one has to verify that the axioms for atopology are satisfied. For closed sets the wording of the axioms is that the


union of two closed and the intersection of any number (finite or infinite) mustbe closed. And of course, both the whole space and the empty set must beclosed. The family of subsets of the form V(a), indeed, satisfies these axioms, aswe prove in the following lemma:

Lemma 2.3 Let A be a ring and assume that taiuiPI is a family of ideals in A. Let aand b be two ideals in A. Then the following three statements hold true:

i) V(aX b) = V(a)YV(b) = V(ab);ii) V(

ř

i ai) =Ş

i V(ai);iii) V(A) = H and V(0) = Spec A.

Proof: Prime ideals are by definition proper ideals, so V(A) = H. Also, thezero-ideal is contained in every ideal, so V(0) = Spec A. This proves the laststatement. The second follows just as easily, since the sum of a family of idealsis contained in an ideal if and only if each of the ideals is.

The first of the three statements needs an argument. The inclusion V(a)Y

V(b)ĎV(aX b) is clear, so we need to show that V(aX b)ĎV(a)YV(b). Letp be a prime ideal such that aX bĎ p. If b Ę p; then there is an element b P b

with b R p. But since ab P aX b for all a P a, and p contains aX b, one has ab P p.The ideal p is prime, so we may deduce that a P p, and consequently one has theinclusion aĎ p. o

Corollary 2.4 The collection tV(a)u where a runs through all the ideals in A, is thefamily of closed sets for a topology on Spec A.

The next lemma is about inclusions between the closed sets of Spec A, and werecognise all the statements from the theory of varieties.

Lemma 2.5 For two ideals a, b Ă A we have

i) V(a)ĎV(b) if and only if?a Ě

?b. In particular, one has V(a) = V(

?a);

ii) V(a) = H if and only if a = A;iii) V(a) = Spec A if and only if aĎ

a

(0).

Proof: The main point is that the radical of an ideal equals the intersection ofall the prime ideals containing it, or expressed with a formula: See Exercise 2.1

?a =

č

pĚa

p. (2.1)

In particular, we see that a and?a are contained in the same prime ideals,

so that V(a) = V(?a). To show the first claim in the lemma, we begin with

assuming that V(a)ĎV(b). From (2.1) we then obtain

?a =

č

pPV(a)

p Ěč

pPV(b)

p =?b.


Now, if p lies in V(a) and?a Ě

?b, the chain of inclusions p Ě

?a Ě

?b Ě b

implies the inclusion p P V(b) as well. This proves i).The second claim follows Lemma ??, since

?a = (1) if and only if a = (1),

and the third holds as V(0) = Spec A. o

Corollary 2.6 The map a ÞÑ V(a) gives a one-to-one correspondence between radicalideals of A and closed subsets of Spec A.

The following corollary is almost a tautology:

Corollary 2.7 (Closure of a subset) The closure of a set SĎ Spec A is given asS = V(a) where a =

Ş

pPS p.

Proof: Let b be the radical ideal with V(b) = S. Then every p P S contains b

and we may infer that bĎ a. On the other hand, V(a) is closed and SĎV(a)

so that SĎV(a). Hence V(b)ĎV(a), and a Ď b by Lemma 2.5. It follows thata = b. o

Exercise 2.1 Let aĎ A be an ideal. Show that?a =

Ş

aĎ p p. Hint: If f R?a theˇ

ideal aA f is a proper ideal in the localization A f , hence contained in a maximalideal. M

Generic pointsThe Zariski topology on Spec A differs greatly from the topology on manifoldswe are used to. For instance, points can fail to be closed. In fact, the next propo-sition implies that a point x P Spec A is closed if and only if the correspondingideal is maximal. Applying Corollary 2.7 to S = tpu, we get:

Proposition 2.8 If p is a prime ideal of A, the closure tpu of the one-point settpu in Spec A equals the closed set V(p).

In general, there are of course typically a lot of prime ideals which arenot maximal. In fact, the rings having the property that every prime ideal ismaximal are quite special; they are the rings of Krull dimension zero, and in theNoetherian case they correspond exactly to the Artinian rings (in which case thespectrum is a finite set of points and has the discrete topology, see Theorem ??in CA).

Definition 2.9 A point x in a closed subset Z of a topological space X is calleda generic point for Z if Z is the closure of the singleton txu; that is, if txu = Z,

In our context, each prime ideal p is a generic point of the closed set V(p).So for example, for an integral domain A, the zero ideal (0) is prime, and itcorresponds to the generic point of X = Spec A = V(0). There are simple


examples of irreducible topological spaces with more than one generic point.For instance, in spaces with the trivial topology, i.e. where the empty set andthe entire set are the only open sets, every point is generic. However, we willsee shortly that in our context (and in the context of schemes in general) genericpoints are unique.

Examples(2.10) Fields. If K is a field, the prime spectrum Spec K has only one element,which corresponds to the only prime ideal in K, the zero ideal. This also holdstrue for local rings A with the property that all elements in the maximal idealsare nilpotent, i.e. the radical

a

(0) of the ring is a maximal ideal (see Exercise2.20). For Noetherian local rings this is equivalent to the ring being an Artinianlocal ring.(2.11) The ring A = C[x]/(x2) is not a field, but it has only one prime ideal(namely the ideal (x)). Note that (0) is not prime, since x2 = 0, but x R (0).(2.12) Artinian rings. More generally, if A is an Artinian ring, then A has onlyfinitely many prime ideals, so Spec A is a finite set. For a Noetherian ring A, theconverse is also true.(2.13) Discrete valuation rings. Consider a discrete valuation ring A, for examplethe power series ring C[[x]], or one of the localizations k[x](x) or Z(p). (SeeSection ?? or Appendix A for background on discrete valuation rings). A hasonly two prime ideals, the maximal ideal m and the zero ideal (0). So its primespectrum Spec A has just two points, and Spec A = tη, xu with x correspondingto the maximal ideal m and η to (0). The point x is closed in Spec A, andtherefore tηu = X´ x is open. So η is an open point! The point η is the genericpoint of Spec A; its closure is the whole Spec A.

The open sets of X are H, X, tηu. In particular Spec A is not Hausdorff, as η

is contained in the only open set containing x, the whole space.

The spectrum of a DVR

(2.14) The spectrum of the integers, Spec Z. There are two types of prime idealsin Z: the zero-ideal and the maximal ideals (p)Z, one for each prime p. Thelatter give closed points in Spec Z, but one has V(0) = Spec Z, so the pointcorresponding to the zero-ideal is a generic point.

The spectrum of the integers

affine spaces 54

The residue field at a closed point (p) is given by k(p) = Zp/(p)Zp = Fp,wheras the residue field at (0) is given by Z(0) = Q.

Each element f of the ring Z gives rise to a ‘regular function’ into thevarious residue fields. Thus for instance f = 17 P Z takes the values f ((0)) =17, f ((2)) = 1, f ((3)) = 2, f ((5)) = 2, f (7) = 3, . . . , in the fields Q, F2, F3, F5,F7, . . . , respectively.

K

2.2 Affine spaces

The most important examples of prime spectra are the affine n-spaces:

Definition 2.15 We define the affine n-space as

An = Spec Z[x1, . . . , xn].

More generally, for a ring A, we define the affine n-space over A by

AnA = Spec A[x1, . . . , xn].

If k is an algebraically closed field, then Ank is the scheme analogue of

the affine n-space An(k) (the variety whose underlying set is kn). In thissetting, Hilbert’s Nullstellensatz tells us that points of An(k) are in 1-1 corre-spondence with maximal ideals in A = k[x1, . . . , xn]; they are all of the form(x1 ´ a1, . . . , xn ´ an) with ai P k. Thus An(k) is naturally a subset of the spec-trum An

k , and the good old Zariski topology on the variety An(k) is the inducedtopology. However, there are many other prime ideals in A than just the maxi-mal ideals; the zero ideal for instance. So for n ě 1, An

k is strictly larger thanAn(k). The differences between An

k and An(k) become even more apparent if kis not algebraically closed.Example 2.16 The affine line A1

k = Spec k[x]. A1k = Spec k[x] is called the Affine line (Den affine

linja)affine

line over k. In the polynomial ring k[x] all ideals are principal, and all non-zeroprime ideals are maximal. In general they are of the form ( f (x)) where f (x)is an irreducible polynomial, hence of the form (x´ a) if we assume that k isalgebraically closed. There is only one non-closed point in Spec k[x], the genericpoint η, which corresponds to the zero-ideal. The closure tηu is the whole lineA1

k . Thus A1C consists of the generic point η, and the closed points (x´ a) for

a P C.An interesting case is when k = R, where A1

R is called the Real affine line (Denreelle affine linja)

real affine line.By the Fundamental Theorem of Algebra a prime ideal p of R[x] is of theform p = ( f (x)) where either f (x) is linear; that is, f (x) = x ´ a for ana P R, or f is quadratic with two conjugate complex and non-real roots; that is,f (x) = (x´ a)(x´ a) with a P C but a R R. This shows that the closed points

distinguished open sets 55

in Spec R[x] may be identified with the set of pairs ta, au with a P C. And ofcourse, there is the generic point η corresponding to the zero ideal.

When k is algebraically closed, the residue fields of A1k are of the form

k(a) = k[x](xá)/(x´ a)k[x](xá) » k,

and k(η) = k[x](0) = k(x). When k is not algebraically closed, we have moreinteresting residue fields; for instance p = (x2 + 1) defines a point in A1

R

with residue field C. In general, a maximal ideal m in k[x] is generated by anirreducible polynomial, say f (t), and defines a point in A1

k whose residue fieldis the extension of k obtained by adjoining a root of f .

K

Example 2.17 The affine plane A2k = Spec k[x1, x2]. When k is algebraically

closed, the maximal ideals of k[x1, x2] are all of the shape (x1 ´ a1, x2 ´ a2) andthese constitute all the closed points of A2

k . There are also the prime idealsof the form p = ( f ) where f (x1, x2) is an irreducible polynomial. The primeideal p is the generic point of the closed subset V( f (x1, x2)). In addition tothe point p, the points of V( f (x1, x2)) are the closed points corresponding toideals (x1 ´ a1, x2 ´ a2) containing f (x1, x2). This condition is equivalent tof (a1, a2) = 0, so the closed points of V( f (x1, x2)) correspond to what one in theworld of varieties would call the curve given by the equation f (x1, x2) = 0. K

2.3 Distinguished open sets

There is no way to describe the open sets in Spec A as simply and elegantlyas the closed sets can be described. However there is a natural basis for thetopology on Spec A whose sets are easily defined, and which turns out to bevery useful. For an element f P A, we let D( f ) be the complement of the closedset V( f ), that is,

D( f ) = t p | f R p u = X´V( f ).

distinguished open sets 56

These are clearly open sets and are called Distinguished opensets (prinsipale åpnemengder)

distinguished open sets.

Lemma 2.18 One has D( f )XD(g) = D( f g).

Proof: If p is a prime ideal, both f R p and g R p hold true if and only if f g R p.o

Lemma 2.19 The open sets D( f ) form a basis for the topology of Spec A when f runsthrough the elements of A.

Proof: We need to show that any open subset U of Spec A can be written asthe union of sets of the form D( f ). Observe that, by definition, the complementUc of U is of the form Uc = V(a), where a Ă A is an ideal, and choose a set t fiu

of generators for a (not necessarily a finite set). Then we have

U = V(a)c = V

(ÿ

i

( fi)

)c

=

(č

i

V( fi)

)c

=ď

i

D( fi).

o

Example 2.20 In A1 every closed set is of the form V( f ) for some polynomialf , so every open set is a distinguished open set D( f ). In A2 = Spec Z[x, y],the set U = A2 ´V(x, y) is open, but not of the form D( f ). Still, we we haveU = D(x)YD(y). K

Exercise 2.2 Show that D( f ) = H if and only if f is nilpotent. Hint: Use thatˇa

(0) =Ş

pPSpec A p. M

We now give some of the basic properties of the distinguished open setsthat will be needed later. The first tells us when the distinguished open setscorresponding to a family t fiuiPI of elements from A cover Spec A.

Lemma 2.21 A family tD( fi)uiPI forms an open covering of Spec A if and only if onemay write 1 =

ř

i ai fi with the ai’s being elements from A only a finite number ofwhich are non-zero.

Proof: One has V(ř

i( fi))c = (

Ş

i V( fi))c =

Ť

i D( fi), so the open sets D( fi)

constitute a covering if and only if V(ř

i( fi)) = H, which happens if and only ifř

i( fi) = (1). But this happens if and only if 1 is a combination of finitely manyof the fi’s. o

These two lemmas tell us that the D( f )’s form a very handy basis for thetopology; any open set U Ă Spec A can be written as a union of finitely manyD( f )’s. Moreover, as the distinguished sets form a basis for the topology, we seethat any open cover may be refined to one whose members all are distinguished,and hence it can be reduced to a finite covering. A topological space with thisproperty is said to be quasi-compact The terminology is a

little bit unfortunate;spaces in which everyopen cover has a finitesubcover are usuallycalled ‘compact’.However, some authorsreserve the term’compact’ forquasi-compact andHausdorff, and thisjargon has caught onin the algebraicgeometry literature.

.

irreducible closed subsets 57

Lemma 2.22 One has D(g)ĎD( f ) if and only if gn P ( f ) for a suitable naturalnumber n. In particular, one has D( f ) = D( f n) for all n.

Proof: The inclusion D(g)ĎD( f ) holds if and only if V( f )ĎV(g), and byLemma 2.5 on page 51 this is true if and only if (g)Ď

a

( f ), i.e. if and only ifgn P ( f ) for a suitable n. o

In fact, the inclusion D(g)ĎD( f ) is equivalent to the condition that the localiza-tion map ρ : A Ñ Ag extends to a map ρ f g : A f Ñ Ag. Indeed, ρ extends if andonly if ρ( f ), i.e. f regarded as an element in Ag, is invertible, which in its turnis equivalent to there being an b P A and an m P N such that gm( f b´ 1) = 0; orin other words, if and only if gm = c f for some c and some m P N. This enablesus to define the localization map by

ρ f g : A f Ñ Ag

a fń ÞÑ acngńm. (2.2)

More generally, for an A-module M, we have Localization maps(localiseringsavbild-ninger)

localization maps

ρ f g : M f Ñ Mg

m fń ÞÑ mcngńm. (2.3)

where m P M.Exercise 2.3 Check that for a nested inclusion D(h)ĎD(g)ĎD( f ), we haveρ f h = ρgh ˝ ρ f g. M

Exercise 2.4 Let A be a ring, let a be an ideal in A and let t fiuiPI be elementsˇ

from a. Show that the open distinguished sets D( fi) cover Spec AzV(a) if andonly if some power of each element f P a lies in the ideal ( fi|i P I) generated bythe fi’s. M

2.4 Irreducible closed subsets

Recall that a topological space X is said to be Irreducible spaces(irreduktible rom)

irreducible if it can not be written asthe union two proper closed subsets; that is, if X = ZY Z1 with Z and Z1 closed,then either Z = X or Z1 = X. Equivalently, the space X is irreducible if and onlyif any two non-empty open subsets has a non-empty intersection: indeed, to saythat U XV = H for two open subsets U and V is to say that Uc YVc = X. Andso if X is irreducible, either Uc = X or Vc = X; that is, either U = H or V = H.A third way of expressing that X is irreducible, is to say that every non-emptyopen subset is dense.Exercise 2.5 Let Z be an irreducible subspace of the topological space X. Showˇ

that the closure Z is irreducible. If f : X Ñ Y is a continuous map to anothertopological space Y, show that f (Z) is irreducible. M

In a topological space X, a maximal, irreducible subset is called an Irreducible component(irreduktiblekomponenter)

irreduciblecomponent. Since the closure of an irreducible set is irreducible, the irreduciblecomponents are automatically closed, and their union equals the entire space X.


Clearly the closure of any singleton is irreducible: if txu is the union of twoclosed sets, x must lie in one of them, and hence that set equals txu.Exercise 2.6 Use Zorn’s lemma to prove that any irreducible subset of aˇ

topological space X is contained in an irreducible component. Show that X isthe union of its irreducible components. M

Exercise 2.7 Let X be a topological space and let ZĎX be an irreduciblecomponent of X. Let U be an open subset of X and assume that U X Z isnonempty. Show that ZXU is an irreducible component of U. M

From the the theory of varieties we know that the coordinate ring of anaffine variety (which by definition is irreducible) is an integral domain, and verysimple examples illustrate that reducibility is closely linked to zero divisors inthe ring of functions (see Example 9 below). In general, we have the following:

Proposition 2.23 Let A be a ring. Then the following statements hold:

i) If pĎ A is a prime ideal, it holds that tpu = V(p), and p is the onlygeneric point of V(p);

ii) A closed subset ZĎ Spec A is irreducible if and only if Z is of theform Z = V(p) for some prime ideal p;

iii) The space Spec A itself is irreducible if and only if A has just oneminimal prime ideal; in other words, if and only if the nilradical

a

(0)is prime.

Proof: Statement i) is just Proposition 2.8 on page 52. For the uniqueness part,when V(p) = V(q), it holds by Lemma 2.5 on page 51 that both pĎ q and qĎ p.

As the closure of any singleton is irreducible and since we just showedthat V(p) = tpu, we know that V(p) irreducible. For the reverse implicationin ii), let V(a)Ď Spec A be a closed subset. Recall that

?a =

Ş

aĎ p p, and if?a is not prime, there are more than one prime involved in the intersection.

We may divide them into two different groups thus representing?a as the

intersection?a = bX b1 where b and b1 are ideals whose radicals are different.

One concludes that V(a) = V(b)YV(b1), so it is not irreducible.For the third statement it suffices to observe that Spec A = V

(a(0)), and

again we rely on Lemma 2.5. o

A consequence of the lemma is that Spec A is irreducible whenever A is anintegral domain, as in that case (0) is a prime ideal. However, contrary to whatholds for coordinate rings of varieties, a ring whose spectrum is irreducible,does not need to an integral domain, but statement iii) tells us that each non-zerodivisor is nilpotent – or, in the spirit of the analogy with functions there arenon-zero functions that vanish everywhere. The ring A = k[t]/(t2) is a simpleexample showing such behaviour. It is not an integral domain, but has only oneprime ideal, namely the principal ideal (t); hence X = Spec A is just a point and


is irreducible. Note also that the lemma tells us that any non-empty closed andirreducible subset of Spec A has a unique generic point.

Recall that a topological space is Connected spaces(sammenhengenderom)

connected if it cannot be written as a disjointunion of two proper open subsets. Connectedness is a weaker topologicalnotion than irreducibility in the sense that an irreducible space is also connected.However, as the examples below show, the converse does not hold. In thedictionary between algebra and geometry, connectedness of Spec A translatesinto absence of non-trivial idempotents in A (see Example 10 below).Example 2.24 The prime spectrum X = Spec k[x, y]/(xy) is a good example ofa space which is connected but not irreducible. The coordinate functions x andy are zero-divisors in the ring k[x, y]/(xy), and their zero-sets V(x) and V(y)show that X has two components. Since these two components intersect at theorigin, X is connected. K

Example 2.25 The spectrum Spec(A1 ˆ A2) of a direct product of two non-trivialrings is not connected. All the ideals are products of the form a1 ˆ a2 of idealsaiĎ Ai, and they are prime if and only if either a1 or a2 is prime and the otheris the entire ring. Hence Spec(A1 ˆ A2) is homeomorphic to the disjoint unionSpec A1 Y Spec A2. Moreover Spec Ai » D(ej) = V(ei) (with ti, ju = t1, 2u), andso both are both open and closed subsets.

See also Exercise 2.13for a generalization

With (1, 0) = e1 and (0, 1) = e2 it holds that 1 = e1 + e2, so that a = ae1 + ae2,and from e2

i = ei, we get that that each ai = aei is an ideal in Ai. Moreover, ase1e2 = 0, it holds that a1 X a2 = 0, in other words, a = a1 ˆ a2. It always holdsthat e1e2 P a, so for a to be prime either e1 or e2 must belong to a; i.e. eithera1 = A1 or a2 = A2, and obviously the proper one must be prime. Clearlyai = Ai if and only if ei P a, so Spec Ai = D(ej) = V(ei). K

Exercise 2.8 Assume that X is a topological space that is not connected. Exhibittwo non-constant orthogonal idempotents with sum unity in the ring of continu-ous functions on X. Hint: The characteristic functions of two disjoint open setswhose union equals X, will do. M

Decomposition into irreducible subsetsFrom commutative algebra we know that ideals in Noetherian rings have aprimary decomposition. This is the Noether-Lasker theorem proven by EmanuelLasker for ideals in polynomial rings in 1905. Some fifteen years later the generalresult, as we know it today, was established by Emmy Noether. It states thatevery ideal a in a Noetherian ring can be expressed as an intersection

a = q1 X q2 X ¨ ¨ ¨ X qr

where the qi’s are primary ideals (See e.g. Theorem ?? in CA). Recall thatprimary ideals have radicals that are primes, so the ideals

?qi are all prime.

They are called the Associated primes(assosierte primidealer)

associated primes to a. Such a decomposition is not alwaysunique, but there are partial uniqueness results. The associated prime ideals


are unique as are the primary components qi whose associated prime idealsare minimal. However, the so-called embedded components A primary component

qi is embedded if?qi contains the

radical ?qj of anothercomponent qj.

are not unique. Forinstance, one has the equality (x2, xy) = (x) X (x2, y), but also the equality(x2, xy) = (x)X (x2, xy, y2) holds true.

Let A be a ring and consider a primary decomposition of an ideal a:

a = q1 X q2 X ¨ ¨ ¨ X qr.

Putting Yi = V(?qi), we find V(a) = Y1 Y Y2 Y ¨ ¨ ¨ Y Yr, where each Yi is an

irreducible closed set in Spec A. If the prime?qi is not minimal among the

associated primes, say ?qjĎ?qi, it holds that YiĎYj, and the component Yi

contributes nothing to union and can be discarded.In a more general context, a decomposition Y = Y1Y ¨ ¨ ¨ YYr of any topologi-

cal space is said to be Redundantdecompositions(redundantedekomposisjoner)

redundant if one can discard one or more of the Yi’s withoutchanging the union. That a component Yj can be omitted, is equivalent to Yj

being contained in the union of the rest; that is, YjĎŤ

i‰j Yi. A decomposition

that is not redundant, is said to be Irredundantdecompositions(irreduntantedekomposisjoner)

irredudant. Translating the Noether–Laskertheorem into geometry we arrive at the following:

Proposition 2.26 If A is a Noetherian ring, every closed subset YĎ Spec Acan be written as an irredundant union

Y = Y1 Y ¨ ¨ ¨ YYr,

where the Yi’s are irreducible closed subsets. The Yi’s are unique up to order.

Notice, that since embedded components do not show up for radical ideals,we get a clear and clean uniqueness statement.

Examples(2.27) Consider the closed subset Y = V( f , g) in A3

k = Spec k[x, y, z], with k al-gebraically closed, where f = x2´ yz and g = xz´ x. A primary decompositionof the ideal I = ( f , g) is given by

I = (x, y)X (x, z)X (y´ x2, z´ 1).

This gives a decomposition of Y as

Y = V(x, z)YV(x, z)YV(y´ x2, z´ 1),

In these situations, it is sometimes easier to use geometric arguments to findthe irreducible components of Y. These arguments are legitimate even if theyinvolve only closed points, since closed points are dense in each V(p) (by theNullstellensatz).

Since g vanishes at a point in Y, we find that either x = 0 or z = 1 at thepoint. If x = 0, then the vanishing of f at the point implies that either y = 0


or z = 0. Thus we find the two components Y1 = V(x, y) and Y2 = V(x, z). Inthe case z = 1, then the vanishing of f implies that y = x2, so we obtain a thirdcomponent Y3 = V(y´ x2, z´ 1). Thus we have find that

Y = V(x, z)YV(x, z)YV(y´ x2, z´ 1),

and we leave to the reader to verify that the three subsets are irreducible andthat the union is irredundant.(2.28) Consider the closed set Y = V(a) Ă A3

k given by the ideal

a = (x2 ´ y, xz´ y2, x3 ´ xz).

Note first that if x = 0 at a point in Y, it follows that y = 0 there, so V(x, y) Ă X.If x ‰ 0 at the point, the third equation gives z = x2 there, and so by the firstand second equations we get xz´ y2 = x3 ´ x4 = 0 at the point; and thus x = 1,y = 1 and z = 1. Hence

X = V(x, y)XV(x´ 1, y´ 1, z´ 1).

That is, X is the union of the z-axis, and the point (1, 1, 1). In fact, a primarydecomposition of a is given by a = q1 X q2 X q3, where

q1 = (x, y), q2 = (x´ 1, y´ 1, z´ 1), q3 = (x2 ´ y, xy, y2, z).

Taking radicals, we find that the primes associated to a are the following:

p1 = (x, y), p2 = (x´ 1, y´ 1, z´ 1), p3 = (x, y, z).

Note that p1 Ă p3, and p3 is thus an embedded component, which does not showup in the decomposition above.(2.29) Consider Y = V(yz, xz, y3, x2y) Ă A3

k . Computing a primary decomposi-tion, we find that

(yz, xz, y3, x2y) = (x, y)X (y, z)X (x2, y3, z),

and it follows that Y has two irreducible components, the two lines V(x, y) andV(y, z), and that there is an embedded component at the origin.

K

Noetherian topological spacesA decomposition result as the one in Proposition 2.16 on the previous pageabove holds for a much broader class of topological spaces than the closedsubsets of spectra. The class in question is the class of the so-called Noetherian topological

spaces (noethersketopologiske rom)

Noetheriantopological spaces; these satisfy the requirement that every descending chain ofclosed subsets is eventually stable. That is, if tXiu is a collection of closed subsetsforming a chain

X1 Ě X2 Ě ¨ ¨ ¨ Ě Xi Ě Xi+1 Ě . . . ,


it holds true that for some index ν one has Xi = Xν for i ě ν. It is easy toestablish that any closed subset of a Noetherian space endowed with the inducedtopology is Noetherian.Exercise 2.9 Let X be a topological space. Show that the following threeˇ

conditions are equivalent:

i) X is Noetherian;

ii) Every open subset of X is quasi-compact; The last statement inthe exercise leads tothe technique calledNoetherianinduction – proving astatement about closedsubsets, one can workwith ‘a minimalcounterexample’. Itenjoys the propertythat the statement tobe proven holds forevery proper closedsubset.

iii) Every non-empty family of closed subsets of X has a minimal member.

M

The Noether–Lasker decomposition of closed subsets in affine space as aunion of irreducibles can be generalized to any Noetherian topological space:

Theorem 2.30 Every closed subset Y of a Noetherian topological space X hasan irredundant decomposition Y = Y1Y ¨ ¨ ¨ YYr where each is Yi is a closed andirreducible subset of X. Furthermore, the decomposition is unique up to order.

The Yi’s that appear in the theorem are the Irreduciblecomponents(Irreduktiblekomponenter)

irreducible components of Y. Theyare maximal among the closed irreducible subsets of Y.Proof: We shall work with the family Σ of those closed subsets of X that cannotbe decomposed into a finite union of irreducible closed subsets; or phrased in adifferent way, the set of counterexamples to the assertion – and of course, weshall prove that it is empty.

Assuming the contrary – that Σ is non-empty – we can find a minimalelement Y in Σ because X by assumption is Noetherian. The set Y itself cannot be irreducible, so Y = Y1 Y Y2 where both the Yi’s are proper subsets ofY and therefore do not belong to Σ. Either is thus a finite union of closedirreducible subsets, and consequently the same is true for their union Y. Wehave a contradiction, and Σ must be empty.

As to uniqueness, assume that we have a counterexample; that is, twoirredundant decomposition such that Y1Y ¨ ¨ ¨ YYr = Z1Y ¨ ¨ ¨ Y Zs and such thatone of the Yi’s, say Y1, does not equal any of the Zk’s.

Since Y1 is irreducible and Y1 =Ť

j(Zj XY1

), it follows that Y1ĎZj for some

index j. A similarly argument gives Zj =Ť

i(Zj XYi

)and Zj being irreducible,

it holds that ZjĎYi for some i, and therefore Y1ĎZjĎYi. Since the union of theYi’s is irredundant, we infer that Y1 = Yi, and hence Y1 = Zj. Contradiction. o

Exercises(2.10) Show that the Zariski topology on Spec A is Hausdorff if and only if everyprime ideal p is maximal. Show that the Zariski topology always is T0.(2.11) Compute a primary decomposition for the following ideals and describeˇ

their corresponding closed subsets.

morphisms between prime spectra 63

a) I = (x2y2, x2z, y2z) in k[x, y, z];b) I = (x2y, y2x) in k[x, y];c) I = (x3y, y4x) in k[x, y];d) I = (x, y, x´ yz) in k[x, y, z];e) I = (x2 + (y´ 1)2 ´ 1, y´ x2) in k[x, y].

(2.12) One says that an ideal n in a ring A is locally nilpotent if each elementˇ

in n is nilpotent. Show that for each ideal a in A it holds that?a+ n =

?a

whenever n is locally nilpotent. Let A Ñ B be a surjection of commutativerings whose kernel is locally nilpotent. Show that the map Spec B Ñ Spec A is ahomeomorphism.(2.13) Direct products of rings. Assume that e1, . . . , er is a complete set of orthog-ˇ

onal idempotents in the ring A, meaning that one has 1 = e1 + e2 + ¨ ¨ ¨+ er, thateiej = 0 when i ‰ j, and that e2

i = ei. Such a set of idempotents correspondsto a decomposition of A as the direct product A = A1 ˆ A2 ˆ . . .ˆ Ar whereAi = ei A for i = 1, . . . , r (each Ai is a subring of A with unit element ei).

a) Let a be an ideal in A and let ai = aei. Show that ai is an ideal in Ai andthat a = a1 + a2 + ¨ ¨ ¨+ ar.

b) Show that a is a prime ideal if and only if ai = Ai for all but one index i0and ai0 is a prime ideal in Ai0 .

c) Show that Spec A is not connected: It holds that Spec A =Ť

i Spec Ai, theunion being disjoint and each Spec Ai being open and closed in Spec A.

(2.14) Lifting of idempotents. Let A Ñ B be a surjective map of rings whoseˇ

kernel a is locally nilpotent; that is, every element of a is nilpotent. Let e be anidempotent in B. The aim of the exercise is to show that there is an idempotent fin A mapping to e. Choose any element x in A that maps to e and let y = 1´ x.

a) Show that xy P a.

b) Let n be such that (xy)n = 0 and define f =ř

iąn (2ni )xiy2ní and g =

ř

iďn (2ni )xiy2ní. Show that 1 = f + g and that f g = 0.

c) Conclude that f is an idempotent in A that maps to e.

d) Show that if Spec A is not connected, the ring A is a non-trivial directproduct; that is, A » Bˆ C for non-zero rings B and C

(2.15) Let tAiuiPI be an infinite sequence of non-trivial rings, and let X be theˇ

disjoint union of the spectra Spec Ai. Show that X is not homeomorphic to aspectrum of a ring.

M

2.5 Morphisms between prime spectra

In the early days of modern algebraic geometry, when varieties were predom-inant, one worked with the space of maximal ideals. In retrospect, this is


somewhat problematic, due to the lack of functoriality. Indeed, the inverseimage of a maximal ideal does not need to be maximal; a simple example is theinclusion of k[x] into k(x); the zero ideal (0) is maximal in the field k(x), butnot in k[x]. The solution was to introduce the full spectrum of all prime ideals.Then there is no issue: The unique point of Spec k(x) is sent to the generic pointof Spec k[x].

Let A and B be two rings and let φ : A Ñ B be a ring homomorphism. Theinverse image φ´1(p) of a prime ideal pĎ B is a prime ideal: that ab P φ´1(p)

means that φ(ab) = φ(a)φ(b) P p, so at least one of φ(a) or φ(b) has to lie in p.Hence sending p to φ´1(p) gives us a well defined map

Φ : Spec B Ñ Spec A

We will sometimes denote this map by Spec(φ). The reason is to emphasizethat Spec defines a contravariant functor from the category Rings of rings tothe category Top of topological spaces. Indeed, in this situation, we have theequality Spec φ ˝ Spec ψ = Spec(ψ ˝ φ), whenever φ and ψ are composable ringhomomorphisms. This is because of the identity φ´1(ψ´1(p)) = (ψφ)´1(p).Also, of course, it holds that Spec idA = idSpec A.

The basic properties of Spec(φ) is summarized in the next two propositions.

Proposition 2.31 Assume that φ : A Ñ B is a map of rings, and let Φ :Spec B Ñ Spec A denote the induced map. Then

i) Φ´1(V(a)) = V(φ(a)B) for each ideal a Ă A. In particular, themap Φ is continuous;

ii) Φ´1(D( f )) = D(φ( f )) for each f P A;iii) Φ(V(b)) = V(φ´1(b)) for each ideal b of B.

Proof: Proof of i): Let aĎ A be an ideal. We have the following series ofequalities

Φ´1(V(a)) = t pĎ B | φ´1(p) Ě a u = t pĎ B | p Ě φ(a) u = V(φ(a)B);

indeed, because φ´1(φ(a)) Ě a, it holds true that p Ě φ(a) if and only if it holdstrue that φ´1(p) Ě a. In particular, the inverse image of any closed subset isagain closed, so Spec φ is continuous.

Proof of ii): Note that for each element f P A we have the following equalities:

Φ´1(D( f )) = t pĎ B | f R φ´1(p) u = t pĎ B | φ( f ) R p u = D(φ( f )).

Proof of iii): According to Corollary 2.7 on page 52 the closure Φ(V(b))

equals V(a) with a the ideal given by p P Φ(V(b)) meansthat p = φ´1(q) forsome q, with q Ě b.

a =č

pPΦ(V(b))

p =č

bĎ q

φ´1(q),


and it holds true that

a =č

bĎ q

φ´1(q) = φ´1( č

bĎ q

q)= φ´1(

?b) =

b

φ´1(b).

Hence V(a) = V(φ´1(b)), which gives the desired identity. o

Proposition 2.32 With the assumptions of Proposition 2.18, we have:

i) If φ is surjective, then Φ is a homeomorphism from Spec A onto theclosed subset V(Ker φ) of Spec A;

ii) If φ is injective, then Φ(Spec B) is dense in Spec A. More precisely,the image Φ(Spec B) is dense in Spec A if and only if Ker φ Ă

?0.

Proof: Proof of i): If φ : A Ñ B is surjective, we may assume B = A/a, wherea = Ker φ. There is an inclusion preserving one-to-one correspondence betweenprime ideals in A/a, and prime ideals in A containing a. This shows that Φ is acontinuous bijection onto the closed subset V(a). It also follows that

Φ(V(b/a)) = tp P Spec A|b/aĎ p/a P Spec(A/a)u = V(b),

so Φ is closed, and hence it is a homeomorphism.Proof of ii): Again, by iii) of Proposition 2.18, the closure of Φ(Spec B) =

Φ(V(0)) equals V(φ´1(0)) = V(Ker φ). So Φ(Spec B) is dense if and onlyif V(Ker φ) = Spec A. But this happens exactly when Ker φĎ p for all p, orequivalently when Ker φĎ

?0. o

We include a few prototypical examples:Example 2.33 The spectrum Spec(A/a) of a quotient. If a Ď A is an ideal, thering homomorphism A Ñ A/a induces a continuous map

f : Spec(A/a)Ñ Spec A.

Note that the prime ideals in A/a pull back bijectively to the prime ideals p in Acontaining a, and this is precisely our closed set V(a). By the above proposition,the map f is a homeomorphism onto the closed subset V(a). This is the standardexample of a Closed immersions

(lukkede immersjoner)closed immersion. We will discuss these in more detail later. K

Example 2.34 The spectrum Spec A f of a localization. For an element f P Aconsider the localization A f of A in which f is inverted and the correspondingring homomorphism A Ñ A f . The prime ideals in the localized ring A f are in anatural one-to-one correspondence with the prime ideals p of A not containing f ;in other words, with the complement D( f ) = Spec A´V( f ). Thus the inducedmap Spec A f Ñ Spec A is a homeomorphism onto the open set D( f ) of Spec A.This is an example of an Open immersions

(åpne immersjoner)open immersion. K

Example 2.35 Reduction mod p. The reduction mod p-map Z Ñ Fp inducesa map Spec Fp Ñ Spec Z. The one and only point in Spec Fp is sent to the


point in Spec Z corresponding to the maximal ideal (p). The inclusion ZĎQ ofthe integers in the field of rational numbers induces likewise a map Spec Q Ñ

Spec Z, that sends the unique point in Spec Q to the generic point η of Spec Z.Every ring A has (since by our convention it contains a unit element) a prime

ring, i.e. the subring generated by 1. Hence there is a canonical (and in fact,unique) map Spec A Ñ Spec Z. This canonical map factors through the mapSpec Fp Ñ Spec Z described above if and only if A is of characteristic p, whichbecomes clear if one considers the diagram on the ring level:

Z Fp

A

and notes that the canonical map Z Ñ A goes via Fp if and only if A is ofcharacteristic p. K

Example 2.36 The twisted cubic. Let k be a field. The ring map φ : k[x, y, z]Ñ k[t]given by x ÞÑ t, y ÞÑ t2, z ÞÑ t3 defines a morphism of spectra

f : A1k Ñ A3

k

The image of f is the twisted cubic curve V(a) Ă A3k defined by the ideal a =

Ker φ = (y´ x2, z´ x3).K

Example 2.37 Let k be a field. The ring map

φ : k[x]Ñ k[x, y]/(xy´ 1)

induces a morphism Spec k[x, y]/(xy´ 1)Ñ A1k . On the level of closed points,

when k is algebraically closed, this maps (a, a´1) to a. Since k[x, y]/(xy´ 1) isan integral domain, it has a unique generic point η, and this maps to the genericpoint of A1

k . Note that Spec k[x, y]/(xy´ 1) » D(x)ĎA1k via this morphism. In

particular, the image is not closed in A1k . K

Example 2.38 A similar example is given by the ring map

φ : k[x, y]Ñ k[x, y, z]/(xz´ y)

which induces h : Spec k[x, y, z]/(xz´ y) Ñ A2k . On the level of closed points

when k is algebraically closed, this maps (a, ab, b) to (a, ab), and the genericpoint maps to the generic point. In this example, the image is neither open norclosed: it equals D(x)YV(x, y). K

The fibres of a morphismThe fibre of the morphism Spec φ : Spec B Ñ Spec A over a prime ideal p of Aconsists of the points of Spec B that map to p; in other words those primes q


in B so that φ´1(q) = p. These are naturally in a one-to-one correspondencewith Spec BbA K(A/p), and so the fibre comes as the spectrum of a ring. Thisspectrum is called the

Scheme-theoretic fibres(skjemateoretiske fibre)

scheme-theoretical fibre of Spec φ, and it is the affine versionof a general constriction we shall come back to later. If A is a domain the fibreover (0) is called

Generic fibres(generiske fibre)

the generic fibre of Spec φ.Example 2.39 The spectrum of the Gaussian integers, Spec(Z[i]). The inclusionZĎZ[i] induces a continuous map

φ : Spec(Z[i])Ñ Spec Z.

We will study Spec(Z[i]) by studying the fibres of this map. If p P Z is a prime,the fibre over (p)Z consists of those primes that contain (p)Z[i]. These come inthree flavours:

i) p stays prime in Z[i] and the fibre over (p)Z has one element, namelythe prime ideal (p)Z[i]. This happens if and only if p ” 3 mod 4; This is related to being

able to write p as asum of squares; ifp = x2 + y2, thenp = (x + iy)(x´ iy),so it is not prime inZ[i].

ii) p splits into a product of two different primes, and the fibre consistsof the corresponding two prime ideals. This happens if and only ifp ” 1 mod 4;

iii) p factors into a product of repeated primes (such a prime is said to‘ramify’). This happens only at the prime (2): note that

(2)Z[i] = (2i)Z[i] = (1 + i)2Z[i],

which is not radical. So the fibre consists of the single prime (1+ i)Z[i].

The following picture shows Spec(Z[i]):

The spectrum Spec(Z[i])

The Galois group G = Gal(Q[i]/Q) » Z/2 acts in this example. This group isgenerated by the complex conjugation map, which permutes the prime idealsin Spec(Z[i]) sitting over any (p) in Spec(Z). So for instance if you look at theprimes sitting over say (5), namely (2 + i) and (2´ i), you see that complexconjugation maps one into the other. Thus we picture Spec(Z[i]) as some curvelying above Spec(Z), with G permuting the points in each fibre (though someare fixed by G). K

Example 2.40 Consider again the real affine line A1R from Example 6. Also

in this example, there is an action by the Galois group Gal(C/R) » Z/2Z.


This group acts on C[t] via complex conjugation map, that is, the map thatsends a polynomial

ř

aiti toř

i aiti. The corresponding map from Spec C[t] toSpec C[t] defines an action of Z/2Z on Spec C[t], and the set Spec R[t] is thequotient space of Spec C[t] by this action. Indeed, by Example 6, this holdsfor closed points, and clearly the generic point of Spec C[t] is invariant andcorresponds to the generic point of Spec R[t]. The quotient map is the mapSpec C[t]Ñ Spec R[t] induced by the inclusion R[t]ĎC[t]. K

Exercises(2.16) In the same vein as Example 16, show that a ring A is a Q-algebra (thatis, it contains a copy of Q) if and only if the canonical map Spec A Ñ Spec Z

factors through the generic point Spec Q Ñ Spec Z.(2.17) With reference to Example 20 on the previous page:ˇ

a) Show that the fibre of φ over a prime ideal (p) is homeomorphic to

Spec Fp[x]/(x2 + 1)

and that dimFp Fp[x]/(x2 + 1) =2. Hint: Use that Z[i] = Z[x]/(x2 + 1).

b) Show that Fp[x]/(x2 + 1) is a field if and only if x2 + 1 does not have a rootin Fp.

c) Show that Fp[x]/(x2 + 1) is a field if and only if (p)Z[i] is a prime ideal.

(2.18) The map Spec Z[i] Ñ Spec Z from Example 20 illustrates a generalˇ

pattern for scheme-theoretical fibres of so-called ‘finite’ morphisms:

a) Assume that A is an algebra over a field k of dimension n as a vector space.Show that Spec A has at most n points. Show that Spec A has exactly npoints if and only if A is the direct product of n copies of k.

b) Let AĎ B be two rings and assume that B is free of rank n as an A-module.Show that the fibres of the induced map Spec B Ñ Spec A has at most npoints.

(2.19) Local rings. Recall that a local ring is a ring A with only one maximalˇ

ideal.

a) Show that A is local if and only if Spec A has a unique closed point.

b) Give examples of local rings A so that Spec A consists of i) one point; ii) twopoints; iii) infinitely many points.

c) A map of rings φ : A Ñ B is said to be local if φ(mA)ĎmB. Show that φ islocal if and only if the induced map Φ : Spec B Ñ Spec A maps the uniqueclosed point of Spec B to that of Spec A.

d) Give an example of a ring homomorphism φ : A Ñ B which is not local.Describe your example in terms of the corresponding map on spectra.


(2.20) Show that the second statement in Example 1 is true. That is, show thatˇ

Spec A has just one element if and only if A is a local ring all whose non-unitsare nilpotent.(2.21) Generalizing Example 6, let KĎ L be a Galois extension of fields withˇ

Galois group G; i.e. G acts on L by K-linear automorphisms and K = LG, thesubfield of invariant elements. This action induces and action on L[t] by lettingthe action of a group element g on the polynomial f =

ř

aiti be g( f ) =ř

g(ai)ti.The map K[t]Ď L[t] induces a map π : Spec L[t]Ñ Spec K[t].

Even more generally, let a group G act on a ring A and denote the action ofg on a by ga. The ring of invariant elements is the subring

AG = t a P A | ga = a for all g P G u.

For instance, in the situation above L[t]G = K[t] because polynomials are de-termined by their coefficients. This action induces an action of G on Spec A byletting g act by the map g : Spec A Ñ Spec A such that g7(a) = ga. Denote byπ : Spec A Ñ Spec AG the map induced by the inclusion AGĎ A.

Show that π has the following universal property: for any ring B and anyG-invariant map f : Spec A Ñ Spec B (that is, it satisfies f ˝ g = f for all g P G),there is a unique map h : Spec AG Ñ Spec B such that f = h ˝ π. The diagramlooks like

Spec A Spec A

Spec B

Spec AG

g

f

π π

f

(2.22) Assume that G is a finite group which acts on the ring A.ˇ

a) Prove that the extension AGĎ A is integral and conclude that the mapSpec A Ñ Spec AG is closed and surjective;

b) Prove that G acts transitively on the fibres of the map π : Spec A Ñ Spec AG;

In other words, one may consider Spec AG to be the ’orbit-space’ of the action:its points are in one-to-one correspondence with the orbits of G, and its topologyis the quotient topology.

M

Chapter 3

Schemes

A scheme has two layers of structures; the bottom one is a topological space ontop of which resides a sheaf of rings, called the structure sheaf. It serves as asubstitute for the sheaf of functions and is introduced axiomatically. Withoutfurther restrictions, such double structures are called ‘ringed spaces’, and theyform a vast collections exceedingly larger than the collection of schemes. Ascheme is a ringed space that is locally affine; i.e. it has an open cover whosemembers are isomorphic to affine schemes. Before proceeding towards thegeneral definition of scheme, as a warm up, we introduce the structure sheaf onthe spectrum of a ring. This makes the spectrum into an affine scheme. Affineschemes then serve as the building blocks of general schemes, and they arecornerstones of the theory. To fully grasp the general definition of a scheme, itis all important to master the mechanics of the affine schemes.

3.1 The structure sheaf on the spectrum of a ring

We have now come to point where we define the structure sheaf on the topo-logical space Spec A. This is a sheaf of rings OSpec A on Spec A whose stalks allare local rings, so that the pair (Spec A,OSpec A) is what one calls a Locally ringed space

(lokalt ringet rom)locally ringed

space.The two most important properties of the structure sheaf OSpec A are the

following:

o Sections over distinguished opens: Γ(D( f ),OSpec A) = A f ;

o Stalks: OSpec A,x = Apx .

These are two properties that one uses all the time when working with thestructure sheaf on an affine scheme. Moreover, as we shortly shall see, they evencharacterize the structure sheaf unambiguously.

the structure sheaf on the spectrum of a ring 71

MotivationThe structure sheaf OSpec A on the prime spectrum Spec A of a ring A is modelledon the sheaf of regular functions on an affine variety X. Before giving the maindefinition we recall that situation.

Let A be the coordinate ring of X; that is, A is the ring of globally definedregular functions on X. The fraction field K of A is the field of rational functionson X, i.e. the functions which are regular in some open subset U Ă X. Fordifferent open sets U the set of functions regular in U form different subringsOX(U) of K, and if VĎU is another open contained in U, the ring of regularfunctions OX(V) on V of course contains the ring OX(U) of those regular onthe bigger set U. The restriction map in this case is nothing but the inclusionOX(U) Ď OX(V); it simply considers the functions in OX(U) to lie in OX(V).

Functions regular on the distinguished open set D( f ) = t x P X | f (x) ‰ 0 uare allowed to have powers of f in the denominator, and they constitute thesubring A f ĎK of elements shaped as a fń with a P A and n a non-negativeinteger. As explained in (2.2), if D(g) is another distinguished open set containedin D( f ), i.e. D(g)ĎD( f ), then one can write gm = c f for some c P A and somesuitable m P N, and hence there is a localization A f Ď Ag (since f´1 = cg´m).

The general situation differs from the situation of varieties in that the ringsOX(U) do not lie naturally as subrings of some field K. More dramatically, thelocalization maps A f Ñ Ag may fail to be injective. This happens already inthe case X = Spec A, for A = k[x, y]/(xy), which corresponds to the union ofx-axis and the y-axis in the affine plane. Since xy = 0, the element x maps to 0

via the localization map A Ñ Ay. Geometrically, the regular function x vanishesidentically on the open D(y) where y ‰ 0, and the regular function y vanisheson D(x). So this is by no means a big mystery, it naturally appears once weallow reducible spaces into the mix.

Definition of the structure sheaf OSpec A

One may straight away write down an explicit definition of the structure sheaf(as several texts do), but many students experience this definition as coming outof the blue. We instead take the approach of defining OSpec A as a B-sheaf whichhas a virtue of being more intuitive. In short, we define OSpec A as the sheafsatisfying the two key properties above – they characterize the sheaf uniquely.

Motivated by the above discussion, it makes sense to require the sections ofthe structure sheaf over D( f ) to be the localized ring A f . There is a subtletyhere, because different f ’s might give identical D( f )’s and to avoid choices, weprefer to use a more canonical localization Of course in the end,

OX(D( f )) will beisomorphic to A f

and replace the multiplicative systemt f nu by its saturation: the multiplicative system

SD( f ) = t s P A | s R p for all p P D( f ) u.


This set does not depend on the particular element f , but only on the set D( f ).Furthermore, SD( f )Ď SD(g) whenever D(g)ĎD( f ), so we get naturally inducedlocalization maps S´1

D( f )A Ñ S´1D(g)A.

Definition 3.1 Let B be the collection of distinguished open subsets D( f ).We define the B-presheaf O by

O(D( f )) = S´1D( f )A,

and for D(g)ĎD( f ) we let the restriction map be localization map S´1D( f )A Ñ

S´1D(g)A.

There is a canonical localization map τ : A f Ñ S´1D( f )A since f P SD( f ). The

following lemma says that, using τ, we may identify the ring O(D( f )) = S´1D( f )A

with the ring A f , as we desired.

Lemma 3.2 The map τ is an isomorphism, permitting us to identify O(D( f )) = A f .

Proof: The point is that each element s P SD( f ) does not lie in p for any p P D( f );in other words, one has D( f )ĎD(s). This is equivalent to

a

( f )Ďa

(s), so onemay write f n = cs for some c P A and n P N. Suppose that a f´m P A f

maps to zero in S´1D( f )A. This means that sa = 0 for some s P SD( f ). But then

f na = csa = 0, and therefore a = 0 in A f . This shows that the map τ isinjective. To see that is surjective, take any as´1 in S´1

D( f )A and write this as

as´1 = ca( f n)´1 = ca fń. o

The notation S´1D( f )A will only be present in the definition of O; we will

usually stick to writing O(D( f )) = A f from now on, bearing in mind that O isdefined in terms of a canonical localization.

Proposition 3.3 O is a B-sheaf of rings.

Unraveling the definitions, this can be rephrased as a concrete statementin commutative algebra. We are given a distinguished set D( f ) and an opencovering D( f ) =

Ť

iPI D( fi), where we by quasi-compactness may assume thatthe index set I is finite. Of course then D( fi)ĎD( f ), and we have localizationmaps ρi : A f Ñ A fi and ρij : A fi Ñ A fi f j . Then since D( fi f j) = D( fi X f j), thestatement in the proposition is then equivalent to the exactness of the followingsequence

0 A fś

i A fi

ś

i,j A fi f jα β

(3.1)

where α(a)i = ρi(a) and β((ai))i,j = (ρij(ai)´ ρji(aj)). It is clear that α ˝ ρ = 0since ρij ˝ ρi = ρji ˝ ρj.

Lemma 3.4 The sequence (3.1) is exact.


Proof: We start by observing that we may assume that A = A f (in other words,that f = 1). Indeed, one has (A f ) fi = A fi and (A f ) fi f j = A fi f j since f ni

i = hi f forsuitable natural numbers ni.

Then to the proof: To say that α(a) = 0 is to say that a is mapped to zero ineach of the localizations A fi . Hence a power of each fi kills a; that is, for eachindex i one has f ni

i a = 0 for an appropriate natural number ni. The open setsD( fi) cover D( f ), which then is covered by the D( f ni

i ) as well. Thus we maywrite 1 =

ř

i bi f nii for some elements bi P A, and upon multiplication by a this

gives

a =ÿ

i

bi f nii a = 0.

Hence α is injective.In down-to-earth terms, the equality Ker β = Im α means the following:

assume given a sequence of elements ai P A fi such that ai and aj are mapped tothe same element in A fi f j for every pair i, j of indices. Then there should be ana P A, such that every ai is the image of a in A fi , i.e. ρi(a) = ai.

Each ai can be written as ai = bi/ f nii where bi P A, and since the indices are

finite in number, one may replace ni with n = maxi ni. That ai and aj induce thesame element in the localization A fi f j means that we have the equations

f Ni f N

j (bi f nj ´ bj f n

i ) = 0, (3.2)

where N a priori depends on i and j, but again due to there being only finitelymany indices, it can be chosen to work for all. Equation (3.2) gives

bi f Ni f m

j ´ bj f Nj f m

i = 0 (3.3)

where m = N + n. Putting b1i = bi f Ni we see that ai equals b1i/ f m

i in A fi , andequation (3.3) takes the form

b1i f mj ´ b1j f m

i = 0. (3.4)

Now D( f mi ) = D( fi), and the distinguished open sets D( f m

i ) form an opencovering of Spec A. Therefore we may also write 1 =

ř

i ci f mi . Letting a =

ř

i cib1i,we find

a f mj =

ÿ

i

cib1i f mj =

ÿ

i

cib1j f mi = b1j

ÿ

i

ci f mi = b1j,

and hence a = b1j/ f mj in A f j . o

Using Proposition 1.30 on page 46 of Chapter 1, we may now make thefollowing definition:

Definition 3.5 We let OSpec A be the unique sheaf extending the B-sheaf O.


Explicitly, the sections of OSpec A over an open set U Ă X, are given by theinverse limit of localizations

OX(U) = limÐÝ

D( f )ĎU

O(D( f )) = limÐÝ

D( f )ĎU

A f . (3.5)

The distinguished open subsets of U form a directed set when ordered by inclu-sion, so the first limit is clear. For the second, which relies on the identificationin Lemma 3.2 and which is the one mostly used, one tacitly chooses one f foreach open D( f ). Thus OX(U) is an A-module, with universal restriction mapsinto each of the localizations in the inverse system

A f

¨ ¨ ¨ Ag ¨ ¨ ¨

A f 1 Ah

¨ ¨ ¨ Ag1. . .

A f 2. . .

That being said, we will basically never need to know the group OX(U) for Uother than a distinguished open set U = D( f ). All that matters is that OSpec A isthe unique sheaf that indeed satisfies the two main properties we want:

Proposition 3.6 The sheaf OSpec A on Spec A as defined above is a sheaf ofrings satisfying the two paramount properties, namely

i) Sections over distinguished opens: Γ(D( f ),OSpec A) = A f ;ii) Stalks: OSpec A,x = Apx .

In particular, Γ(Spec A,OSpec A) = A.

Proof: We defined OSpec A so that the first property would hold. The secondfollows from Example 13 on page 26. The last statement follows from i) bytaking f = 1. o

It is worthwhile to consider the special case when A has no zero-divisors. Inthat case, all the localizations A f are subrings of the fraction field K(A) of A andthe localization maps Ag Ñ A f are simply inclusions. Then the intuition fromvarieties is correct: we can think of elements in OSpec A(U) simply as elementsg/h in the fraction field of A, so that h(x) ‰ 0 for every x P U.Example 3.7 Consider X = Spec Z. Then OSpec Z(D(p)) = Z[1/p] for eachprime number p. The stalks of OSpec Z are given by OSpec Z,p = Z(p) in each of

locally ringed spaces 75

the closed points (p) and OSpec Z,(0) = Z(0) = Q at the generic point. K

Example 3.8 Let us continue Example 4 about spectra of dvr’s. In X = Spec A,the spectrum of a dvr, we have three open sets H, η, and X. The structure sheaftakes the following values at these opens:

OX(H) = 0, OX(X) = A, OX(η) = Ax = K,

where K denotes the fraction field of A. The stalks are given by OX,x = A(x) = Aand OX,η = A(0) = K. K

Example 3.9 While it is straightforward to compute OX over an open setU = D( f ), one can sometimes use the sheaf exact sequence to compute overother open sets. See Section 5.2 for an explicit computation illustrating this. K

Example 3.10 One may use the structure sheaf to give another proof that X =

Spec A being disconnected implies that A is a direct product of rings. SupposeX = U1 YU2, where U1, U2 are open and closed subsets and U1 XU2 = H. Thesheaf exact sequence takes the form

0 OX(X) = A OX(U1)ÔX(U2) OX(U1 XU2) = 0,

and we deduce that A » OX(U1)ÔX(U2). K

Exercise 3.1 (A-module structure on OSpec A(U).) Let a P A, show that there isa map of sheaves [a] : OSpec A Ñ OSpec A, inducing multiplication by a both onOSpec A(D( f )) = A f and on the stalks Ap. Hint: For each distinguished opensubset D( f ) of Spec A define [a] : OSpec A(D( f )) = A f Ñ OSpec A(D( f )) = A f

as the multiplication by a map; verify that this is a map of B-sheaves. M

3.2 Locally ringed spaces

We would like to define a scheme to be a space that locally looks like Spec A forsome rings A. To be able to make sense of such a definition, we need a suitablecategory of spaces to work with. We will rely on the two pieces of data we have;the topological space Spec A together with its sheaf of rings OSpec A.

Definition 3.11 A ringed space is a pair (X,OX) where X is a topologicalspace and OX is a sheaf of rings on X. A morphism of ringed spaces is a pair

( f , f 7) : (X,OX)Ñ (Y,OY)

where f : X Ñ Y is continuous, and

f 7 : OY Ñ f˚OX

is a map of sheaves of rings on Y.

This means that the mappings f 7(U) are all maps of rings

f 7(U) : OY(U)Ñ OX( f´1U),


and we require that they commute with the restriction maps:

OY(U) OX( f´1U)

OY(V) OY( f´1V)

ρU,V

f 7(U)

ρ f´1U, f´1V

f 7(V)

(3.6)

A map (g, g7) : (Y,OY)Ñ (Z,OZ) of ringed spaces, whose sources is the targetof ( f , f 7), may be composed with ( f , f 7). The underlying map X Ñ Z on thetopological spaces is naturally the composition g ˝ f . Since the direct imagesatisfies (g ˝ f )˚OX = g˚ f˚OX, we may define (g ˝ f )7 as g7 ˝ f 7.

If X and Y are two ringed spaces, an Isomorphisms ofringed spaces(isomorfier av ringederom)

isomorphism from X to Y is a morphismφ : X Ñ Y that has an inverse morphism; in other words, there is a morphismψ : Y Ñ X such that ψ ˝ φ = idX and φ ˝ ψ = idY. In more concrete terms,this boils down to φ being a homeomorphism such that f 7(U) : OY(U) Ñ

OX( f´1(U)) is an isomorphism for every open U Ă Y.

The intuition behind this definition again comes from varieties, where wewould like to think of f 7 as a way of “pulling back" a regular function g : U Ñ kdefined on some open subset U Ă Y to f ˝ g : f´1(U)Ñ k. For general ringedspaces however, we do not have the luxury of speaking about functions intosome fixed field k, so the ring maps f 7 have to be specified as part of the data.

In some sense, this gives us too much freedom in the above definition; thesheaf OX can be any sheaf of rings and the maps f 7 above can also be completelyarbitrary ring maps, as long as they are compatible with the restrictions. Ifwe really want the analogy with affine varieties to hold, we should add someadditional requirements.

It turns out that there is a single additional requirement that makes every-thing work. To explain it, let us again consider what happens in the case ofvarieties in more detail.

Let X and Y be affine varieties with coordinate rings A(X) and A(Y) re-spectively. Given a morphism of affine varieties f : X Ñ Y there is an inducedpullback map f ˚ : A(Y)Ñ A(X) on the coordinate rings. Let us consider whathappens to functions locally: for a point x P X the ring of regular functions at xis the local ring A(X)mx with maximal ideal mx (the functions which vanish atx), and similarly for A(Y)my , where y = f (x) P Y. Note that f ˝ g vanishes aty = f (x) if g vanishes at y. This means that the pullback

f ˚x : A(Y)my Ñ A(X)mx

maps the maximal ideal my into the maximal ideal mx. In other words f ˚x is amap of local rings. A map of local rings is

a ring homomorphismwhich maps themaximal ideal to themaximal ideal.

For a morphism of ringed spaces, we do have an induced map between thestalks; if x P X and y = f (x), the map f 7 induces

f 7y : OY,y Ñ OX,x . (3.7)


To see this, let sy P OY,y be the germ of a section (s, V) with V Ă Y. Thent = f 7(s) is a well-defined section of OX( f´1(V)), and since f´1(V) containsx, we can form its germ tx P OX,x. Morphisms of ringed

spaces induce ringmaps on stalks.

The diagram (3.6) shows that this operationcommutes with the restriction maps, so we get an induced map of rings as in(3.7). However, without any hypothesis on the ring maps f 7(U), there is noguarantee that this will behave as we want; that is, that f 7x(s) vanishes at x ifs vanishes at y. For this to be the case, the maximal ideal my in OY,y must bemapped into the maximal ideal mx in OX,x. We therefore make the followingdefinition:

Definition 3.12 A locally ringed space is a ringed space (X,OX) with theadditional requirement that for every point x P X the stalk OX,x is a local ring.

A morphism of locally ringed spaces is a morphism ( f , f 7) : (X,OX) Ñ

(Y,OY) of ringed spaces with the additional requirement that for every x P Xand y = f (x), the map on stalks

f 7x : OY,y Ñ OX,x

is a map of local rings; that is,

f 7x(my) Ď mx.

Example 3.13

K

Note that the last condition is equivalent to ( f 7x)´1(mx) = my. It is clear thatthe composition of two morphisms is a morphism; so locally ringed spaces forma category. Recall: ring maps are

required to send 1 to 1.The example we are most interested in at the moment is of course the

spectrum of a ring:


Proposition 3.14 For a ring A, the pair (X,OX) = (Spec A,OSpec A) is alocally ringed space.For a map of rings φ : A Ñ B, there is an induced map of locally ringed spaces

Spec(φ) = (h, h7) : (Spec B,OSpec B)Ñ (Spec A,OSpec A)

satisfying the two properties

i) Distinguished open sets: The map h7(D( f )), which fits into thediagram

OSpec A(D( f )) OSpec B(D(φ( f ))

A f Bφ( f )

h7(U)

is the natural localization of the map φ; that is, it is given by theassignment a fń ÞÑ φ(a)φ( f )ń.

ii) Stalks: The map induced by h7 on stalks at p P Spec B and φ´1(p) isthe localization Aφ´1(p) Ñ Bp of φ.

Earlier we defined a continuous map Spec φ : Spec A Ñ Spec B which nowwill be the topological part, named h, of the new map Spec φ, and which we shallenrich by a sheafy part h7. Note that the two last assertions of the propositionreflect the two statements in Proposition 3.6. The slogan is: on affine schemes,both OX and f 7 are just given by the localizations.

Proof: The pair (X,OX) is clearly a ringed space. Moreover, we defined thestructure sheaf on X = Spec A so that OX,x = Apx at each point for each x P X.In particular, the stalks are local rings, so we have a locally ringed space.

For the second claim, assume that φ : A Ñ B is a morphism of rings and leth : Spec B Ñ Spec A be the induced map given by h(p) = φ´1(p). We want toassociate to φ a map of sheaves of rings

h7 : OSpec A Ñ h˚OSpec B.

By Proposition 1.30, it suffices to tell what φ7 should do to the sections over thedistinguished open sets D( f ). Here we recall Lemma 2.18, which tells us that

h´1(D( f )) = D(φ( f )).

This means that we have the equality Γ(D( f ), h˚OSpec B) = Bφ( f ), and we knowthat Γ(D( f ),OSpec A) = A f . The original map of rings φ : A Ñ B now localizesto a map A f Ñ Bφ( f ), sending a fń to φ(a)φ( f )ń, and this shall be the map h7

on sections over the distinguished open set D( f ).

affine schemes 79

To prove that h7 is well defined, we need to check that it is compatible withthe restriction maps among distinguished open sets: indeed, when there is aninclusion D(g)ĎD( f ), we write as usual gm = c f , and the localization mapA f Ñ Ag will then send a fń to acngńm. One has φ(g)m = φ(c)φ( f ), whichmakes the diagram below commutative:

A f Ag

Bφ( f ) Bφ(g)

and this is exactly the required compatibility.For p P Spec B, with image q = φ´1(p) P Spec A, the map h7 induces a map

of stalksh7p : OSpec A,q Ñ OSpec B,p

which is just the localization map Aφ´1(p) Ñ Bp. Thus the preimage of themaximal ideal of Aφ´1(p) equals the maximal ideal in Bp, making h7p a map oflocal rings. Hence (h, h7) is a morphism of locally ringed spaces. o

Example 3.15 Cuspidal cubic. Let k be an algebraically closed field, A =

k[x, y]/(y2 ´ x3) and B = k[t], and let φ : A Ñ B be the ring map given byx ÞÑ t2, y ÞÑ t3. This induces a morphism of locally ringed spaces

f : A1k Ñ X = Spec A

On the level of closed points, this sends (t´ a) P Spec k[t] to (x´ a2, y´ a3) P

Spec A.If p = (t) P A1

k denotes the origin, the stalk induced map f 7p : OA1k ,p Ñ

OX, f (p), is given by map of localizations

φ(x,y) :(k[x, y]/(y2 ´ x3)

)(x,y) Ñ k[t](t) (3.8)

K

Exercise 3.2 Show that that the composition of two morphisms of locally ringedspaces is again a morphism. M

3.3 Affine schemes

Definition 3.16 An affine scheme is a locally ringed space (X,OX) which isisomorphic to (Spec A,OSpec A) for some ring A.

Affine schemes form a category AffSch, a subcategory of the category oflocally ringed spaces. This category is closely linked to the category of rings, aswe will see next.

affine schemes 80

By Proposition 3.9, the assignment A ÞÑ Spec A gives a contravariant func-tor from the category Rings of rings to the category AffSch. There is alsoa contravariant functor going the other way: taking the global sections ofthe structure sheaf OSpec A gives us the ring A back. Furthermore, a map ofaffine schemes f : Spec B Ñ Spec A, comes equipped with a map of sheavesf 7 : OSpec A Ñ f˚OSpec B. Taking global sections gives a ring map

A = Γ(Spec A,OSpec A)Ñ Γ(Spec A, φ˚OSpec B) = B.

We can define a canonical map

Γ : HomAffSch(X, Y)Ñ HomRings(OY(Y),OX(X)) (3.9)

which sends ( f , f 7) to the map f 7(Y) : OY(Y)Ñ OX(X).

Proposition 3.17 If X and Y are affine, the map Γ is bijective.

Proof: Write X = Spec B and Y = Spec A. By construction, we have A = OY(Y)and B = OX(X).

If φ : A Ñ B is a ring homomorphism, it follows from Proposition 3.9i), that Γ(Spec φ) = φ. So to establish the bijection, we just need to showthat Spec (Γ( f )) = f for a given a morphism f : X Ñ Y. We let φ = Γ( f ) =

f 7(Y) : A Ñ B. Note: Bzq is the subsetof elements in B thatbecome invertible inBq.

Let x P X be a point, corresponding to the prime ideal q Ď B, andlet p Ď A be the prime ideal corresponding to f (x) P Y. There is a commutativediagram

A B

Ap Bq

φ

f 7x

where the vertical maps are the localization maps. By the commutativity of thediagram we have φ(Azp) Ď Bzq, so φ´1(q) Ď p. Now f 7x is a local homomor-phism, so in fact φ´1(q) = p. Here we are using that

the stalk maps f 7x aremaps of local rings!

This means that Spec φ induces the same map asf on the underlying topological spaces. Moreover, we have two morphisms ofsheaves OY Ñ f˚OX, one induced by f and one induced by Spec φ. For each x,the induced stalk maps f 7x and (Spec φ)7 coincide with the map Ap Ñ Bq above,so also f 7 = (Spec φ)7 as maps of sheaves. o

We have established the following important theorem:

Theorem 3.18 The two functors Spec and Γ are mutually inverse and definean equivalence between the categories Rings and AffSch.

In summary, affine schemes X are completely characterized by their rings ofglobal sections Γ(X,OX), and morphisms between affine schemes X Ñ Y are inbijective correspondence with ring homomorphisms Γ(Y,OY)Ñ Γ(X,OX). In

schemes in general 81

particular, a map f between two affine schemes is an isomorphism if and only ifthe corresponding ring map f 7 is an isomorphism.Example 3.19 The morphism f : A1 Ñ Spec A from Example 6 is a homeomor-phism, but it is not an isomorphism.

Indeed, note that f sends (t´ a) P Spec k[t] (injectively) to (x´ a2, y´ a3) P

Spec A. By the Nullstellensatz any maximal ideal of A is of that form so it isalso surjective. Thus, since f maps the generic point (0) of A1

k to the genericpoint (0) of Spec A, we see that f is a bijection. It is finally a homeomorphismbecause it is closed; any closed subset of A1

k is a finite set of points.To see it is not an isomorphism, we simply note, that it is induced by

φ : k[t]Ñ k[x, y]/(y2 ´ x3), which is not an isomorphism (the ring on the righthand side is not even a UFD). In fact, the same argument shows that the stalkmap at the origin f 7p : OA1

k ,p Ñ OX, f (p) given by (3.8) is not even an isomorphism.This confirms our intuition that the cuspidal cubic is not even ’locally isomorphic’to A1

k near the origin.K

3.4 Schemes in general

Finally, we can give the definition of a scheme.

Definition 3.20 A scheme is a locally ringed space (X,OX) which is locallyisomorphic to an affine scheme, i.e., there is an open cover Ui of X such that each(Ui,OX|Ui) is isomorphic to some affine scheme (Spec A,OSpec A).

So as before, a scheme has two components: a topological space X, which iscovered by open sets of the form Spec A and the structure sheaf OX, which is asheaf of rings.

For a point x P X, we define The local ring at apoint (Den lokaleringen i et punkt)

the local ring at x as the stalk OX,x. Note that,the affine open subsets form a basis for the topology of X, so when computingthe direct limit that gives the stalk, we may use distinguished affine subsets. Soif x is contained in U = Spec A and corresponds to p Ă A, we have a naturalisomorphism

OX,x = limÝÑ

D( f )Qp

OX(D( f )) = limÝÑfRp

A f = Ap.

As before, we think of elements in OX,x as ‘functions defined at x’, even if thisis strictly only true for well-behaved schemes (see Proposition 3.21).

In the local ring OX,x, we also have the maximal ideal mx = pAp and thecorresponding Residue field

(Residykropp)residue field k(x) = Ap/pAp.

A Morphism of schemes(Morfi av skjemaer)

morphism, or map for short, between two schemes X and Y is simply amap f between X and Y regarded as locally ringed spaces. This also has two


components: a continuous map, which we shall denote by f as well, and a mapof sheaves of rings

f 7 : OY Ñ f˚OX

with the additional requirement that f 7 induces a map of local rings on thestalks.

In this way the schemes form a category, which we shall denote by Sch. Thiscontains the category of affine schemes, denoted by AffSch, as a full subcategory.

Relative schemesThere is also the notion of Relative schemes

(relative skjemaer)relative schemes where a base scheme S is chosen. A

scheme over S is a scheme X together with a morphism f : X Ñ S, which we callthe structure map or the structure morphism. If two schemes over S are given, sayX Ñ S and Y Ñ S, then a map between them is a map X Ñ Y compatible withthe two structure maps; that is, such that the diagram below is commutative

X Y

S

.

The schemes over S form a category Sch/S, and the set of morphisms as definedabove is denoted by HomS(X, Y). Composition of maps makes it a functor inboth X and Y, from Sch/S to Sets, contravariant in X and covariant in Y.

If the base scheme S is affine, say S = Spec A, we say that X is a scheme overA. A short hand notation for Sch/Spec A is Sch/A. To say that an affine schemeSpec B is a scheme over Spec A is just a rephrasing of B being an A-algebra;giving a structure map f : Spec B Ñ Spec A is equivalent to giving the map ofrings f 7 : A Ñ B. As there is a canonical map from any scheme X to Spec Z,every scheme is a Z-scheme. See Exercise ??. On the level of categories one may express this asSch = Sch/Z.

The concept of relative schemes can be thought of as a vast generalization of‘varieties over k’. However, extending this to more general rings or even schemesturns out to be conceptually very fruitful, e.g. when discussing properties ofmorphisms or fibre products (in Chapter 7).

Definition 3.21 Let X/S be a scheme over S with structure morphism f : X ÑS. One says that

i) X/S is of locally finite type if S has a cover consisting of open affinesubsets Vi = Spec Ai such that each f´1(Vi) can be covered by affinesubsets of the form Spec Bij, where each Bij is finitely generated as aAi-algebra;

ii) f is of finite type if, in i), one can do with a finite number of Spec Bij.


In case S = Spec A, one says that a scheme over A is of locally finite type(respectively of finite type) over A if the morphism X Ñ Spec A is locally of finitetype (respectively of finite type).

Schemes which are of finite type over a field will be our main concern in thisbook. This is the class of schemes which are closest to varieties; but we allownon-algebraically closed fields, as well as zero divisors in the structure sheaf.

There is another related, but much stronger finiteness property a morphismcan have:

Definition 3.22 Let f : X Ñ S be a scheme over S. We say that

i) f is affine if there is a covering Vi = Spec Ai of S such that eachinverse image f´1(Vi) is affine;

ii) f is finite if it is affine, and in the notation above, if f´1(Vi) =

Spec Bi, the Ai-algebra Bi is a finitely generated Ai-module.

To underline the huge difference between the the two finiteness condition,note that a scheme X which is finite over a field k in particular has a finite anddiscrete underlying topological space. X being of finite type on the other handsimply means it is covered by affine schemes of the form Spec k[x1, . . . , xr]/a.Thus, for instance, A1

k is of finite type over k, but it is not finite. An infinitedisjoint union, such as

š8i=1 Spec k is locally of finite type, but not of finite type.

The definitions above reference a single open covering of affine schemesVi = Spec Ai over which the morphism has the indicated properties. A usefulfact is that it in fact holds for every affine covering, once it holds for one. This proposition is not

particularly deep, butthe proof is rather longand tedious, so weomit it.

Proposition 3.23 For each of the properties in Definitions 3.14 and 3.15, if theindicated condition holds for one affine covering, then it holds for every affinecovering.

Open immersions and open subschemesIf X is a scheme and U Ď X is an open subset, the restriction OX|U is a sheaf onU making (U,OX|U) into a locally ringed space. This is even a scheme, sinceif X is covered by affines Vi = Spec Ai, then each U XVi is open in Vi, hencecan be covered by affine schemes. It follows that there is a canonical schemestructure on U, and we call (U,OX|U) an Open subschemes

(åpne underskjemaer)open subscheme of X and say that U

has the induced scheme structure. We say that a morphism of schemes ι : V Ñ Xis an Open immersions

(åpne immersjoner)open immersion if it is an isomorphism onto an open subscheme of X.

Example 3.24 The open set U = A1k ´V(x) is an open subscheme of the affine

line A1k = Spec k[x]. Note that there is an isomorphism U » Spec k[x, x´1] =


Spec k[x, y]/(xy´ 1), as schemes. K

Example 3.25 More generally, consider V = Spec A f and the map ι : V Ñ

Spec A = X, induced by the localization map A Ñ A f . This is an open immer-sion onto the open set U = D( f ) Ă X. Indeed, we saw in Example 15 that ι is ahomeomorphism onto U, and it follows from the definition of the sheaf OX thatthe restriction OX|U coincides with the structure sheaf on Spec A f . K

A word of warning: anopen subscheme of anaffine scheme mightnot itself be affine (wewill see an example ofthis in Chapter 5).

Closed immersions and closed subschemesIf X is a scheme, we would like to define what it means for a closed subsetZ Ă X to be a closed subscheme of X. This is a little bit more subtle than the casefor open subsets, because for a given closed subset Z Ă X, there is no canonicallocally ringed space structure on Z.

The prototypical example of a closed subscheme is Spec(A/a), which as wehave seen, embeds in a natural way as the closed subset V(a) of Spec A (Example14). Here the scheme structure is evident. Thus we have a clear intuitive pictureof what a closed subscheme should be in general: it is a scheme (Z,OZ) with amorphism ι : Z Ñ X, which looks locally like the map Spec(A/a)Ñ Spec A.

Definition 3.26 Let X and Z be schemes and ι : Z Ñ X a morphism.

i) The map ι is called a Closed immersions(lukkete immersjoner)

closed immersion if there is an affine coverUi = Spec Ai of X and ideals ai in Ai such that for each i it holdsthat ι´1(Ui) » Spec(Ai/ai) as schemes over Ui.

ii) A Closed subschemes(lukkedeunderskjemaer)

closed subscheme is a scheme (Z,OZ) together with a closed im-mersion ι of (Z,OZ) in (X,OX), with ι : Z Ñ X being the inclusionof a closed subset, and ι7 inducing an isomorphism ι˚OZ » OX/I

for some sheaf of ideals I ĎOX.

In clear text, the isomorphism ι´1Ui » Spec(Ai/ai) being an isomorphismover Ui, means that the diagram

ι´1Ui Ui = Spec Ai

Spec(Ai/ai)

ι

»

commutes, where the skew map is the canonical closed immersion.Closed subschemes and closed immersions are (almost) equivalent pieces

of information. For each closed subscheme Z Ď X, we get a closed immersionι : Z Ñ X given by the inclusion map. Conversely, a closed immersion ι : Z Ñ Xinduces an isomorphism of Z with its target, which is a closed subscheme ofX, but ι is only determined up to automorphisms of Z, hence the reservation‘almost’.


The ideal sheaf I fits into an exact sequence

0 Ñ I Ñ OX Ñ i˚OZ Ñ 0. (3.10)

In fact, this ideal sheaf alone determines the subscheme because of the equalityZ = (Supp(OX/I ),OX/I ). Recall: Supp(F ) is

the set of points xwhere Fx ‰ 0

Unfortunately, it is not true that all ideal sheavesarise from closed subschemes (see Exercise 3.6). The precise relationship betweenclosed subschemes and ideal sheaves will be made clear in Chapter 10: the idealsheaf I will define a closed subscheme if and only if it is ‘quasi-coherent’.

In fact, classifying closed subschemes according to the above definition is notso easy even for affine schemes. Of course, for each ideal a Ă A, we get a closedimmersion Spec(A/a) Ñ Spec A and therefore a desired closed subscheme ofSpec A with underlying topological space V(a). However, because the definitionreferences a certain affine covering, it is a priori not obvious that all closedsubschemes arise from an ideal a Ă A in this way, or even that every closedsubscheme of Spec A is an affine scheme. This is nevertheless true, but we willhave to postpone the proof until Chapter 10.

Proposition 3.27 Let X = Spec A be an affine scheme. The map a ÞÑ

Spec(A/a) induces a one-to-one correspondence between the set of ideals ofA and the set of closed subschemes of X. In particular, any closed subscheme ofan affine scheme is also affine.

Example 3.28 The schemes Spec k[x]/(x), Spec k[x]/(x2), Spec k[x]/(x3), . . .give different subschemes of A1

k , where k is a field. Still, the underlying topo-logical space consists of the origin only. Thus these spectra are homeomorphic,but not isomorphic as schemes, because they have non-isomorphic structuresheaves. K

Example 3.29 Consider the affine 4-space A4k = Spec A, with A = k[x, y, z, w].

Then the three ideals

I1 = (x, y), I2 = (x2, y) and I3 = (x2, xy, y2, xw´ yz),

have the same radical, and thus give rise to the same closed subset V(x, y) Ă A4k ,

but they give different closed subschemes of A4k . K

Exercise 3.3 Show that being a closed immersion is a property which is localon the target. In clear text: Assume given a morphism f : X Ñ Y and an opencover tUiu of Y. Let Vi = f´1(Ui) and assume that each restriction f |Vi : Vi Ñ Ui

is a closed immersion. Prove that then also f is a closed immersion. M

Exercise 3.4 Let f : X Ñ Y and g : Y Ñ Z be two morphisms of schemes. Provethat if both f and g are closed immersions then g ˝ f is one as well. M

Exercise 3.5 Show that a morphism ι : Z Ñ X is a closed immerion if and onlyif it is affine, and the sheaf map i7 : OX Ñ ι˚OZ is surjective. M

Exercise 3.6 Let X = A1k = Spec k[x] and let Z Ă X be a subset.

properties of the scheme structure 86

i) Show that the presheaf I defined by

I (U) =

#

OX(U) if ZXU ‰ H

0 otherwise

is an ideal sheaf and that Supp(OX/I ) = Z.

ii) Conclude that not all ideal sheaves give rise to closed subschemes.

M

3.5 Properties of the scheme structure

In the previous section, we noted that there might be several scheme structureson the same topological space. In this section, we discuss this phenomenon alittle bit further.

Recall that a ring A is said to be reduced if it has no nilpotent elements. Ascheme (X,OX) is said to be Reduced scheme

(redusert skjema)reduced if for every x P X, the local ring OX,x is

reduced. This condition holds if and only if for every open U Ď X, the ringOX(U) has no nilpotents: any non-zero nilpotent element in a ring OX(U)

would have a non-zero germ in at least one local ring OX,x, which would thennot be reduced. For the reverse implication, note that any non-zero nilpotentsx element in OX,x is induced from some section s of OX(V) over some openneighbourhood V of x. Since the support of powers of s are closed, s will benilpotent in OX,x(W) for some neighbourhood W of x (possibly smaller than V).

If X = Spec A is an affine scheme, the scheme Xred = Spec(A/?

0) is,by definition, a reduced scheme. Moreover, there is a canonical morphismXred Ñ X, a closed immersion, induced by the quotient map A Ñ A/

?0. This

map has the following universal property: Any morphism Y Ñ X, where Y isreduced, factors through a map Y Ñ Xred. This follows from

Theorem 3.12 for Yaffine, and Theorem4.6 in general.

In fact, one can for any scheme X, define its associated reduced scheme Xred,together with a closed immersion Xred Ñ X satisfying the above universalproperty. Xred has the same underlying topological space as X, but the structuresheaf has been modified to kill all nilpotent elements; it is obtained by "gluingtogether" each Spec A/

?0 for each affine subset Spec A. We will postpone the

details of this construction until Chapter 10.Example 3.30 For A = k[x]/(xn) and X = Spec A, we have Xred = Spec A/

a

(0) =Spec k. K

Example 3.31 For X = Spec A, with A = k[x, y, z, w]/I where

I = (x2, xy, y2, xw´ yz),

we have Xred = Spec k[x, y, z, w]/(x, y) » A2. K


Integral schemes and function fieldsWe say that a scheme is Integral scheme (Helt

skjema)integral if it is both irreducible and reduced. For an affine

scheme, X = Spec A is integral if and only if A is an integral domain. Indeed,X is reduced if and only if A has no nilpotents – that is, the nilradical vanishes– and X is irreducible if and only if the nilradical

?0 is prime. These two

statements imply that the zero-ideal is prime, and so A is an integral domain.Moreover, it is not so hard to prove the following:

Proposition 3.32 A scheme X is integral if and only if OX(U) is an integraldomain for each open U Ď X.

Any irreducible subset of a scheme has a unique generic point. In particular,an integral scheme X has a unique generic point η. It is the only point which isdense in X and it belongs to every open non-empty subset of X. In particular, ifU = Spec A Ď X is an open affine, η P U and corresponds to the zero ideal in A(which is prime). The local ring OX,η is thus equal to the field of fractions of A.We define the Function field

(Funksjonskroppen)function field k(X) of X to be the stalk OX,η at the generic point.

Example 3.33 The function field of Spec Z is OSpec Z,(0) = Z(0) = Q. K

Example 3.34 The function field of Ank = Spec k[x1, . . . , xn] is k(x1, . . . , xn). K

Example 3.35 The quadratic cone. The quadric cone Q = Spec k[x, y, z]/(x2 ´ yz)is integral, as it is the spectrum of an integral domain. The function field of Q isgiven by K(k[x, y, z]/(x2 ´ yz)) » k(x, y). Since y is invertible,

we can use z = y´1x2

to eliminate z.

K

For an integral scheme X, it is sometimes fruitful to consider sections ofOX as elements in k(X); one may heuristically think about them as ‘rationalfunctions’ on X, thus pushing the analogy with function just before Definition 2.2in Section 2.1 further. This is legitimate in view of the following. For any non-empty open U the generic point η belongs to U, and there is a canonical ‘stalkmap’ Γ(U,OX)Ñ OX,η = k(X) which is easily seen to be injective. These mapsare compatible with restrictions, i.e. all diagrams

Γ(U,OX) k(X)

Γ(V,OX)

ρUV

where VĎU are two open subsets, commute. Identifying the Γ(U,OX) withtheir images in k(X), the restriction maps just become inclusions. The samereasoning applies to the other stalks OX,x: they all lie in k(X). Moreover, it holdstrue that x P U if and only if Γ(U,OX)ĎOX,x. We say that an element f P k(X)

is defined in the point x if f P OX,x.

Lemma 3.36 Let X be an integral scheme and let f P k(X). The set U f = t x P X |

f P OX,x u where f is defined, is open.


Proof: Let x P U f and let Spec A be an affine neighbourhood of x. Considerthe ideal a f = t b P A | b f P A u. If p is a prime in A, then f P Ap if and only ifa f Ę p; that is, V(a f ) is the complement of U f X Spec A. o

Proposition 3.37 Let X be an integral scheme with function field K. Then

OX(U) =č

xPU

OX,x =

#

f P K

ˇ

ˇ

ˇ

ˇ

ˇ

for each point x P U, f can berepresented as g/h where h(x) ‰ 0

+

Ă K.

Proof: There are two equalities to prove here. To prove the first, assume firstthat U is affine, say U = Spec A. Then A =

Ş

Ap where the intersection extendsover all prime ideals in A; indeed, if the ideal a f is proper, it will be containedin a maximal ideal m, hence f R Am. If U is general, the statement follows sinceΓ(U,OX) =

Ş

Γ(V,OX), where the intersection extends over all non-emptyopen affine subsets VĎU. Indeed, in general Γ(U,OX) equals the inverse limitΓ(U,OX) = lim

ÐÝΓ(V,OX), and this inverse limit becomes the intersection when

all rings are identified with subrings of K.To prove the second, let x P X be a point, and let Spec A Q x be an affine

open containing x. Then an element f P K = K(A) lies in OX,x = Ap Ă K if andonly if it can be expressed as f = a/s where s R p. o

Example 3.38 Non-reduced schemes appear frequently when two schemes Xand Y intersect. For instance, consider the parabola X = Spec k[x, y]/(y´ x2)

and the line Y = Spec k[x, y]/(y). The intersection of these is given by theideal I = (y´ x2, y) = (x2, y), which is a non-radical ideal. Thus the nilpotentelements of k[x, y]/(x2, y) in some sense accounts for the "tangency" of theintersection XXY. K

Example 3.39 Here is a similar example in A3k . Consider

X = Spec k[x, y, z]/(x´ y2z)

which defines a closed subscheme of A3 (a cubic surface). The intersection of Xwith the plane given by x = 0 is given by the ideal I = (x, y2z) = (x, y2)X (x, z).Thus the intersection Z = Spec k[x, y, z]/I is neither irreducible nor reduced:It has two irreducible components corresponding to the lines x = y = 0 andx = z = 0; the former has "multiplicity 2", since the plane is tangent to X alongthat line. K

Example 3.40 Schemes of matrices. Let k be an algebraically closed field andconsider the k-scheme

Mn = An2

k = Spec k[xij]1ďi,jďn.

As the notation suggests, this is a scheme whose closed points parameterizesnˆ n-matrices with entries in k. Mn contains several interesting subschemes:


The general linear group, GLn(k) Ă Mn defined as the open subscheme ofinvertible matrices; this is the distinguished open set D(det M), where det Mis the determinant of the matrix of variables M = (xij). GLn(k) is an exampleof a group scheme, i.e., it is a scheme G equipped with scheme morphismsm : GˆG Ñ G and i : G Ñ G satisfying the usual axioms of being a group. Assuch, GLn(k) also contains lots of other closed subschemes which are also groupschemes: the special linear group SLn(k), which is defined by V(det M´ 1) Ă Mn;and the orthogonal group O(n) (defined in Mn by the ideal generated by therelations Mt M = I, which are polynomial in the xij); and the special orthogonalgroup SO(n), which is defined by Mt M = I and det M = 1. K

Example 3.41 Nilpotent matrices. A particularly interesting example is given bythe set of nilpotent matrices, i.e., matrices A such that Ak = 0 for some k ą 0.We can put a scheme structure on this set by noting that an nˆ n-matrix A isnilpotent if and only if An = 0. The equation Mn = 0 gives n2 degree n relationsin the variables xij, and the ideal J they generate define a closed subschemeN = Spec(k[xij]/J) of Mn. Interestingly, the subscheme N is typically non-reduced. Indeed, note that the trace of a nilpotent matrix is always zero, so theequation TrM = 0 gives a linear polynomial in the xij which vanishes on all theclosed points of N. So if k is algebraically closed, the Nullstellensatz impliesthat Tr M lies in

?J, and so J is not radical.

We can put a different scheme structure on the set of nilpotent matrices, usingthe fact that a matrix A is nilpotent if and only if it has characteristic polynomialequal to xn. Note that the coefficients of the characteristic polynomial

det(tI ´ A) = xn ´ c1(A)xn´1 + ¨ ¨ ¨+ (´1)ncn(A)

are polynomials in the entries of A, so we see that we get n equations c1(M) =

¨ ¨ ¨ = cn(M) = 0, that define a subscheme in Mn with the same underlyingtopological space as N. In fact, it is not too hard to check that the ideal Igenerated by the ci(M) is radical, so that Nred = Spec

(k[xij]/I

). K

Exercises(3.7) Give an example of a scheme X, a field K, and a morphism of ringedspaces Spec K Ñ X which is not a morphism of schemes.(3.8) Show that any irreducible subset of a scheme has a unique generic point.(3.9) Which of the topologies on a set with three points is the underlyingˇ

topology of a scheme?(3.10) Let X be a scheme.ˇ

a) Show that any irreducible and closed subset ZĎX has a unique genericpoint. Such topological spaces are called sober. Hint: Reduce to the affinecase.


b) Show that in general schemes are not Hausdorff (they are not T1 in thetopologists’s jargon). What are the possible underlying topologies of affineschemes that are Hausdorff?

c) Show that X satisfies the zeroth separation axiom (they are T0); that is, giventwo points x and y in X, there is an open subset of X containing one of thembut not the other.

d) Show that every quasi-compact and sober topological space has a closedpoint. (However, there are schemes that are not quasi-compact withoutclosed points, see Proposition 24.5 on page 412).

(3.11) The Frobenius morphism. Let p be a prime number and let A be a ringof characteristic p. The ring map FA : A Ñ A given by a ÞÑ ap is called theFrobenius map on A.

a) Show that FA induces the identity map on Spec A;

b) Show that if A is local, then FA is a local homomorphism;

c) For a scheme X over Fp, define the Frobenius morphism F : X Ñ X by theidentity on the underlying topological space and with F7 : OX Ñ OX givenby g ÞÑ gp. Show that FX is a morphism of schemes;

d) Show that FX is natural in the sense that if f : X Ñ Y is a morphism ofschemes over Fp, we have f ˝ FX = FY ˝ f .

In particular, this exercise shows that for a morphism of schemes f : X Ñ Y,in order to check that f is an isomorphism, is not enough to check that f is ahomeomorphism on the level of topological spaces; also the map f 7 must be anisomorphism.(3.12) Show that for an affine scheme X = Spec A, the nilradical presheaf J (U) =?

0 Ă OX(U) is a sheaf of ideals. Show that (X,OX/J ) defines a schemeisomorphic to Xred. (There is a subtle point here: Remember that there is asheafification involved in forming quotients).(3.13) Let A be a ring and X an integral scheme over A. Let f P k(X). Showthat there is a morphism φ : U f Ñ A1

A such that φ7 : A[t]Ñ Γ(U f ,OX) is givenby t ÞÑ f .(3.14) Prove Proposition 3.19. That is, prove that a scheme X is integral if andonly if OX(U) is an integral domain for each open U Ď X.(3.15) For each of the following rings A, decide whether the correspondingmorphism Spec A Ñ Spec Z is finite or finite type:

a) Z[i]b) Z[ 1

p ]

c) Z(p)

d) ZˆZ

e) Z[x]

(3.16) Let X be a scheme and let x P X be a point. Show that x is a closed point


if and only if the corresponding morphism Spec k(x)Ñ X is finite.M

Chapter 4

Gluing and first results on schemes

It is sometimes said that ‘algebraic geometry is the study of the geometry ofzero sets of polynomials’. After Grothendieck, perhaps a more precise slogan isthat ‘algebraic geometry is the geometry of rings’.

While this is true, the theory of schemes is much richer than just the spectraof rings. This is essentially due to the enormous flexibility we have in gluing: weare allowed to glue together new schemes out of old ones, as well as sheaves onthem, and also morphisms between these. The aim of this chapter is to explainthe conditions under which this can be done. We begin with gluing togethersheaves (which is the easiest case and which works over any topological space),and then move on to schemes and morphisms. In the final part of the chapterwe outline some applications of these constructions to the study of schemes.

4.1 Gluing maps of sheaves

This is the easiest gluing situation we encounter in this chapter. The setting is asfollows. We are given two sheaves F and G on the topological space X and anopen covering tUiuiPI of X. On each open set Ui we are given a map of sheavesφi : F |Ui Ñ G|Ui , and we assume that the following gluing condition holds onthe overlaps:

o φi|Uij = φj|Uij

for all i, j P I where Uij denotes the intersection Uij = Ui XUj. Then we have

Proposition 4.1 (Gluing morphisms of sheaves) Under the assumptionsabove, there exists a unique map of sheaves φ : F Ñ G such that φ|Ui = φi.

Proof: Take a section s P F (V) where VĎX is open, and let Vi = Ui X V.Then φi(s|Vi) is a well defined element in G(Vi), and it holds true that φi(s|Vij) =

φj(s|Vij) by the gluing condition. Hence the sections φi(s|Vi)’s of the G|Vi ’s glue

gluing sheaves 93

together to a section of G over V, which we define to be φ(s). This gives thedesired map of sheaves.

The uniqueness also follows: if φ and ψ are two morphisms of sheaves sothat φ(s)|Ui = ψ(s)|Ui for all i P I then φ(s) = ψ(s), by the Locality axiom for G,and hence φ = ψ. o

4.2 Gluing sheaves

The setting in this section is a topological space X and an open covering tUiuiPI

of X with a sheaf Fi on each open subset Ui. We want to “glue” the Fi

together; that is, we search for a global sheaf F restricting to Fi on each Ui. As usual, the sheavescan take values in anycategory, but the mainsituation we have inmind is when thesheaves are sheaves ofabelian groups.

Theintersections Ui XUj are denoted by Uij, and triple intersections Ui XUj XUk

are written as Uijk.The gluing data consists of isomorphisms τji : Fi|Uij Ñ Fj|Uij . The idea is to

identify sections of Fi|Uij with Fj|Uij using the isomorphisms τij. For the gluingprocess to be possible, the τij’s must satisfy the three conditions

i) τii = idFi

ii) τji = τ´1ij

iii) τki = τkj ˝ τji

where the last identity takes place where it makes sense: on the triple intersectionUijk. Observe that the three conditions parallel the three requirements for arelation being an equivalence relation; the first reflects reflectivity, the secondsymmetry and the third transitivity.

The third requirement is obviously necessary in order to glue togethersections: a section si of Fi|Uijk will be identified with its image sj = τji(si) inFj|Uijk , and in its turn, sj is going to be equal to sk = τkj(sj). Then, of course, si

and sk are also identified, which means that τki = τkj ˝ τji.

Proposition 4.2 (Gluing sheaves) In the setting as above there exists a sheafF on X, unique up to isomorphism, such that there are isomorphisms νi : F |Ui Ñ

Fi satisfying νj = τji ˝ νi over the intersections Uij.

Proof: If VĎX is an open set, we write Vi = Ui XV and Vij = Uij XV. We aregoing to define the sections of F over V, and they are of course obtained by glu-ing togehter sections of the Fi’s over Vi’s along the Vij’s using the isomorphismsτji. We define

F (V) = t (si)iPI | τji(si|Vij) = sj|Vij uĎź

iPI

Fi(Vi). (4.1)

The τji’s are maps of sheaves and therefore are compatible with all restric-tion maps, so if WĎV is another open set, we have τji(si|Wij) = sj|Wij if

gluing sheaves 94

τji(si|Vij) = sj|Vij . Because of this, the defining condition (4.1) is compatiblewith componentwise restrictions, and they can therefore be used as the restric-tion maps in F . We have thus defined a presheaf on X.

The first step in what remains of the proof, is to establish the isomorphismsνi : F |Ui Ñ Fi. To avoid getting confused by the names of the indices, we shallwork with a fixed index α P I. Suppose VĎUα is an open set. Then naturallyone has V = Vα, and projecting from the product

ś

i Fi(Vi) onto the componentFα(V) = Fα(Vα) gives us1 a map να : F |Vα Ñ Fα. This map sends the section(si)iPI to sα. The situation is summarized in the following commutative diagram

F (V)ś

iPI Fi(Vi)

Fα(V).να

pα

We proceed to show that the να’s give the desired isomorphisms.To begin with, we note that on the intersections Vαβ the requirement in

the proposition, that νβ = τβα ˝ να, is fulfilled. This follows directly from thedefinition in (4.1) that sβ|Vαβ

= τβα(sα|Vαβ).

The map να is surjective: take a section σ P Fα(V) over some VĎUα anddefine s = (τiα(σ|Viα))iPI . Then s satisfies the condition in (4.1) and is an honestelement of F (V). Indeed, by the third gluing condition we obtain

τji(τiα(σ|Vjiα)) = τjα(σ|Vjiα)

for every i, j P I, and that is just the condition in (4.1). As ταα(σ|Vαα) = σ by thefirst gluing request, the element s projects to the section σ of Fα.

The map να is injective: this is clear, since if sα = 0 it follows that si|Viα =

τiα(sα) = 0 for all i P I. Now Fα is a sheaf and the Viα constitute an opencovering of Vα, so we may conclude that s = 0 by the Locality axiom for sheaves.

The final step is to show that F is a sheaf, and we start with the Gluingaxiom: so suppose that tVαu is an open covering of VĎX and that sα P F (Vα)

is a bunch of sections matching on the intersections Vαβ. Since F |UiXV is asheaf (we just checked that F |Ui is isomorphic to Fi) the sections sα|VαXUi patchtogether to give sections si in F (Ui XV) matching on the overlaps Uij XV. Thislast condition means that τij(si) = sj. By definition, (si)iPI defines a section inF (V) restricting to si, and we are done.

The Locality axiom is easier: given a section s = (si)iPI in F (V) and acovering W = tWjujPJ of V such that s|Wj = 0 for all j P J, then also s|WjXVi = 0,and since tWj X ViujPJ forms a covering of Vi, we must have s|Vi = 0 as well,

1Since restrictions operate componentwise, it is straightforward to verify that this map is compatiblewith restrictions.

gluing schemes 95

since F |Vi = Fi is a sheaf. But then from the description (4.1), we thus see thats = 0. o

Exercise 4.1 Show the uniqueness statement in the proposition. M

Exercise 4.2 Let tUiuiPI be an open cover of X. Let B be the collection of opensets V so that V Ă Ui for some i. Show that B is a basis for the topology, anduse this to give another proof of Proposition 4.2 on page 93. M

4.3 Gluing schemes

The possibility of gluing different schemes together along open subschemes isa fundamental property of schemes. It gives a plethora of new examples; themost prominent ones being the projective spaces. The gluing process is also animportant part in many general existence proofs, like in the construction of thefibre product, which as we are going to show, exists without any restrictions inthe category of schemes. This is in stark contrast with the theory of varieties,where two varieties that are glued together, easily can fail to be a variety.

In the present setting of ‘scheme gluing’ we are given a family tXiuiPI ofschemes indexed by a set I. In each of the schemes Xi we are given a collectionof open subschemes Xij, where the indices i and j run through I. Their role is toform the glue lines in the process, i.e. the contacting surfaces that are to be gluedtogether: in the glued scheme they will be identified and will be equal to theintersections of Xi and Xj. The identifications of the different pairs of the Xij’sare encoded by a family of scheme isomorphisms τji : Xij Ñ Xji. Furthermore,we let Xijk = XikXXij (these are the various triple intersections before the gluinghas been done), and we have to assume that τji(Xijk) = Xjik. Notice that Xijk isan open subscheme of Xi.

The three following gluing conditions, very much alike the ones we saw forsheaves, must be satisfied for the gluing to be possible:

i) τii = idXi ;

ii) τij = τ´1ji ;

gluing schemes 96

iii) The isomorphism τij takes Xijk into Xjik and one has τki = τkj ˝ τji overXijk.

Proposition 4.3 (Gluing schemes) Given gluing data Xi, τij as above, thereexists a scheme X with open immersions ψi : Xi Ñ X such that ψi|Xij = ψj|Xji ˝

τji, and such that the images ψi(Xi) form an open covering of X. Furthermore,one has ψi(Xij) = ψi(Xi)X ψj(Xj). The scheme X is uniquely characterized bythese properties up to a unique isomorphism.

Proof: To build the scheme X, we first build the underlying topological spaceX and subsequently equip it with a sheaf of rings. For the latter, we rely onthe gluing technique for sheaves presented in Proposition 4.2. And finally, weneed to show that X is locally affine; this follows however immediately once theimmersions ψi are in place – the Xi’s are schemes and therefore locally affine.

On the level of topological spaces, we start out with the disjoint unionš

i Xi

and proceed by introducing an equivalence relation on it. We declare two pointsx P Xij and x1 P Xji to be equivalent when x1 = τji(x). Observe that if the point xdoes not lie in any Xij with i ‰ j, we leave it alone, and it will not be equivalentto any other point.

The three gluing conditions imply readily that we obtain an equivalencerelation in this way. The first requirement entails that the relation is reflexive, thesecond that it is symmetric, and the third ensures it is transitive. The topologicalspace X is then defined to be the quotient of

š

i Xi by this relation equippedwith the quotient topology: if π :

š

i Xi Ñ X denotes the quotient map, a subsetU of X is open if and only if π´1(U) is open.

Topologically, the maps ψi : Xi Ñ X are just the maps induced by the openinclusions of the Xi’s in the disjoint union

š

i Xi. They are clearly injectivesince a point x P Xi is never equivalent to another point in Xi. Now, X has thequotient topology so a subset U of X is open if and only if π´1(U) is open, andthis holds if and only if ψ´1

i (U) = Xi X π´1(U) is open for all i. In view of theformula

π´1(ψi(U)) =ď

j

τji(U X Xij)

we conclude that each ψi is an open map, hence a homeomorphism onto itsimage.

To simplify notation, we now write Xi for ψi(Xi), which is in accordancewith our intuitive picture of X as being the union of the Xi’s with points inthe Xij’s identified according to the τij’s. Then Xij becomes Xi X Xj and Xijk

becomes the triple intersection Xi X Xj X Xk.On Xij, we have the isomorphisms τ7ji : OXj |Xij Ñ OXi |Xij ; the sheaf maps of

the scheme isomorphisms τji : Xij Ñ Xji. In view of the third gluing conditionτki = τkj ˝ τji, valid on Xijk, we obviously have τ7ki = τ7ji ˝ τ7kj. The two first gluing

gluing morphisms of schemes 97

conditions translate into τ7ii = id and τ7ji = (τ7ij)´1. The conclusion is that the

gluing properties needed to apply Proposition 4.2 are satisfied, and we areallowed to glue the different OXi ’s together and thus to equip X with a sheaf ofrings. This sheaf of rings restricts to OXi on each of the open subsets Xi, andtherefore its stalks are local rings. So (X,OX) is a locally ringed space that islocally affine, hence a scheme.

We leave it to the reader to prove the uniqueness statement in the proposition.o

Exercise 4.3 Prove the uniqueness part in the above proposition. M

Global sections of glued schemesThe standard exact sequence for computing global sections from an open cover-ing is a valuable tool in the setting of glued schemes. If X is obtained by gluingthe open subschemes Xi along isomorphisms τji : Xij Ñ Xji, it reads:

0 Γ(X,OX)À

i Γ(Xi,OXi)À

i,j Γ(Xij,OXij)α ρ

(4.2)

where ρ((si)iPI

)=(si|Xij ´ τ7ij(sj|Xji)

)iPI and α(s) = (ψ7i (s))i,jPI .

4.4 Gluing morphisms of schemes

Suppose we are given schemes X and Y and an open covering tUiuiPI of X.Assume further that there is given a family of morphisms φi : Ui Ñ Y whichmatch on the intersections Uij = Ui XUj. The aim of this paragraph is to showthat the φi’s can be glued together to give a morphism X Ñ Y:

Proposition 4.4 (Gluing morphisms of schemes) Assume given gluingdata φi as above, there exists a unique map of schemes φ : X Ñ Y such thatφ|Ui = φi.

Proof: Clearly the map on topological spaces is well defined and continuous,so if UĎY is an open set, we have to define φ7 : Γ(U,OY) Ñ Γ(U, φ˚OX) =

Γ(φ´1U,OX). So take any section s P OY(U) over U. This gives sectionsti = φ7i (s) of OX(Ui). But since φ7i and φ7j restrict to the same map on Uij, wehave ti|Uij = tj|Uij . The ti therefore patch together to a section t P OX(φ

´1U),which is the section we are aiming at: we may define φ7(s) to be t. Proving theuniqueness statement is again left to the student. o

Exercise 4.4 Let X and Y be schemes and let B be a base for the topology onX. Suppose that there is a collection of morphisms φU : U Ñ Y, one for eachU P B, such that if V P B satisfies V Ă U, we have

φU|V = φV .

universal properties of maps into affine schemes 98

Show that there exists a unique morphism of schemes φ : X Ñ Y such thatφ|U = φU . M

4.5 Universal properties of maps into affine schemes

For a general scheme X, one may very well consider the associated affinescheme Spec Γ(X,OX). This is however in general very different from X (forinstance, for the projective line P1

k introduced in the next chapter, Spec Γ(X,OX)

will be just a point). There is however still a canonically defined morphismX Ñ Spec Γ(X,OX), which satisfies the following universal property:

Proposition 4.5 Let X be any scheme. Then there is a canonical map ofschemes ψ : X Ñ Spec Γ(X,OX) inducing the identity on global sections of thestructure sheaves.

There is in fact an even stronger relationship between maps of affine schemesand ring homomorphisms:

Theorem 4.6 (Maps into affine schemes) For any scheme X, the canonicalmap

ΦX : HomSch(X, Spec A)Ñ HomRings(A, Γ(X,OX))

given by ( f , f 7) ÞÑ f #(X) is bijective.

Proof: Let tUiu be an affine covering of X. By the affine schemes case (Theorem3.12), we know that each ΦUi is bijective. This also gives that ΦX is injective: Ifwe are given two morphisms φ, ψ : X Ñ Spec A, that map to the same ring mapβ : A Ñ OX(X), we get morphisms φi : Ui Ñ Spec A and ψi : Ui Ñ Spec A. Foreach i these both correspond to the ring map given by composing β with therestriction βi : A Ñ OX(X) Ñ OX(Ui), thus φi = ψi, by the bijectivity of ΦUi .Then φ = ψ by the uniqueness part of Proposition 4.4, so ΦX is injective.

To show that ΦX is surjective let β : A Ñ Γ(X,OX) be a ring homomorphism.The maps induced by restriction, βi : A Ñ Γ(X,OX) Ñ Γ(Ui,OX), and inducealso morphisms of schemes fi : Ui Ñ Spec A. We claim that the fi’s may beglued together to a map f : X Ñ Spec A. This is a consequence of the followingdiagram being commutative where VĎUi XUj is an open affine:

Γ(Ui,OUi)

A Γ(X,OX) Γ(Ui XUj,OX) Γ(V,OX).

Γ(Uj,OUj)

βi

β j

β


Indeed, note that for V Ď UiXUj affine, the diagram implies that the restrictionsfi|V and f j|V induce the same element in HomRings(A, Γ(V,OX)), and so theyare equal on V (according to Theorem 3.12). Since this is true for any V, thefi are equal on all of Ui XUj. So by gluing the fi’s, we obtain a morphismf : X Ñ Spec A. It must hold that ΦX( f ) = β, since ΦX is injective (which wejust proved) and since f |Ui maps to βi via ΦUi for each i. This completes theproof. o

Proposition 4.5 above follows immediately by applying the theorem toA = Γ(X,OX).

Corollary 4.7 The canonical map ψ : X Ñ Spec Γ(X,OX) is universal among themaps from X to affine schemes; i.e. any map φ : X Ñ Spec A factors as φ = η ˝ ψ for aunique map η : Spec Γ(X,OX)Ñ Spec A.

Proof: In the theorem above, ψ corresponds to the identity map idΓ(X,OX)

on the right hand side. The morphism η is the map of Spec’s induced bythe ring map φ7 : A Ñ Γ(X,OX). We check that it factors φ: the morphism(η ˝ψ) : X Ñ Spec A satisfies (η ˝ψ)# = ψ# ˝ η# = φ# and hence it coincides withφ by the above theorem. o

As a special case, we note that there is a bijection

HomSch(X, Spec Z) » HomRings(Z, Γ(X,OX)).

Since ring maps always preserve the unit element, the set on the right is clearlya one-point set. So there exist precisely one morphism of schemes X Ñ Spec Z.In categorical terms this means that Spec Z is a final object in the category ofschemes Sch.

The category Sch also has an initial object, the empty scheme; it equalsthe spectrum of the zero ring, Spec 0, which has the empty set as underlyingtopological space. Given any scheme X there is clearly a unique morphismSpec 0 Ñ X, which on the level of sheaves sends every section of OX to zero.

R-valued pointsFor a scheme X, it makes sense to study morphisms Spec R Ñ X from affineschemes into it. We call such morphisms R-valued points, and the set of all suchwill be denoted by X(R); that is, X(R) = HomSch(Spec R, X). The jargon hereis justified from the following:Example 4.8 Recall the ‘absolute’ affine space An = Spec Z[x1, . . . , xn]. AnR-valued point of An is a morphism g : Spec R Ñ Spec Z[x1, . . . , xn], whichdetermines and is determined by the n-tuple (ai) = (g˚(xi)) of elements in R.Hence,

An(R) = Rn.


Now, let X = Spec Z[x1, . . . , xn]/a where a = ( f1, . . . , fr) is an ideal. The set ofR-points of X can be found similarly: any morphism

g : Spec R Ñ Spec Z[x1, . . . , xn]/a

is determined by the n-tuple (ai) = (g˚(xi)), and the n-tuples that occur areexactly those such that f ÞÑ f (a1, . . . , an) defines a homomorphism

Z[x1, . . . , xn]/aÑ R.

In other words, the n-tuples (ai) in Rn which are solutions of the equationsf1 = ¨ ¨ ¨ = fr = 0. K

Example 4.9 More generally, one may replace Z in the example above by anyring A and R by any A-algebra: if a = ( f1, . . . , fr) is an ideal in A[x1, . . . , xn] andX = Spec A[x1, . . . , xn]/a the set HomSch/A(Spec R, X) consists of tuples (ai) inRn that are solutions of the equations f1 = ¨ ¨ ¨ = fr = 0. K

Example 4.10 A conic with no real points. Let X = Spec A, where A is the realalgebra A = R[x, y]/(x2 + y2 + 1). Note that the conic x2 + y2 + 1 = 0 has noreal points, so X(R) = H. However, A has infinitely many maximal ideals, sothat X as well as X(C) are infinite. K

Example 4.11 Pythagorean triples. Consider X = Spec Z[x, y]/(x2 + y2 ´ 1). ThenX(R) is the unit circle in R2 while X(Q) consists of pairs (m/r, n/r) in Q2 cor-responding to the ‘Pythagorean triples’ of integers; i.e. triples m, n and r so thatm2 + n2 = r2. K

The sets X(R) of points over R are obviously important in number theory, asthey naturally generalize the solution set of the polynomials f1 = ¨ ¨ ¨ = fr = 0.Of course, even when R is a field, it can be very difficult to describe the setX(K) of K-valued points Spec K Ñ X, or even determining whether X(K) ‰ H(a most spectacular example is Spec Z[x, y]/(xn + yn ´ 1), n ě 3 for K = Q).

When K is a field, the underlying topological Spec K is very simple; it is just asingleton since the only prime ideal in K is the zero ideal. However, the structuresheaf OSpec K on Spec K carries more information: morphisms Spec L Ñ Spec K(i.e. the elements of X(L)) correspond exactly to field extensions L Ě K. Inparticular, Spec K and Spec L are isomorphic if and only if K » L.

The residue fields play an important role here. The following result saysthat they satisfy a universal property with respect to morphisms from spectra offields to X.

Proposition 4.12 Let X be a scheme and let K be a field. Then to give amorphism of schemes Spec K Ñ X is equivalent to giving a point x P X plus anembedding k(x)Ñ K.

More generally, one may for a fixed scheme S define X(S) to be the set of allmorphisms S Ñ X; the so-called S-valued points of X. In the example above, we

brave new varieties 101

have for any scheme S,

An(S) = HomSch(S, An) = Γ(S,OS)n.

In fancy terms, this says that An represents the functor taking a scheme to n-tuples of elements of Γ(S,OS). We shall see a similar functorial characterizationof projective space Pn later in the book.Exercise 4.5 Prove Proposition 4.8 Mˇ

Exercise 4.6 Be aware that there may be a huge difference between abolutescheme isomorphisms and relative ones; e.g. the notion of a scheme isomor-phism between to fields K and L can be elusive. Show that (Spec C)(C) =

HomSch(Spec C, Spec C) is infinite (of cardinality of the continuum) where asHomSch/C(Spec C, Spec C) = tidu and HomSch/R(Spec C, Spec C) » Z/2Z. M

4.6 Brave new varieties

We have already said that schemes are generalizations of algebraic varieties,but on the other hand, we have also seen that even the simplest schemes, e.g.A2

k = Spec k[x, y], behave differently than varieties in the sense that they usuallyhave many non-closed points. Thus for this generalization to make sense, weshould expect there to be a canonical way to ‘add non-closed points’ to analgebraic variety so that the resulting topological space has the structure of ascheme. Let us explain what this means more precisely.

Let k be an algebraically closed field and let V be a variety over k. We firstconsider the case where V is affine. Each affine variety has a coordinate ringA = A(V); it is canonically attached to V being the ring of regular functionson V. From A(V), we can build Vs = Spec A, which is an affine scheme whoseclosed points are in bijection with the points of V (that is, Vs(k) = V) accordingto the Nullstellensatz. Thus the ‘new points’ correspond to the non-maximalideals of A; in fact, one way of thinking about this is that the scheme Vs in somesense is the collection of all subvarieties of the variety V.

Moreover, the fundamental theorem of affine varieties tells us that mapsφ : V Ñ W between two affine varieties are in one-one-correspondence withk-algebra homomorphisms φ7 : A(W) Ñ A(V), which exactly parallels ourTheorem 4.6. Hence putting φs = Spec φ7, we obtain a morphism φs : Vs Ñ Ws

which extends φ. As φ7 is a map of k-algebras, the morphism φs is a morphismof schemes over Spec k.

Summing up, we have thus defined a functor s : AffVar/k Ñ Sch/k, whereAffVar/k denotes the category of affine varities over k. As morphisms of k-varieties V Ñ W and affine k-schemes Vs Ñ Ws are both in canonical bijectionwith k-algebra homomorphisms A(W)Ñ A(V), the functor s is therefore fullyfaithful, in the sense that the assignment φ ÞÑ φs is a bijection

HomAffVar/k(V, W) » HomSch/k(Vs, Ws).

brave new varieties 102

In the general case, a variety V has an open cover by affine varieties Vi, andgluing can be performed in both the category of varieties as well as in thecategories of schemes, and it is a matter of straightforward checking that thisgives a well-defined scheme Vs containing each Vs

i as an open subscheme. Thegluing works equally well for morphisms, so we again obtain a functor, whichwe denote

s : Var/k Ñ Sch/k,

where now Var/k is the category of all varieties over k. Once again this functoris fully faithful, in the sense that the induced maps between HomVar/k(V, W)

and HomSch/k(Vs, Ws) are bijective. So two varieties give rise to isomorphicschemes over k if and only if they are isomorphic as varieties, and each schemeisomorphism is unambiguously determined by the variety isomorphism. Inparticular, this tells us that the category of varieties Var/k is equivalent to a fullsubcategory of Sch/k. We have already seen that s is far from being surjective,e.g. Spec k[x]/(x2) does not come from a variety.

New definition of a varietyLet k be an algebraically closed field. We say that a scheme X/k over k is an

Affine varieties (affinvarietet)

affine variety over k if it is isomorphic to the spectrum of the coordinate ring ofan affine k-variety. In other words, X = Spec A, where A is a finitely generatedk-algebra with no zero divisors. Thus, an affine variety is an integral scheme offinite type over an algebraically closed field.

The terminology one finds in the literature at this point, is varying; someauthors do not require varieties to be irreducible (but they are always reduced),and some allow varieties over fields that are not algebraically closed.Example 4.13 The schemes

i) A1Q= Spec Q[t];

ii) Spec C[x, y]/(x2 ´ y3) ;

iii) Spec Fp[x, y, z]/(x2 ´ yz)

are affine varieties, whereas

i) Spec Q[t]/t2;

ii) Spec C[x, y]/(xy);iii) Spec Q[x]

are not. K

A Prevarieties(prevarieteter)

prevariety over k is an irreducible scheme of finite type over k which has afinite affine covering consisting of affine varieties.

A prevariety X/k is a Varieties (varieteter)variety over k if it is in the image of the functor sabove. Thus, a variety X is a scheme over k which is glued together by affinek-varieties. Varieties in the classical sense satisfy the ‘Hausdorff axiom’, so thereare restrictions on how this gluing can be done. The scheme analogue of theHausdorff axiom is called ‘separatedness’; we will explore this concept in detailin Chapter 8.

Chapter 5

Examples constructed by gluing

5.1 Gluing two schemes together

To concretify the gluing techniques introduced in Chapter 4, we will in moredetail study the simple case of schemes obtained by gluing together just twoschemes.

We start out with two schemes X1 and X2 with respective open subsetsX12ĎX1 and X21ĎX2; these are open subschemes equipped with their canon-ical induced scheme structures obtained by restricting the structure sheaves.Furthermore, we assume given an isomorphism τ : X21 Ñ X12. The gluingconditions are trivially fulfilled, and these data allow us to glue together X1 andX2 along X12 and X21, and we thus construct a new scheme X.

On the level of topological spaces X is obtained from the disjoint unionX1

š

X2 by forming the quotient modulo the equivalence relation with x „ τ(x)for x P X21 Ď X2 and giving X the quotient topology. Moreover, each morphismιj : Xj Ñ X is an open immersion, allowing us to view each Xj as an open subsetof X.

The sections of the sheaf OX over an open UĎX is determined by the exactsequence (4.2), which in the present context takes the shape below. We insistson being precise and write Uj = ι´1

j U (which is identified with Xj XU) and

Uji = ι´1j (U1 XU2) (which is identified with X1 X X2 XU; this can be done in

two ways related by the ‘glue’ τ).

0 Γ(U,OX) Γ(U X X1,OX)ˆ Γ(U X X2,OX) Γ(U X X1 X X2,OX)

Γ(U1,OX1)ˆ Γ(U2,OX2) Γ(U21,OX2)

» »

ρ

The components of the left map are just the restrictions, i.e. it maps s to the pair(ι71s|UXX1 , ι72s|UXX2) (which after identification is nothing but (s|UXX1 , s|UXX2)),and the second sends a pair (s, t) to ρ(s, t) = t|U21 ´ τ7(s|U12).

a scheme that is not affine 104

The main example to keep in mind is when X1 and X2 are both affine,say X1 = Spec R and X2 = Spec S, and they are glued together along twodistinguished open subsets D(u) and D(v) for some u P R and v P S. The ‘glue’τ is induced from a ring isomorphism between localizations

φ : Ru Ñ Sv.

The geometric picture is as in the following diagram of schemes

Spec Ru = D(u) D(v) = Spec Sv

Spec R Spec S.

τ»

The global sections of OX is computed by the standard sequence on the previouspage, which in the present staging takes the form

0 Γ(X,OX) Rˆ S Sv.ρ

(5.1)

Here ρ(r, s) = s/1´ φ(r/1) with s/1 and r/1 denoting the images of s and rrespectively in Sv and Ru. In other words, elements in OX(X) correspond topairs (r, s) P Rˆ S such that s/1 = φ(r/1) in the localized ring Sv.

Many important examples arise from this basic construction. We will nowsurvey a few of these.

5.2 A scheme that is not affine

The first application is to see that the affine plane minus the origin is not anaffine scheme.

Let k be a field, let A2k = Spec A where A = k[u, v], and consider the open

subset U = A2k ´V(u, v); this is precisely the affine plane with the closed point

corresponding to the origin removed. Since U is an open set of A2k , there is a

canonical scheme structure on U as described in Section 3.4 on page 83. Wecontend that U can not be isomorphic to an affine scheme, the key point beingthat the restriction map Γ(A2

k ,OA2k)Ñ Γ(U,OU) is an isomorphism. Indeed, if

U were an affine scheme, Theorem 4.6 on page 98 would then imply that theinclusion map U Ñ A2

k were an isomorphism, which obviously is not true as ite.g. is not surjective.

Let us check that the restriction map really is an isomorphism. The twodistinguished open sets D(u) = Spec Au and D(v) = Spec Av form an openaffine covering of U, and the exact sequence (5.1) takes the following form:

0 Γ(U,OU) Au ˆ Av Auv,

A

ρ

i7

the projective line 105

where ρ is the difference between the two localization maps; that is, it mapsa pair (au´m, bvń) to au´m ´ bvń considered as an element in Auv. We haveincluded the restriction map i7 in the diagram, which is the just the map comingfrom the inclusion map i : U Ñ A2

k . It sends an element a P A to the pair(a/1, a/1) in Au ˆ Av.

Elements of Γ(U,OU) correspond to a pairs (au´m, bvń) in the kernel of ρ,and for such a pair the relation

avn = bum

holds in A = k[u, v]. Since A is a ufd, it follows that there is an element c P Awith a = cvm and b = cun; that is, au´m = bvń, and this shows that i7 issurjective. Hence it is an isomorphism since it obviously is injective.Exercise 5.1 Show that A1

k minus the origin is an affine scheme. M

Exercise 5.2 Let X = Spec k[x, y, z, w]/(xw ´ yz) and consider the open setU = X´V(x, y). Use the above strategy to compute OX(U). Conclude that Uis not affine. M

5.3 The projective line

In elementary courses on complex analysis one encounters the Riemann sphere.This is the complex plane C with one point added, the point at infinity. If zis the complex coordinate centered at the origin, the inverse w = z´1 is thecoordinate centered at infinity. Another name for the Riemann sphere is thecomplex projective line, denoted CP1.

The construction of CP1 can be vastly generalized and works in fact overany ring A. Let u be a variable (‘the coordinate function at the origin’) and letU0 = Spec A[u]. The inverse u´1 is a variable as good as u (‘the coordinate atinfinity’), and we let U1 = Spec A[u´1]. Both are copies of the affine line A1

Aover A.

Inside U0 we have the distinguished open set U01 = D(u), which is canoni-cally isomorphic to the prime spectrum Spec A[u, u´1], the isomorphism comingfrom the inclusion A[u]Ď A[u, u´1]. In the same way, inside U1 there is thedistinguished open set U10 = D(u´1). This is also canonically isomorphic tothe spectrum Spec A[u´1, u], the isomorphism being induced by the inclusionA[u´1]Ď A[u´1, u]. Hence U01 and U10 are isomorphic schemes (even canoni-cally), and we may glue U0 to U1 along U01. The result is called the projectiveline over A and is denoted by P1

A.


Gluing two affine lines to get P1A

Note that the complement of U1 equals V(u)ĎU0 = Spec A[u], which is isomor-phic to Spec A, so when A = k is a field, P1

k is a ‘one-point ’compactification’ ofU0. Of course, a similar statement holds true for the complement of U0.

The following computation is very important.

Proposition 5.1 We have Γ(P1A,OP1

A) = A.

Proof: The projective line P1A is covered by the two open affines U0 and U1,

and the standard exact sequence (5.1) above takes the form

Γ(P1A,OP1

A) Γ(U0,OP1

A)ˆ Γ(U1,OP1

A) Γ(U01,OP1

A)

A[u]ˆ A[u´1] A[u, u´1],

» »

ρ

where the map ρ sends a pair ( f (u), g(u´1)) of polynomials with coefficientsin A, one in the variable u and one in u´1, to the difference g(u´1)´ f (u). Weclaim that the kernel of ρ equals A; i.e. the polynomials f and g must both beconstants.

So assume that f (u) = g(u´1). Write f (u) = aun + lower terms in u, and ina similar way, write g(u´1) = bu´m + lower terms in u´1, where both a ‰ 0 andb ‰ 0. Without loss of generality we may assume that m ě n. Now, supposethat m ě 1. Upon multiplication by um we obtain b + uh(u) = um f (u) for somepolynomial h(u), and putting u = 0 we get b = 0, which is a contradiction.Hence m = n = 0 and we are done. o

In particular, the global sections of OX=P1C

is just C, which is a special case ofLiouville’s theorem that the only global holomorphic functions are the constants.We note that we also have got yet another example of a scheme which is notaffine: if P1

A were affine, it would have to be isomorphic to Spec C accordingto Theorem 3.12 on page 80. But this is clearly not the case, as P1

C containsinfinitely many closed points (e.g. it contains A1

C as an open subset).Another morale to extract from this is that the group Γ(X,OX) does not give

much information about X for general schemes, in contrast to the case whenX = Spec A is affine.


Exercise 5.3 Let X = Spec A be an affine scheme over a field k. Show thatˇ

every morphism P1A Ñ X is constant. M

The projective line P1 as a quotientThe two examples we have constructed are in fact closely related. In particular,there is a natural morphism between them:

π : A2A ´V(u, v)Ñ P1

A,

which we shall construct by gluing together the two morphisms φ1 : D(u) =

Spec A[u, v]u Ñ Spec A[u´1v] and φ2 : D(v) = Spec A[u, v]v Ñ Spec A[uv´1]

induced by the inclusions A[u´1v] Ă A[u, v]u and A[uv´1] Ă A[u, v]v. Witht = uv´1 the two targets yields P1

A when glued together along Spec A[t, t´1] =

Spec A[uv´1, vu´1]; the union of the sources equals D(u)YD(v) = V(u, v)c andD(u)XD(v) = D(uv) = Spec A[u, v]uv. The gluing condition is satisfied as wesee by applying Spec to the following commutative diagram (which, in fact, is adiagram of inclusion between subrings of A[u, v]uv = A[u, u´1, v, v´1]):

A[u, v]u A[u´1v]

A[u, v]uv A[u´1v, uv´1]

A[u, v]v A[uv´1]

On the level of closed points, when A = k is an algebraically closed field, themorphism π is exactly the quotient morphism used in the construction of theprojective line as a quotient space.

A family of sheaves on P1

The projective spaces, in particular P1A, carry a family of sheaves, there is one

for each integer, which play a foremost role in algebraic geometry. We shallconstruct these sheaves on the projective line P1

A by the gluing techniques so farexplained; they will be denoted by OP1

A(m) with m P Z.

Let X = P1A and let U0 = Spec A[u] and U1 = Spec A[u´1] be the usual

covering. Consider the intersection U0 XU1 = Spec R, where R = A[u, u´1].Multiplication by um gives an isomorphism

R R,[um]

and by Exercise 3.1 this induces an isomorphism of sheaves

τ : OU1 |U0XU1 Ñ OU0 |U0XU1 .


Now, we define a sheaf OP1A(m) by gluing OU1 to OU0 along U0 XU1 via this

isomorphism. Note that the direction of τ is very important; we could of coursehave used the multiplication map the other way round, but would then hadobtained another sheaf, namely OP1

A(´m).

By construction, the sheaf OP1A(m) has the property that OP1

A(m)|U0 » OU0

and O(m)P1A|U1 » OU1 . In the jargon of Chapter 11 it is a ‘locally free sheaf’. Let

us, however, show that when m ‰ 0, the sheaf OP1A(m) is not isomorphic to the

structure sheaf OP1A. In particular, we see that a sheaf is not determined by its

stalks alone.To prove this claim, we compute global sections using the standard sequence:

0 Γ(OP1A,OP1

A(m)) A[t]‘ A[t´1] A[t, t´1]

ρ

where ρ(p(u), q(u´1)) = umq(u´1)´ p(u). If m ă 0, clearly there are no non-trivial polynomials p and q satisfying umq(u´1) = p(u), and we infer thatΓ(P1

A,OP1A(m)) = Ker ρ = 0; hence OP1

A(m) cannot be isomorphic to OP1

A. For

m ě 0 there are, however, such polynomials: every polynomial p(u) of degreeat most m is on the form umq(u´1), and q is uniquely determined by p. We haveshown the following:

Proposition 5.2 For m ě 0 we have

Γ(P1A,OP1

A(m)) = A‘ Au‘ ¨ ¨ ¨ ‘ Aum.

Closed subschemes of P1

Let us have a closer look at the sheaf OP1A(´1), which is usually called the

tautological sheaf on P1. We claim that there is a map of sheaves

φ : OP1A(´1)Ñ OP1 ,

which makes OP1A(´1) into a subsheaf of OP1 . We shall define φ locally on

each of the open sets U0 and U1 and subsequently apply the gluing lemma formorphisms of sheaves. On the open set U0 = Spec A[u] we use the multiplicationby u as the φ0 : OU0 Ñ OU0 . Likewise, we define φ1 : OU1 Ñ OU1 by the identitymap. To be able to glue them together, we need to verify that the two agreeon the intersection U0 XU1 = Spec A[u, u´1], but this follows directly from thecommutative diagram

OU0 |U0XU1 OU0 |U0XU1

OU1 |U0XU1 OU1 |U0XU1

u

=

u´1 =

the affine line with a doubled origin 109

Here all four sheaves are equal to OU0XU1 . The right vertical map is the gluing-map for the sheaf OP1

Aand the left one that for OP1

A(´1), whilst the horizontal

maps are the restrictions φ0|U0XU1 and φ1|U0XU1 . Thus we have the desired mapφ : OP1

A(´1)Ñ OP1 . This map is injective, because it is injective over U0 and U1

(by Lemma 1.17).The map φ lives in the short exact sequence

0 Ñ OP1A(´1)Ñ OP1

AÑ Coker φ Ñ 0. (5.2)

We claim that Coker φ » i˚OSpec A, where i : Spec A Ñ P1A is the closed immer-

sion given by Spec(A[u]/u)Ñ Spec A[u] over U0. To check this, we work locallywith the covering U0, U1. Over U0, the sequence (5.2) is exactly the ideal sheafsequence

0 Ñ A[u] uÝÑ A[u]Ñ A[u]/(u)Ñ 0,

associated to the closed immersion Spec(A[u]/(u))Ñ U0. Over U1, φ restrictsto the identity of OU1 , so Coker(φ)|U1 = 0. Note that the image of i does notmeet U1 = Spec A[u´1] (we have U0 XU1 = Spec A[u, u´1] and u generates theunit ideal in A[u, u´1]); hence also i˚OSpec A restricts to the zero sheaf there.

When A = k is a field, the sequence above reduces to

0 Ñ OP1k(´1)Ñ OP1

kÑ k(p)Ñ 0. (5.3)

where k(p) is the skyscraper sheaf at the origin p.Exercise 5.4 (Closed subschemes of P1

k .)

a) Imitate the construction above to define a map of sheaves φ : OP1A(´m)Ñ

OP1A

for each integer m ě 1.

b) There are in fact many maps φ : OP1A(´m) Ñ OP1

A. Show that for each

polynomial p P A[u] of degree at most m, there is such a map. Describe thecokernel of these maps. Hint: Glue together the multiplication map by pon U0 and the one by q on U1, where q is as in Proposition 5.2.

c) Let A = k be a field. Show that for each non-empty closed subschemeZ Ă P1

k there is a φ with Coker φ = OZ.

M

5.4 The affine line with a doubled origin

We intend to glue together two copies X1 and X2 of the affine line A1k = Spec k[u]

over a field k along their common open subset X12 = Spec k[u, u´1] with theidentity morphism φ : k[u, u´1] Ñ k[u, u´1] as glue. The resulting schemecontains two A1

k’s which overlap outside the origin. But since the the gluingprocess does nothing over the origins of each A1

k , there are now two points in Xthat replace the origin; X is sometimes called the affine line with two origins.

semi-local rings 110

This scheme is not affine: the sheaf sequence from before takes the form

0 Γ(X,OX) Γ(A1k ,OA1

k)‘ Γ(A1

k ,OA1k) Γ(X12,OX12)

k[u]‘ k[u] k[u, u´1]

= =

ρ

where now ρ(a1, a2) = a1 ´ a2, and it follows that either open inclusion ι : A1k Ñ

X induces an isomorphism Γ(X,OX) » Γ(A1k ,OA1

k) = k[u]. However, the open

inclusion ι : A1k = Spec k[u] Ñ X is not an isomorphism (it is not surjective,

since the image misses one of the two origins).Exercise 5.5 Imitate the construction of the sheaves OP1

k(n) on P1

k to form afamily of sheaves on X. M

5.5 Semi-local rings

Example 5.3 Semi-local rings. The rings Z(2) and Z(3) are both discrete valuationrings whose maximal ideals are (2) and (3) respectively. Their fraction fields areboth equal to Q. Let X1 = Spec Z(2) and X2 = Spec Z(3). Both have a genericpoint that is open, so there is a canonical open immersion Spec Q Ñ Xi fori = 1, 2. Hence we can glue the two along their generic points and thus obtaina scheme X with one open point η and two closed points. Let us compute theglobal sections of OX using the now classical sequence for the open coveringtX1, X2u:

Γ(X,OX) Γ(X1,OX)ˆ Γ(X2,OX) Γ(X1 X X2,OX)

Z(2) ˆZ(3) Q.

= =

ρ

The map ρ sends a pair (an´1, bm´1) to the difference an´1´ bm´1, hence thekernel consists of the diagonal, so to speak, in Z(2) ˆZ(3), which is isomorphicto the intersection Z(2) XZ(3). This is a semi-local ring with the two maximalideals (2) and (3). Hence there is a map X Ñ Spec Z(2) XZ(3) and it is left asan exercise to show that this is an isomorphism. K

Example 5.4 More semi-local rings. More generally, if P = tp1, . . . , pru is a finiteset of distinct prime numbers, one may let Xp = Spec Z(p) for p P P. There is, asin the previous case, a canonical open embedding Spec Q Ñ Xp. Let the imagebe tηpu. Obviously conditions for gluing the ηp’s together are all satisfied (the

the blow-up of the affine plane 111

transition maps are all equal to idSpec Q and Xpq = tηpu for all p). We do thegluing and obtain a scheme X. Again, to compute the global sections of thestructure sheaf, we use the standard sequence

Γ(X,OX)ś

pPP Γ(Xp,OX)ś

p,qPP Γ(Xp X Xq,OX)

ś

pPP Z(p)ś

p,qPP Q.

= =

ρ

The map ρ sends a sequence (ap)pPP to the sequence (ap´ aq)p,qPP, and it followsthat the kernel of ρ equals the intersection AP =

Ş

pPP Z(p). This is a semi-localring whose maximal ideals are the (p)Ap’s for p P P. There is a canonicalmorphism X Ñ Spec AP, and again we leave it to the industrious student toverify that this is an isomorphism. K

Exercise 5.6 Verify the claims in Examples 1 and 2 above that X is isomorphicrespectively to Spec Z2 XZ3 and to Spec AP. Hint: Use the unicity statementin Proposition 4.3 on page 96. M

Exercise 5.7 Glue Spec Z(2) to itself along the generic point to obtain a schemeX. Show that X is not affine. Hint: Show that Γ(X,OX) = Z(2). M

5.6 The blow-up of the affine plane

In this section, we will construct the blow-up of A2 at the origin, by gluing togethertwo affine schemes. We begin by recalling the classical construction for varieties.To be precise, we write A2(k) for the variety, and A2

k for the scheme, etc.

The blow-up as a varietyLet k be an algebraically closed field, and consider the affine plane A2(k). Thereis a rational map f : A2(k) 99K P1(k) that sends a point (x, y) to the point (x : y)(in homogeneous coordinates on P1(k)). This map is not defined at the origin(0, 0), but we can still associate with it the closure X in A2(k)ˆP1(k) of itsgraph, which lies in

(A2(k)´ (0, 0)

)ˆP1(k).

To describe the graph in more detail it is better to rename the homogenouscoordinates on P1(k) to (s : t). If the coordinate t ‰ 0, it holds that (s : t) =

(st´1 : 1), so the part of the graph where y ‰ 0, is given by the equationxy´1 = st´1; or in other words, by xt´ ys = 0. And similarly, the same equationgives the part where x ‰ 0 since there yx´1 = ts´1. Hence X is defined inA(k)2 ˆP1(k) by the single equation

X = Z(xt´ ys) Ă A2(k)ˆP1(k).

We also have two projection maps p : X Ñ A2(k) and q : X Ñ P1(k). Let us


Figure 5.1: The blow-up of the plane at a point

analyze the fibres of these two maps. The fibres of p are easy to describe. If(x, y) P A2(k) is not the origin, then p´1(x, y) consists of a single point: theequation xt = ys allows us to determine the point (s : t) uniquely since eitherx ‰ 0 or y ‰ 0. However, when (x, y) = (0, 0), any choices of s and t satisfythe equation, so p´1(0, 0) = (0, 0) ˆ P1(k). In particular, this inverse imageis one-dimensional; it is called the exceptional divisor of X, and is frequentlydenoted by E.

Similarly, if (s : t) P P1(k) is a point, the fibre

q´1(s : t) = t(x, y)ˆ (s : t) | xt = ysu Ă A(k)2 ˆ (s : t)

is the line in A2(k) with sx´ ty = 0 as equation, s and t being the coefficients.The map q is an example of a line bundle; all of its fibres are affine lines; that is,A1(k)’s. We will see these again later on in the book.

The standard covering of P1(k) as a union of two A1(k)’s gives an affinecover of X: If U Ă P1(k) is the open set where s ‰ 0, we can normalize by settings = 1, and the equation xt = sy becomes y = tx. Hence x and t may serveas affine coordinates on q´1(U), and q´1(U) » A2(k). In these coordinates,the morphism p : X Ñ A2

k restricts to the map A2(k) Ñ A2(k) given by(x, t) ÞÑ (x, xt). Similarly, if V denotes the open set where t ‰ 0, it holds thatq´1(V) = A2(k) with affine coordinates y and s, and the map p is given here as(y, s) ÞÑ (sy, y).

The blow-up as a schemeInspired by the above discussion we proceed to define the scheme-analogue ofthe blow-up of A2

k at a point. It will be defined as a scheme over Z rather thanover a field k (we get a blow-up of A2

A for any ring A by tensorizing everythingbelow by A). Also, in addition to the scheme X, we also want a morphisms ofschemes p : X Ñ A2 and q : X Ñ P1 having similar properties to the morphismsin the example above.


Consider the affine plane A2 = Spec Z[x, y]. The prime ideal p = (x, y) ĂZ[x, y] corresponds to the origin of A2(k) in the analogy with the situationabove. Consider the diagram

Z[x, y]

Z[x, t] Z[y, s]

R = Z[x, y, s, t]/(xt´ y, st´ 1)

Here the two diagonal maps in the upper part are given by x ÞÑ x, y ÞÑ xtand y ÞÑ y, x ÞÑ ys respectively, and the two others are induced by obviousinclusions.

Note that the ring R is isomorphic to Z[x, s, t]/(st ´ 1) = Z[x, t, t´1] aswell as to Z[y, s, t]/(st ´ 1) = Z[y, s, s´1]. Since this ring is a localization ofboth Z[x, t] and Z[y, s], we can identify its spectrum both as an open subset ofSpec Z[x, t] and as an open subset of Spec Z[y, s]. From this we get a diagram

Spec Z[x, y]

U = Spec Z[x, t] Spec Z[y, s] = V

Spec R

where the bottom diagonal maps are the two open immersions. Hence wecan glue these two affine spaces together along Spec R to obtain a new schemeX. By construction, the restriction of the maps Spec Z[x, t] Ñ Spec Z[x, y]and Spec Z[y, s] Ñ Spec Z[x, y] to Spec R coincide with the map Spec R Ñ

Spec Z[x, y] which is induced by Z[x, y] Ñ R. Therefore they may be gluedtogether to a morphism (the ‘blow-up morphism’)

p : X Ñ A2 = Spec Z[x, y].

To complete the discussion, we should define the corresponding morphismq : X Ñ P1. Again we work locally. On the affine open U = Spec Z[x, t] wehave a map U Ñ A1 = Spec Z[t] induced by the inclusion Z[t] Ă Z[x, t].Similarly, on V = Spec Z[y, s] we have a map V Ñ A1 = Spec Z[s]. Checkingif they can be glued together, amounts to seeing what happens on the overlapUXV = Spec R. However, on Spec R it holds that t = s´1, so using the standarddescription of P1 as being glued together of two affine lines, we see that themaps Z[t]Ñ R and Z[s]Ñ R induce the desired morphism q : X Ñ P1.

projective spaces 114

Exercise 5.8 Compute the space Γ(X,OX) of global sections and describe theˇ

canonical map X Ñ Spec Γ(X,OX). M

Exercise 5.9 Imitate the construction above to define the blow-up of AnZ along

a codimension 2 linear space V(x, y). M

5.7 Projective spaces

We now give examples of some more involved gluings where more than twoparts are joined and we shall construct the all important projective spaces overa ring A which are omnipresent in algebraic geometry. So let A be a ring, andconsider the subrings of A[x0, x´1

0 , . . . xn, x´1n ] given by

Ri = A[x0x´1i , . . . , xnx´1

i ]

for i = 0, . . . , n. Each Ri is isomorphic to a polynomial ring in n variables overA, and Ui = Spec Ri is an affine space An

A. Note that we have equalities

Ri[xix´1j ] = Rj[xjxi

´1] (5.4)

for each i and j; indeed, this follows from the identities xlx´1i = xlx´1

j ¨ xjx´1i

valid for all i, j and l. Each Uij = Spec Ri[xixj´1] is the standard open D(xjx´1

i )

in Ui, and using the equalities from (5.4) as transition functions, we can gluetogether the affine spaces Ui = Spec Ri » An

A along the Uij’s (the transition-function are identities, and the gluing condition are trivially fulfilled, and notethat the triple intersections are Uijl = Spec Ri[xix´1

j , xix´1l ]). In this way we

construct a scheme which we shall denote by PnA, and this is the The projective n-space

(det projektiven-rommet)

projective n-spaceover A.

Note that each Spec Ri comes with a canonical map Spec Ri Ñ Spec A,induced by the inclusion A Ă Ri. Moreover, the isomorphisms used as glueabove are all ‘over A’; that is, they are A-algebra homomorphisms, and thus theyare compatible with the inclusions A Ă Ri. Hence they may be glued togetherto form a morphism Pn

A Ñ Spec A.Note in particular, that for n = 1 we obtain the projective line P1

A constructedearlier. An argument similar to that in Proposition 5.1 gives

Proposition 5.5 Γ(PnA,OPn

A) = A.

Example 5.6 The projective plane. The projective plane P2k is formed by gluing

together the three copies of the affine plane A2k . The notation quickly becomes

cluttered when working with the fractions as in the previous paragraph, andone way to avoid this is to single out one of the variables, say x0, and ‘set itequal to one’, or expressed more seriously, renaming x1x´1

0 to x and x2x´10 to y.

The other fractions follow suit; for instance, x1x´12 = xy´1. With this convention

projective spaces 115

the three affine pieces become:

U0 = Spec k[x, y], U1 = Spec k[x´1, yx´1], U2 = Spec k[y´1, xy´1].

K

Gluing three affine planes to P2

We proceed by giving some examples.Example 5.7 A triangle of reference. In the standard affine open U0 = Spec k[x, y]the subset V(x) = Spec k[y] is an affine line A1

k , and the same is true forthe subset V(xy´1) = Spec k[y´1] in U2 = Spec k[y´1, xy´1]. The two affinelines match in the overlap U0 X U2 = Spec k[x, y, y´1] because the equality(x) = (xy´1) of principal ideals holds in k[x, y, y´1]. Note that the lines meetU0 XU2 in the closed subset equal to Spec k[y, y´1]. Hence when the Ui’s areglued together to form P2

k , the two affine lines are glued together to a projectiveline; indeed, the gluing setup is given by the inclusions k[y] Ă k[y, y´1] Ą k[y´1],which is precisely the recipe for P1

k .Returning to the coordinates x0, x1 and x2, the projective line just constructed

is denoted V(x1). In a completely symmetric way we find two other lines in P2,one is V(x0) and the other V(x2). They constitute what one calls a ‘triangle ofreference’. K

Example 5.8 Consider the three ideals

I0 = (y2 ´ x3) Ă k[x, y];

I1 = (x´1(yx´1)2 ´ 1) Ă k[x´1, yx´1];

I2 = (y´1 ´ (xy´1)3) Ă k[y´1, xy´1].

Each ideal Ii defines a closed subscheme of the corresponding affine pieceUi = A2

k , and it is readily checked that they agree on the overlaps Ui XUj. Forinstance, in U0 XU1 = Spec k[x, x´1, y], we have(

(x´1)(yx´1)2 ´ 1)=(x´3(y2 ´ x3)

)= (y2 ´ x3),

since x is invertible in k[x, x´1, y]. Thus the three subschemes glue together to aclosed subscheme Z Ă P2

k .

line bundles on1

116

In Chapter 9, we will see that there is a much more economic way ofspecifying subschemes of Pn using graded ideals. In fact, the above subschemeis defined by a single homogeneous polynomial, F = x0x2

2 ´ x31. K

Exercise 5.10 Prove Proposition 5.3. (A more general result will be proved inChapter 14). M

5.8 Line bundles on P1

The sheaves OP1k(n) which we constructed in Example 5.3 have a geometric alter

ego, the so-called line bundles Ln. (Here we work with A = k, a field). The twoconcepts are closely related and the connection between them will be furtherexplored in Chapter 11. Here we contend ourselves to give the construction anda closer description of a few of the Ln’s.

The sheaves OP1k(n) were obtained by gluing OU0 and OU1 via the multiplica-

tion by tn map on OU0XU1 , where U0 = Spec k[u], U1 = Spec k[u´1] form the stan-dard affine cover of P1

k , and their intersection equals U0 XU1 = Spec k[u, u´1].The new schemes Ln will be constructed essentially by the same gluing process,but schemes and not sheaves will be joined together; two copies of the affineplane, Spec k[u, s] and Spec k[u´1, t] will be glued together with glue being ‘mul-tiplication of one coordinate by a power of the other’. The schemes Ln comeequipped with a canonical morphism π : Ln Ñ P1

k .To be precise, consider the following commutative diagram of ring maps

k[u, s] k[u, u´1, s] k[u, u´1, t] k[u´1, t]

k[u] k[u, u´1] k[u´1]

ρ

»

where the map ρ : k[u, u´1, s]Ñ k[u, u´1, t] is defined by sending s to unt and uto u, and the others are the obvious inclusions* ˚The attentive student

will observe a changeof sign compared to thesheaf case; this is notmerely an annoyingconvention, but isdictated by a deeperbut p.t. mysteriousprinciple.

. Applying Spec to this diagramresults in the gluing diagram

A2 = Spec k[u, s] D(u) » D(u´1) A2 = Spec k[u´1, t]

U0 U0 XU1 U1.

The gluing conditions are fulfilled, and hence we obtain scheme Ln admitting amorphism π : Ln Ñ P1. Note that if x P P1 is a closed point, say x P U0, thenthe fibre π´1(x) is isomorphic to the affine line A1

k(x). This explains the term‘line bundle’: intuitively Ln is a family of affine lines parameterized by the basespace P1

k .

line bundles on1

117

There is a copy of P1k embedded in Ln, called the The zero section

(nullseksjonen)zero section of Ln; that

is, there is a closed immersion ι : P1k Ñ Ln with image a closed subscheme

C Ă Ln, having the property that CX π´1(x) is the origin in each fibre A1k(x)

of π. Intuitively, it is defined by the equation s = 0 or t = 0 in each fibre. Theproper definition is as follows: Inside each A2

k we have the closed subschemesA1

k » V(s) Ă Spec k[u, s] and A1k » V(t) Ă Spec k[u´1, t], and the salient point

is that when the two A2k’s are glued together, the subschemes V(s) and V(t)

match up and are glued together to form a P1k . Indeed, the gluing data on the

algebraic level are expressed in the diagram

k[u, s] k[u, u´1, s] k[u, u´1, t] k[u´1, t]

k[u, s]/(s) k[u, u´1, s]/(s) k[u, u´1, t]/(t) k[u´1, t]/(t)

k[u] k[u, u´1]) k[u, u´1]

ρ

»

»

»

» »»

where we note that the bottom row precisely expresses the recipe for P1k , so Spec

of the vertical maps may be glued together to a map ι : P1k Ñ Ln.

On each affine open, ι is given by factoring out some ideal, hence is aclosed immersion. Moreover, each morphism k[u]Ñ k[u, s]Ñ k[u, s]/(s) » k[u]and k[u´1] Ñ k[u´1, t] Ñ k[u´1, t]/(t) » k[u´1] is the identity. It follows thatπ ˝ ι = idP1

k, and ι is a ‘section’ of π.

A few particular casesThe schemes Ln give a rich source of examples in algebraic geometry, and wewill come back to them frequently in the book. For now let us compute andstudy a few explicit examples.

L0

The scheme L0 is glued together of two copies of A2k with the help of the

inclusions k[u, t] Ñ k[u, u´1, u] Ð k[u´1, t]. In addition to π, the bundleL0 admits a morphism L0 Ñ A1

k obtained by gluing together the two mapsSpec k[t, u]Ñ Spec k[t] and Spec k[u´1, t]Ñ Spec k[t]. Anticipating the notion of

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118

a fibre product (which we will study in detail in Chapter 7), this makes, in fact,L0 isomorphic to the fiber product P1 ˆk A1

k . It is the scheme associated to theproduct variety P1(k)Â1(k).

L1

We claim that the scheme L1 is isomorphic to the complement of a closedpoint P in P2, i.e. Y = P2ztPu. Indeed, choose coordinates x0, x1 and x2 inthe projective plane and consider the two distinguished open sets D+(x0) =

Spec k[x1x´10 , x2x´1

0 ] and D+(x1) = Spec k[x0x´11 , x2x´1

1 ]. Their union in P2k

equals the complement of the closed point P = (0 : 0 : 1). Renaming thevariables u = x0x´1

1 , s = x2x´11 and t = x2x´1

0 we find that D+(x0) = Spec k[u, s]and D+(x1) = k[u´1, t] and the identity x2x´1

1 = x0x´11 ¨ x2x´1

0 turns into theequality s = ut, which is precisely the gluing data for L1.

Geometrically the morphism P2kztPu Ñ P1

k is given by ‘projection from thepoint P’. We will make this more precise in Chapter 16. The fibres of π are thelines in P2

k through P (with the point P removed) and the zero section equalsthe line ‘at infinity’; i.e. the line V(x2).

L´1

We have in fact seen the scheme L´1 before: it is isomorphic to the blow-upof A2

k at the origin. To see this, one needs only check that the gluing mapsare exactly the same. In particular, the map π : L1 Ñ P1

k , is exactly the mapconstructed in Section 5.6, and the zero-section C corresponds to the exceptionaldivisor E. See Exercise 5.11 below.

L´2

The scheme L´2 is quite interesting. It is the so-called resolution of a quadratic cone.There is a surjective morphism σ : L´2 Ñ Spec R, where R = k[x, y, z]/(y2´ xz),and Spec R is singular whereas L´2 is not. To construct σ, we need to constructthe maps with source R in the diagram

R

k[u, s] k[u, u´1, s] k[u, u´1, t] k[u´1, t]

k[u] k[u, u´1] k[u´1]

ρ

where the rest of the maps form the gluing data for the bundle L´2; in particular,the gluing map ρ : k[u, u´1, s] Ñ k[u, u´1, t] will be the isomorphism so thats ÞÑ u´2t. We define the homomorphism R Ñ k[u, s] by the assignments x ÞÑ s,y ÞÑ us and z ÞÑ u2s, and R Ñ k[u´1, t] by the assignments x ÞÑ u´2t, y ÞÑ u´1t

line bundles on1

119

and z ÞÑ t. This makes all three upper triangles commute. Applying Spec to thediagram we get a morphism σ : L´2 Ñ Spec R.

Let us study the fibers of this morphism, and we begin by figuring out whathappens over the open set U = Spec k[u, s], where σ is given by the map

σU : Spec k[u, s]Ñ Spec R

induced by the upper left map in the diagram. Consider the prime ideal p =

(x, y, z) Ă R with zero-set the origin. We have σ´1(V(p)) = V(s, su, u2s) = V(s),which means that the whole ’u-axis’ V(s) in A2 = Spec k[u, s] is collapsedonto the origin P in Spec R. Likewise, the ’u´1-axis’ in A2 = Spec k[u´1, t] iscollapsed to the origin. It follows that the whole zero-section C in L´2 is mappedto the origin. This is consistent with

the fact that anymorphismP1

k Ñ Spec R to anaffine scheme isconstant.

In fact, the zero-section C is the only subscheme of L´2 which iscontracted by this map; σ is an isomorphism outside C.

Proposition 5.9 σ restricts to an isomorphism L´2 ´ C Ñ Spec RztPu.

Proof: The complement Spec RztPu of the origin is covered by the two dis-tinguished open sets D(x), D(z) (note that D(y) = D(y2) = D(xz) by thequadratic relation defining R). Likewise, the complement L´2zC of the zero-section is covered by the distinguished open subsets D(s) Ă Spec k[u, s] andD(t) Ă Spec k[u´1, t]. It holds that σ´1

U (V(x)) = V(s) Ă Spec k[u, s], and thismeans that the restriction σ|U = σU maps D(s) onto D(x). In fact, using the iden-tification D(x) = Spec Rx, and the identity Rx = (k[x, y, z]/(y2´ xz))x » k[x, y]xwe see that σ|U is the map

Spec k[u, s]s Ñ Spec k[x, y]x

induced by the ring map which is defined by x ÞÑ s and y ÞÑ us; this is anisomorphism since we have inverted s. Hence σ|U is an isomorphism over D(x).A symmetric argument shows that σ|V is an isomorphism over D(z); and alltogether, σ is an isomorphism outside C. o

Exercises(5.11) Check that L´1 is indeed the blow-up constructed in Section 5.6.ˇ

(5.12) Show that for n ě 0, the scheme Lń admits a morphism σ : Lń Ñ Ycontracting the zero-section C to a point.(5.13) For the morphism π : Ln Ñ P1

k , show that

π˚OLn = OP1k.

M

hyperelliptic curves 120

5.9 Hyperelliptic curves

Let k be a field. Let g be a non-negative integer, and consider the two affineschemes X1 = Spec A and X2 = Spec B, where

A =k[x, y]

(y2 ´ a2g+1x2g+1 ´ ¨ ¨ ¨ ´ a1x)and B =

k[u, v](v2 ´ a2g+1u´ ¨ ¨ ¨ ´ a1u2g+1)

with scalars a1, . . . , a2g+1 P k. The two distinguished open sets D(x) = Spec Ax

and D(u) = Spec Bu are isomorphic: the assignments φ(u) = x´1 and φ(v) =x´g´1y give an isomorphism φ : Bu Ñ Ax. It is well-defined as the little calcula-tion

φ(v2 ´ a2g+1u´ ¨ ¨ ¨ ´ a1u2g+1) = y2x´2g´2 ´ a2g+1x´1 ´ ¨ ¨ ¨ ´ a1x´2g+1

= x´2g´2(y2 ´ a2g+1x2g+1 ¨ ¨ ¨ ´ a1x)

shows that the defining ideal for Bu maps into the one defining Ax, and oneverifies effortlessly that the inverse homomorphism is given as x ÞÑ u´1 andy ÞÑ vu´g´1. We can thus glue X1 and X2 together along the open subsets D(x)and D(u).

The resulting scheme X is what is called a Hyperelliptic curves(hyperelliptisk kurve)

hyperelliptic curve or a doublecover of P1

k . In the case g = 1, X is an example of an elliptic curve. Here is anillustration of the real points of one of the affine charts for g = 2:

Notice that the gluing is very similar to the schemes Ln introduced in Section5.8. In fact, X is naturally a closed subscheme of L´g´1, since L´g´1 is gluedtogether by U = Spec k[x, y] and V = Spec[u, v] using the same gluing maps,and X is locally given by the quotient maps k[x, y]Ñ A and k[u, v]Ñ B on thetwo affine open sets.

In particular, X admits a morphism f : X Ñ P1k . In more detail, we can

see the map as the map glued together by the two maps Spec A Ñ Spec k[x]and Spec B Ñ Spec k[u], induced by the two natural inclusions k[x] Ă A andk[u] Ă B. The ‘gluing diagram’

k[u] k[x]

Bu Ax

u ÞÑx´1

φ

double covers ofna 121

commutes, so by the Gluing Lemma for morphisms (Proposition 4.4 on page 97)the inclusions match on the overlap and patch together to the desired mapX Ñ P1

k . Note that the correspondence u ÞÑ x´1 gives the standard constructionof P1

k by gluing together the two affine lines Spec k[x] and Spec k[u].It is the morphism f : X Ñ P1

k that lies behind the name ‘double cover’: thegeneral fibre f´1(q) consists of two points, when k is algebraically closed and ofcharacteristic different from two.Exercise 5.14 Assume that k is algebraically closed. Let a2g+1 = 1 and a1 = ´1ˇ

and ai = 0 for the other indices. Determine the image of D(x) and D(u) in P1k .

Find all points in P1k where the fibre of the double cover f does not consist of

exactly two points. How many are there? M

Exercise 5.15 (Adapted from New Zealand Mathematical Olympiad 2019 Problem 5.)ˇ

Consider the hyperelliptic curve X = X1 Y X2, where

X1 = Spec Z[x, y]/(x4 ´ x3 + 3x2 + 5´ y2)

andX2 = Spec Z[u, v]/(1´ u + 3u2 + 5u4 ´ v2)

and u = x´1, v = x2y´1 on the overlap. Determine the set of Z-points, X(Z).M

5.10 Double covers of PnA

Algebraic geometry is not only about polynomials, but algebraic functions aswell. A simple example of this is the square root of a polynomial; for instance,of f (x1, . . . , xn) P R = A[x1, . . . , xn]. Due to the sign ambiguity, this is not afunction on An

A, and this leads to the following construction of a scheme which isa double cover of An

A = Spec R: the closed subscheme X = V(t2´ f )Ď Spec R[t]maps in a two-to-one-fashion onto Spec R (with a liberal interpretation of two-to-one when A is of even characteristic). By definition, the coordinate t is asquare root of f defined on the double cover X.

We intend to generalize this to double covers of projective spaces by gluingtogether local construction like the one we just did. This generalizes also theexample of hyperelliptic curves to higher dimensions.

Let A be a ring and let R = A[x0, . . . , xn] with the usual grading. Let f P Rbe a homogeneous polynomial of degree 2d, and for each 0 ď i ď n let

Si = A[x0x´1i , . . . , xnx´1

i , yx´di ]/

((yx´d

i )2 ´ f (x0x´1i , . . . , xnx´1

i ))

For each pair i, j letting Sij = Si[xix´1j ], one checks that Sij = Sji; indeed, this

reduces to the identity

x2di x´2d

j

((yx´d

i )2 ´ f (x0x´1i , . . . , xnx´1

i ))= (yx´d

j )2 ´ f (x0x´1j , . . . , xnx´1

j ).

hirzebruch surfaces 122

It is then straightforward to verify that the Spec Si’s glue together along theopen subschemes Spec Sij’s to a scheme X. Moreover, keeping the notation Ri

from the previous section, the morphisms Spec Si Ñ Spec Ri, induced by theinclusions Ri Ñ Si, glue together to a morphism π : X Ñ Pn

A.Example 5.10 A Del Pezzo surface. Let us consider the case f (x0, x1, x2) =

x41 + x3

0x1 + x22(x2 ´ x0)2. Note that

S0 » k[u, v, y]/(y2 ´ u3 ´ u + v2(v2 ´ 1)

)via the identifications u = x1x´1

0 and v = x2x´10 . So the scheme X is a surface

glued together of three open sets, each isomorphic to a quartic surface in A3k .

The ‘double cover’ morphism is given by π : Spec S0 Ñ Spec k[u, v].The closed subset V(u) is interesting: Note that

(y2 ´ u4 ´ u + v2(v´ 1)2, u) = (y + v(v´ 1), u)X (y´ v(v´ 1), u)

So the preimage π´1(V(u)) consists of two components, each mapping isomor-phically to V(u). K

5.11 Hirzebruch surfaces

FriedrichHirzebruch(1927–2012)

Let r ě 0 be an integer and consider the scheme X which is glued together bythe four affine scheme charts

U00 = Spec k[x, y] U01 = Spec k[x, y´1]

U10 = Spec k[x´1, xry] U11 = Spec k[x´1, x´ry´1](5.5)

When k = C, these are the so-called Hirzebruch surfaces. In many ways, thesesurfaces behave as the ’Möbius strips’ in algebraic geometry. We will studythese surfaces in several contexts in the book. For now, we will explain that Xadmits a morphism to P1

k .The inclusions

k[x] Ă k[x, y] k[x] Ă k[x, y´1]

k[x´1] Ă k[x´1, xry] k[x´1] Ă k[x´1, x´ry´1](5.6)

induce morphisms Uij Ñ A1k . Moreover, these agree over the various intersec-

tions Uij XUjl , and so we obtain a morphism X Ñ P1k .

Exercise 5.16 Show that when r = 1, the surface above is isomorphic to theblow-up of P2 at a point. (Hint: Show that the latter can be described using fouraffine charts). M

Chapter 6

Geometric properties of schemes

In this chapter we survey some of the main geometric properties of schemes.We have already seen a few such properties in Chapter 3: these were propertiesthat could be formulated just in terms of the underlying topological spaces. Forinstance, a scheme (X,OX) was said to be connected if the topological space X isconnected. Recall that this means that X can not be written as a disjoint unionof two proper, open subsets. In particular, for X = Spec A, being connected wasreflected in the algebraic condition that A was not the direct product of twonon-zero rings B and C. Likewise, we said that a scheme X is irreducible if itsunderlying topological space is irreducible; that is, it is not the union of twodifferent proper, closed subsets. For X = Spec A this amounts to saying that Ahas a single minimal prime.

In this chapter we will survey more subtle geometric properties of schemes.These properties reflect both the underlying topological space, as well as thestructure sheaf OX.

6.1 Noetherian schemes

By the correspondence between irreducible subsets of Spec A and prime idealsof A, we immediately see that if A is a Noetherian ring, the prime spectrumSpec A is a Noetherian topological space, i.e. all descending chains of irreducibleclosed subsets stabilize. Indeed, such a chain is of the form V(a1) Ě V(a2) Ě ¨ ¨ ¨ ,where we may assume that the ideals an are prime. Then the condition thatV(an) is decreasing, corresponds to the sequence (an) being increasing, and soit has to be stationary because A is Noetherian.

The converse however, is not true. A simple example is the following:Example 6.1 Consider the polynomial ring A = k[t1, t2, . . . ] in countably manyvariables ti and mod out by the square m2 of the maximal ideal generated by thevariables, m = (t1, t2, . . . ). The resulting ring A has just one prime ideal, the onegenerated by the ti’s. So Spec A has just one point, and hence is a Noetherian

noetherian schemes 124

space. The ring A, however, is clearly not Noetherian; the sole prime idealrequires infinitely many generators, namely all the ti’s. K

In light of this example, we take a different route to define Noetherianschemes:

Definition 6.2 i) A scheme is locally Noetherian if it can be coveredby open affine subsets Spec Ai where each Ai is a Noetherian ring.

ii) A scheme is Noetherian if it is both locally Noetherian and quasi-compact.

Recall from Chapter 3 that a scheme X is quasi-compact if every open cover ofX has a finite subcover. We also showed that affine schemes were quasi-compact:Any open covering can be refined to a covering by distinguished open sets D( fi),and when Spec A =

Ť

i D( fi), the ideal generated by the fi’s contains 1, and thefinitely many D( fi)’s with fi occurring in an expansion of 1, will do.

From the definition, it follows that a general scheme is Noetherian if andonly if it can be covered by finitely many open affines Spec Ai where each Ai isNoetherian.

In fact, with the new definition, we now have

Proposition 6.3 Spec A is Noetherian (as a scheme) if and only if A is Noethe-rian.

We can see this as a purely algebraic fact: Refining the cover, we may assumethat each Ai = A fi . By a theorem in Commutative Algebra [?], a ring A isNoetherian provided that each localization A fi is Noetherian and 1 P ( f1, . . . , fr).

Proposition 6.4 If X is a Noetherian scheme, then its underlying topologicalspace is Noetherian.

Proof: Since X is quasi-compact it may be covered by a finite number of openaffine subsets, and since a descending chain stabilizes if the intersection witheach of those open sets stabilizes, we reduce to showing the proposition forX = Spec A with A a Noetherian ring, which is clear. o

Proposition 6.5 Let X be a (locally) Noetherian scheme. Then any closed oropen subscheme of X is also (locally) Noetherian.

Proof: Without loss of generality, we may assume that X is Noetherian. LettXiuiPI be a finite open cover with Xi = Spec Ai and assume that each Ai isNoetherian. Let Y Ă X be an open or closed subscheme. We will show that eachY X Xi is Noetherian. In particular, since Y X Xi is a closed subscheme of an

noetherian schemes 125

affine scheme, we reduce to considering the case where X = Spec A where A isNoetherian.

First case, Y is open. Then there are elements f1, . . . , fn P A such thatY =

Ťni=1 D( fi) =

Ťni=1 Spec(A fi). If A is Noetherian, then so is each of the

localizations A fi , so Y is also Noetherian.Second case, Y is closed. Then Y = V(a) for some ideal s Ă A. If A is

Noetherian, then so is Spec(A/a), and again Y is Noetherian. o

Example 6.6 All of the examples from Chapter 5 are Noetherian. This followsbecause they are glued together by finitely many schemes of the form Spec Rwhere R is a Noetherian ring. K

Example 6.7 Let k be a field. The following schemes are not Noetherian:

i)š8

i=1 A1k ;

ii) SpecÀ8

i=1 k[x];iii) Spec

ś8i=1 k[x],

where the union is the disjoint union. Indeed, the disjoint unionš8

i=1 A1k

is not quasi-compact (thus not affine). The latter two are affine (thus quasi-compact), but non-isomorphic, because their rings of global sections OX(X) arenon-isomorphic.

The ideal structure of infinite products of rings can in fact be very compli-cated. For instance Spec

ś

iPN k is described by all ultrafilters on N (if you knowwhat such exotic creatures are); anyhow, it is a compact, totally disconnectedand Hausdorff(!) topological space. It is called the Stone–Cech compactification ofN. K

Example 6.8 In Example 2 on page 110 we worked with a finite set of primes,but the hypothesises of the gluing theorem impose no restrictions on the numberof schemes to be glued together, and we are free to take P infinite, for examplewe can use the set P of all primes! The glued scheme XP is a peculiar animal:it is neither affine nor Noetherian, but it is locally Noetherian. There is a mapφ : XP Ñ Spec Z which is bijective and continuous, but not a homeomorphism,and it has the property that for all open subsets UĎ Spec Z the map induced onsections φ7 : Γ(U,OSpec Z)Ñ Γ(φ´1U,OXP ) is an isomorphism, in other words,φ7 : OSpec Z Ñ φ˚(OXP ) is an isomorphism!

As before we construct the scheme XP by gluing the different Spec Z(p)’stogether along the generic points. However, when computing the global sec-tions, we see things changing. The kernel of ρ is still

Ş

pPP Z(p), but now thisintersection equals Z: indeed, a rational number α = a/b lies in Z(p) preciselywhen the denominator b does not have p as factor, so lying in all Z(p), meansthat b has no non-trivial prime-factor. That is, b = ˘1, and hence α P Z.

There is a morphism XP Ñ Spec Z which one may think about as follows.Each of the schemes Spec Z(p) maps in a natural way into Spec Z, the mapping

the dimension of a scheme 126

being induced by the inclusions ZĎZ(p). The generic points of the Spec Zp’sare all being mapped to the generic point of Spec Z. Hence they patch togetherto give a map XP Ñ Spec Z. This is a continuous bijection by construction, butit is not a homeomorphism: indeed, the subsets Spec Z(p) are open in XP by thegluing construction, but they are not open in Spec Z, since their complementsare infinite, and the closed sets in Spec Z are just the finite sets of maximalideals.

The topology of the scheme XP is not Noetherian since the subschemesSpec Z(p) form an open cover that obviously can not be reduced to a finitecover. However, it is locally Noetherian, as the open subschemes Spec Z(p) areNoetherian. The sets Up = XPzt(p)u map bijectively to D(p)Ď Spec Z andΓ(Up,OXP ) = Zp, but Up and D(p) are not isomorphic. K

6.2 The dimension of a scheme

Recall that the Krull dimension of a ring A is defined as the supremum of thelength of all chains of prime ideals in A. For a scheme, we make the followingsimilar definition, which works for any topological space.

Definition 6.9 Let X be a topological space. The dimension of X is the supre-mum of all integers n such that there exists a chain

Y0 Ă Y1 Ă ¨ ¨ ¨ Ă Yn

of distinct irreducible closed subsets of X.

Note that this supremum might not be a finite number, in which case wesay that dim X = 8. If X is a scheme, we define the dimension of X as thedimension of the underlying topological space. In particular, dim X = dim Xred.

Here are a few basic properties of the dimension:

Proposition 6.10 i) If Y Ă X is any subset, then dim Y ď dim X.ii) If X is covered by open sets Ui, then dim X = supi dim Ui.

iii) If X is covered by closed sets Zi, then dim X = supi dim Zi.iv) If dim X is finite, and Y Ă X is a closed irreducible subset such that

dim Y = dim X, then Y = X.

In the case where X = Spec A is affine, we know that the closed irreduciblesubsets are of the form V(p) where p is a prime ideal. Using this observationwe find

Proposition 6.11 The dimension of X = Spec A equals the Krull dimensionof A.


Example 6.12

i) dim Spec Z is 1. All maximal chains have the form V(p) Ă V(0) =

Spec Z.

ii) dim Spec(k[ε]/ε2) = dim Spec k = 0.

iii) The dimension of AnA = Spec A[x1, . . . , xn] is n + dim A when A is a

Noetherian ring (for general rings A, dim AnA takes values between

dim A + n and dim A + 2n, and all values are possible). In particular,when A = k is a field, An

k has dimension n. A maximal chain isV(x1) Ą V(x1, x2) Ą ¨ ¨ ¨ Ą V(x1, . . . , xn). For A1

Z, the dimension is 2,and a maximal chain is given by ( f (x), p) sup(p) sup(0), where f (x)is an irreducible polynomial, and p is a prime number.

K

Having finite dimension does not guarantee that the scheme is Noetherian(see Example 1). More seriously, there are even Noetherian rings whose Krulldimension is infinite. The first example was constructed by Masayoshi Nagata,the great master of counterexamples in algebra. Although each maximal chainof prime ideals in a Noetherian ring will be of finite length, since the primeideals satisfy the descending chain condition, there can be arbitrary long ones(see Problem 6.5).Example 6.13 Zero-dimensional schemes. The schemes Spec Z/p, Spec C/xn andSpec C[x, y]/(x2, xy, y3) have dimension 0. More generally, the spectrum of anArtinian ring has dimension 0 (and in the case where A is Noetherian, Spec Ahas dimension 0 if and only if A is Artinian). However, there are non-Noetherianrings, e.g.,

ś8i=1 F2 which have dimension 0 and even infinitely many points. K

Exercise 6.1 Show that A =ś8

i=1 Z/2 has dimension 0, but it is not Noetherian.M

Exercise 6.2 The ring A =ś8

n=1 Z/2nZ has infinite Krull dimension 1. Showthat Spec A is still Noetherian as a topological space. M

CodimensionFor a closed subset Y Ă X, we can define dim Y and dim X using closed subsetscontained in Y and X respectively. There is also a relative notion, the codimensionof Y inside X, defined in terms of closed subsets of X containing Y. Thesethree numbers will in many important cases be related by the relationshipdim Y + codim Y = dim X (although this formula does not hold in general).

1This was shown in R. Gilmer, W. Heinzer, Products of commutative rings and zero-dimensionality. Trans.Amer. Math. Soc. 331 (1992), 663–680.


Definition 6.14 Let Y Ď X be a closed subset of X. We define the codimen-sion of Y as the supremum of all integers n such that there exists a chain

Y = Y0 Ă Y1 Ă ¨ ¨ ¨ Ă Yn

of distinct irreducible closed subsets of X.

By the correspondence between closed subsets and prime ideals, the codi-mension of the closed subset V(p) in Spec A is the height of the prime p in A,that is, the maximal length of a chain of prime ideals =p0Ď p1Ď ¨ ¨ ¨ Ď pr, orequivalently, dim Ap.

Proposition 6.15 Let X be a scheme. Let x P X be a point and set Z = txu.Then dimOX,x = codim(Z, X).

Proof: Take a chain Z Ă Z1 Ă . . . Zn of distinct irreducible closed subsets .Then for any open neighborhood U of x the generic points η1, . . . , ηn of the Zi’sare contained in U. Thus if U = Spec A is an affine open neigborhood of x, thenthe generic points correspoind to prime ideals px Ą p1 Ą ¨ ¨ ¨ Ą pn in A. Takingthe supremum gives the claim. o

Dimension theory for schemes of finite type over a fieldOne should have in mind that codimension can be counterintuitive even forNoetherian schemes; for instance, there are Noetherian affine schemes of anydimension with closed points being of codimension one; we shall see a two-dimensional one in Proposition 24.12.

For integral schemes of finite type over fields however, the theory is simpler,and we can study the dimension in terms of the function field: Recall that the

transcendencedegree of a fieldextension K/k is themaximal number ofelements of K whichare algebraicallyindependent.

Theorem 6.16 Let X be an integral scheme of finite type over a field k, withfunction field K. Then

i) The dimension dim X equals the trancendence degree of K over k (inparticular, dim X ă 8);

ii) For each U Ď X open, dim U = dim X;iii) If Y Ă X is a closed subset, then codim Y = inftdimOX,p|p P Yu

anddim Y + codim Y = dim X.

In particular, for a closed point p P X, dim X = dimOX,p.

Proof: To prove i) we may assume that X = Spec A is affine. The hypothesison X gives that A is a finitely generated k-algebra with quotient field K. In this


case, i) is a consequence of the Noether normalization lemma (see Chapter 11 ofAtiyah–MacDonald).

Note that statement ii) follows from i) since X and an open subset U havethe same function field.

To prove iii), we may again assume that X = Spec A, and use the formula

dim A/p+ ht p = dim A

which holds for prime ideals in finitely generated k-algebras. o

Example 6.17 The scheme Pnk satisfies the conditions of the theorem. It’s

dimension is n, which follows because Pnk contains An

k as an open dense subset,and An

k has dimension n. K

Example 6.18 The quadric cone Q = Spec k[x, y, z]/(x2 ´ yz) of Example 16 onpage 87 has dimension 2. This follows because the function field K = k(Q) isisomorphic to k(x, y), which has transcendence degree 2 over k. Alternatively,we can use the morphism A2

k Ñ Q which are isomorphisms over an open setU Ă A2

k (which thus also has dimension 2). K

Example 6.19 It’s important to note that the formula dim Y + codim Y = dim Xdoes not always hold, even if X is the spectrum of a very nice ring. Indeed, letX = Spec A where A = R[t] and R is any dvr with generator t of the maximalideal (for instance, the localization R = k[t](t)). The prime p = (tu ´ 1) hasheight one, but A/p » R[1/t] is a field, hence of dimension zero. However,dim A = dim R + 1 = 2. K

For schemes which aren’t integral but still of finite type, we still have agood control over the dimension. First of all, the dimension of X is the sameas of Xred, so we may assume that X is reduced. Then, if X =

Ť

Xi is thedecomposition into irreducible components, we have that Xi is integral, anddim X is the supremum of all dim Xi.Example 6.20 Consider X = A3

k = Spec k[x, y, z] and Y = V(a) where a is theideal

a =(

x y´ x, x2, y2z´ z, y3 ´ y, x y2 ´ x y)= (z, y, x)X

(y´ 1, x2)X (y + 1, x)

The associated primes of a are p1 = (x, y + 1), p2 = (x, y´ 1) and p3 = (x, y, z).So Y has three components: L = V(x, y + 1), M = V(x, y ´ 1) (two lines),and p = V(x, y, z) (the origin). The dimension of Y equals the largest ofthe dimension of each component, and dim L = 1, dim M = 1, dim p = 0, sodim Y = 1. The codimension of Y in X equals the maximum of the heights ofthe associated primes of a, i.e. ht(p1) = 2. So the codimension of Y equals 2. K

Exercise 6.3 (The dimension of a product.) For two integral schemes of finiteˇ

type over k, show that

dim Xˆk Y = dim X + dim Y.


M

Exercise 6.4 (A polynomial ring of excess dimension.) The aim of this exercise isto exhibit a one-dimensional local ring R such that the Krull dimension of thepolynomial ring R[T] equals three; that is dim R[T] = dim R + 2.The ring R is no more exotic than the ring of rational functions f (x, y) in twovariables over an algebraically closed field k that are defined and constant on they-axis. The elements of R, when written in lowest terms, have a denominatornot divisible by x, and f (0, y), which is then meaningful, lies in k. The examplewas originally constructed by Krull as an example of a non-Noetherian domainwith just one non-zero prime ideal.

a) Show that the ideals ar = (x, xy´1, . . . , xy´r) with r P N form an acsendingchain that does not stabilize. Conclude that R is not Noetherian.

b) Show that R is local with the set m of elements f P R that vanish along they-axis as the maximal ideal.

c) * Prove that there are no other primes than m and (0) in R. Hint: Showfirst that any element in R is of the form xiyjα where i ě 0, j P Z and α is aunit in R.

d) Prove that q = t F(T) P R[T] | F(y) = 0 u is a non-zero prime ideal strictlycontained in mR[T]. Conclude that dim R[T] ě 3. Hint: Consider the chain(0) Ă q Ă mR[T] Ă mR[T] + (T).

M

Exercise 6.5 (Nagata’s example.) Let B = k[x1, x2, . . . ] be the polynomial ring incountably many variables, and decompose the set of natural numbers into aunion N =

Ť

i Ji of disjoint finite sets Ji whose cardinality tends to infinity withi. Any such partition will do, but a specific example can be the following. Thefirst set J1 has 1 as its sole element, J2 concists of the next two integers, J3 of thenext three etc.

Roger Godement(1927–2008)

Let ni be the ideal in B generated by the xj’s for which j P Ji, and let S be themultiplicative closed subset

Ş

i Bzni of B; i.e. the set of elements from B not lyingin any of the ni’s. Nagata’s example is the localized ring A = S´1B. The aim ofthe exercise is to prove that A is Noetherian, but of infinite Krull dimension. Welet mi denote ni A; the ideal in A generated by the xj’s with j P Ji.

We shall need the rational function field Ki = k(xj|j R Ji) in the variables xj

whose index does not lie in Ji, and the polynomial ring Ki[xj|j P Ji] over Ki inthe remaining variables; that is, those xj for which j P Ji. Moreover, the ideal ai

will be the ideal in Ki[xj|j P Ii] generated by the latter; that is, ai = (xj|j P Ji).

a) Show that Bni » Ki[xj|j P Ii]ai .

b) Prove that Ami= Bni and conclude that each local ring Ami is Noetherian

with dim Ami = #Ji and hence that dim A = 8.

normal schemes and normalization 131

c) * Show that A is Noetherian. Hint: Any ideal is contained in finitely manyof the of the mi’s, and therefore finitely generated.

M

6.3 Normal schemes and normalization

Recall that a ring is called normal if each of it’s localizations Ap is integrallyclosed in their fraction field. We will be mostly interested in the case where Ais an integral domain; in this case the condition is equivalent to A itself beingintegrally closed in its fraction field K. Motivated by all the desirable algebraicproperties of such rings, we make the following definition:

Definition 6.21 Let X be an integral scheme with fraction field K. We saythat X is normal at a point x P X if the ring OX,x is integrally closed (viewedas a subring of K).

Example 6.22 Ank and Pn

k are normal schemes, because the local rings isisomorphic to k[x1, . . . , xn](x1,...,xn) which is a localization of an UFD, hencenormal. K

Example 6.23 More generally, a scheme which is locally factorial (meaning thatall stalks OX,x are ufd’s), is also normal. [CA notes chapter 7]. In particular,all regular schemes are normal. (This follows from the algebraic fact that localregular rings are UFDs (Atiyah–MacDonald)). K

For an integral scheme X, we will define a new scheme X which is a normalscheme, and a morphism π : X Ñ X. There are many schemes with thisproperty (take Spec K Ñ X for instance), so to get something more canonical,we want X and π to satisfy a certain universal property.

Dominant mapsWe say that a morphism of schemes f : X Ñ Y is dominant if the image of fis dense in Y. When X and Y are integral, this is equivalent to saying that thegeneric point of X maps to the generic point of Y. This means the f 7 induces amap between the stalks f 7 : OY,ε Ñ OX,η where η and ε are the generic pointsin X and Y. But the stalks at the generic points are the function fields k(X) andk(Y); hence we obtain a map φ7 : k(Y) Ñ k(X), which is injective as any ringmap between fields is.

Lemma 6.24 Let f : X Ñ Y be a morphism of integral schemes. Then the following areequivalent:

i) f is dominant;ii) For all affine open sets U Ă X, V Ă Y with f (U) Ă V, the ring map

OY(V)Ñ OX(U) is injective;


iii) For one affine open set U Ă X, V Ă Y with f (U) Ă V, the ring mapOY(V)Ñ OX(U) is injective

iv) For all x P X, the local homomorphism f 7x : OY, f (x) Ñ OX,x is injective;

v) For one x P X, the local homomorphism f 7x : OY, f (x) Ñ OX,x is injective.

Proof: i) ô ii): We may assume that U = X = Spec B and V = Y = Spec A,and that f is induced by a map φ : A Ñ B of integral domains. We see that fmaps the generic point to the generic point if and only if φ´1(0) = (0) whichholds true if and only if φ is injective.

The equivalence of the remaining implications follows by a similar argumentand is left to the reader. o

X

Y X

π

g

hTheorem 6.25 Let X be an integral scheme, then there is a normal scheme X,and a morphism π : X Ñ X satisfying the following universal property: Forany dominant morphism g : Y Ñ X from a normal scheme Y, there is a uniquemorphism h : Y Ñ X such that g = π ˝ h.

Proof: The uniqueness part follows from the universal property. We thereforeonly need to check the existence.

Suppose first that X = Spec A is affine. Let A1 be the normalization of A inthe fraction field K.

Let Y be a normal scheme and let B = OY(Y). For a dominant morphismg : Y Ñ X, the map g7(X) : A Ñ B is injective, so it factors through a uniquemorphism A Ñ A1 Ñ B, by the universal property of normalization of rings.Hence g factors via a unique morphism g1 : Y Ñ Spec A1. In particular, thecanonical map π : Spec A1 Ñ Spec A satisfies the universal property in thetheorem.

Now let X be an arbitrary integral scheme, and let Ui = Spec Ai be an affinecover. Note that there are normalization morphisms πi : U1

i Ñ Ui defined by theinclusions Ai Ă A1i. Consider the open set Uij = Ui XUj, which is an open set inboth Ui and Uj. As πi|π´1(Uij)

: π´1(Uij) Ñ Uij and πj|π´1(Uij): π´1(Uij) Ñ Uij

are both normalizations of Uij, they must coincide by the uniqueness. Henceby the Gluing lemma for morphisms, the morphisms πi glue, so we obtain ascheme X1 and a morphism π : X1 Ñ X. o

The X-scheme X is called the normalization of X.

Corollary 6.26 The normalization X has the following properties:

i) π : X Ñ X is surjective;ii) There is an open subset U Ă X so that π restricted to π´1(U) is an

isomorphism;iii) X and X have the same dimension;iv) If X is of finite type over a field, then π : X Ñ X is a finite morphism.


Proof: The proof relies on some of the basic properties of the integral closure.Statement i) follows from the Going-Up theorem (or the Lying-Over theorem).Statement ii) holds true because being normal is a generic property; that is, for a

finitely generated integral domain A, the localization Ap is normal for all p P Uin a non-empty open subset U.

Statement iii) follows from the Going-Up theorem.Finally, statement iv) follows from the fact that if A is an integral domain

which is finitely generated over a field, then the normalization rA in the fractionfield K of A is a finite A-module. (This statement is essentially a consequence ofNoether’s normalization lemma.) o

In general, the normalization map π : X Ñ X need not be finite in the sense ofDefinition 3.15 on page 83. Nagata found the first example of a local Noetherianintegral domain A such that the integral closure is not Noetherian (in particularnot finite over A). See also Exercise 12.10 in [CA].

We perform normalization because normal schemes have better propertiesthan arbitrary ones. For example in normal varieties regular functions definedoutside a closed subvariety of codimension ě 2 can be extended to regularfunctions defined everywhere ("Algebraic Hartogs theorem") . For instance, acurve is non-singular if and only if it is normal, so that normalization is thesame as desingularization for curves.Example 6.27 Cuspidal cubic. Let k be a field, and let X = Spec A whereA = k[x, y]/(y2 ´ x3). This is the cuspidal cubic curve in A2

k .

There is an isomorphism of k-algebras A k[t2, t3]» given by sending

x ÞÑ t2 and y ÞÑ t3. It is clear that k[t2, t3] is an integral domain with fractionfield K = k(t). Moreover, the normalization of A equals A = k[t]. The inclusionA Ă A induces the normalization morphism π : A1

k Ñ X, and this is anisomorphism over the open set D(t) Ă A1

k where t is inverible. K

Example 6.28 Nodal cubic. Let now X = Spec A with A being the ring A =

k[x, y]/(y2 ´ x3 ´ x2), where k now is a field whose characteristic is not two (ifthe characteristic is two, we are back in previous cuspidal case). This is the nodalcubic curve in A2

k . Here it is a little bit tricker to find the normalization, but ithelps to think about it geometrically.

If we think of the corresponding affine variety t (x, y) | y2 = x3 + x2 u Ă

A2(k), we see that the origin (0, 0) is a special point: a line l Ă A2k through the

closed point (0, 0) P X (with equation y = tx) will intersect X at (0, 0) and atone more point (with x = t2 ´ 1), and this gives a parameterization of the curve,which is generically one-to-one.

Back in the scheme world, we imitate this by introducing the parametert = yx´1 in the function field K of X, the equation y2 = x3 ´ x2 then reduces tot2 = 1 + x after being divided by x2. Moreover, the element t is integral, since it


satisfies the monic equation T2 ´ x´ 1 = 0 (which has coefficients in A). Sincex = t2 ´ 1 and y = x ¨ y/x = t3 ´ t, we see that

A = k[t2 ´ 1, t3 ´ t]Ď k[t]ĎK = k(t),

and since k[t] is integrally closed, any element in K which is integral over A,can be written as a polynomial in t. So A = k[t] is the integral closure of A ink(t). The normalization map π : Spec A Ñ Spec A is an isomorphism outsidethe origin (0, 0) P X. Geometrically the map π identifies two points (t + 1) and(t´ 1) in A1

k to the origin in X.K

Example 6.29 The quadratic cone. Consider the affine scheme X = Spec A whereA = C[x, y, z]/(xy´ z2). Note that this is not a factorial scheme (A is not a ufd

as xy = z2 and x, y and z all are irreducible elements), so we cannot immediatelyconclude that A is normal. However, there are a few ways to see that it is in factso:

i) There is an isomorphism of rings

φ : A Ñ C[u2, uv, v2],

and the latter algebra is normal in k(u, v).ii) Let B = C[x, y], so that A = B[z]/(z2 ´ xy). Then B Ă A is a ring

extension making A into a finite B-module, in fact, it is a free moduleof rank two with basis 1, z. We get an inclusion of fields K(B) =

C(x, y) Ă K(A) obtained by adjoining the element z, which equals thesquare root

?xy, to C(x, y). Each element of K(A) can be written asw = u + vz where u, v P C(x, y). If this element is integral over A, itwill also be integral over B. In fact, w satisfies the minimal polynomial

T2 ´ 2uT + (u2 ´ xyv2) = 0,

and if this is a polynomial integral over B, we must have 2u P C[x, y],and hence u P C[x, y]. Moreover, it ensues that the element (u2´ xyv2)

belongs to C[x, y], so also that xyv2 P C[x, y], from which it readilyfollows that v P C[x, y] since a potential denominator of v2 will be asquare and can therefore not be neutralized by the non-square elementxy.

K

Example 6.30

Here is an example of a non-normal surface with an isolated singularity. Welet X be the scheme obtained by identifying two points in A2

k ; X is the affinevariety given by the k-algebra

A = t f P k[x, y] | f (0, 0) = f (0, 1)u.


Then the normalization X is the affine plane. K

Exercise 6.6 Let k be a field and consider the k-algebra

A = k[x, xy, y2, y3] Ă k[x, y]

Show that X = Spec(A) is a surface with the maximal ideal m = xx, xy, y2, y3y Ă

A the unique point x P X so that OX,x is not normal. M

Exercise 6.7 (A consequence of Noether’s normalization lemma.) Let X = Spec Aˇ

be an affine scheme of dimension n, of finite type over a field k.(i) Show that there is a finite morphism

X Ñ Ank

Such a morphism is called a Noether normalization of X.(ii) Find a Noether normalization for the scheme

X = Spec C[x, y, z]/I

where I = (xy´ z2, x2y´ xy3 + z4 ´ 1). M

Chapter 7

Fibre products

7.1 Introduction

From the theory of varieties we know that we can construct the Cartesianproduct X ˆ Y of two varieties X and Y. If X = Z( f1, . . . , fs) Ă An(k) andY = Z(g1, . . . , gt) Ă Am(k) are two affine varieties, then their product XˆY isthe affine variety Z( f1, . . . , fs, g1, . . . , gt) Ă Am+n(k), and departing from this,the general case is handled by a gluing process.

In this chapter, we will consider a vast generalization of this construction. Forany scheme S and any two S-schemes X Ñ S and Y Ñ S we will construct a newscheme, denoted XˆS Y, equipped with projection morphisms πX : XˆS Y Ñ Xand πY : X ˆS Y Ñ Y satisfying a certain universal property. The aim of thischapter is to prove the fundamental theorem (which certainly is ‘Cartesian’ inspirit): Fibre products of schemes exist.

The fact that all fibre products exist is one of the most important propertiesof the category of schemes, and one can argue that this is the definitive reasonfor transitioning from varieties to schemes: the fibre product of two varieties isin general not a variety, but it is a scheme.

The general fibre product is moreover extremely useful in many situationsand takes on astonishingly versatile roles. At the end of the chapter we shallexplain some of the various contexts where fibre products appear, including basechange and scheme theoretic fibres. We begin the chapter by recalling the definitionof the fibre product of sets, then transition into a very general situation todiscuss fibre products in general categories, and then finally, return to thecontext of schemes. The strategy for proving existence will be similar to whatone does for varieties; one constructs the fibre product first when X, Y and S areaffine schemes, and subsequently by using several gluing constructions shows itexists in general. The majority of the chapter will be devoted to going throughthe steps of this gluing procedure. Towards the end, we will treat the mainapplications and see a series of examples.

introduction 137

Fibre products of sets.As a warming up, we recall the fibre product in the category of sets, Sets. Thepoints of departure is two sets X and Y both equipped with a map to a third setS; i.e. we are given a diagram

X Y

SφX φY

The fibre product XˆS Y is the subset of the cartesian product XˆY consistingof the pairs whose two components have the same image in S; that is,

XˆS Y = t (x, y) | φX(x) = φY(y) u.

Clearly the diagram below where πX and πY denote the restrictions of the twoprojections to the fibre product (in other words, πX(x, y) = x and πY(x, y) = y),is commutative.

XˆS Y

X Y

S

πYπX

φX φY

(7.1)

And more is true: the fibre product enjoys a universal property. Given any twomaps ψX : Z Ñ X and ψY : Z Ñ Y such that φX ˝ ψX = φY ˝ ψY there is a uniquemap ψ : Z Ñ XˆS Y satisfying πX ˝ ψ = ψX and πY ˝ ψ = ψY. Defining such aψ is easy, just use the map whose two components are ψX and ψY and observethat it takes values in XˆS Y since the relation φX ˝ ψX = φY ˝ ψY holds.

XˆS Y

X Z Y

S

πYπX

φX

ψX ψY

φY

(7.2)

Giving the two ψ’s is to give a commutative diagram like (7.1) above with Zreplacing the product X ˆS Y (as in the lower part of (7.2)), and the universalproperty is to say that 7.1 is universal, or in categorical terms, final among suchdiagrams. One also says that (7.1) is a Cartesian diagram

(Kartesisk diagram)Cartesian diagram or a Cartesian square.

Speaking of names, the reason for the name ‘fibre product’ is that the fibres ofthe map XˆS Y Ñ S are the product of the fibres of the two maps X Ñ S andY Ñ S.Exercise 7.1 Show that if Y is a subset of S, then the fibre product XˆS Y equalsthe preimage φ´1

X (Y). More strikingly: assume that also X is a subset of S, show

introduction 138

that the fibre product‚ X ˆS Y, then will be equal to the intersection X XY. IfS is reduced to a singleton, show that X ˆS Y is just the good old Cartesianproduct XˆY. M

The fibre product in general categoriesThe notion of a fibre product formulated as the solution to a universal problemas above, is mutatis mutandis meaningful in any category C. Given any twoarrows φX : X Ñ S and φY : Y Ñ S in the category C, an object, which we shalldenote X ˆS Y, is said to be a Fibre product

(Fiberproduktet)fibre product of the objects X and Y, or more

precisely of the two arrows φX and φY, if the following two conditions arefulfilled:

o There are two arrows πX : X ˆS Y Ñ X and πY : X ˆS Y Ñ Y in C suchthat φX ˝ πX = φY ˝ πY (called the projections);

o For any two arrows ψX : Z Ñ X and ψY : Z Ñ Y in C such that φX ˝ ψX =

φY ˝ ψY, there is a unique arrow ψ : Z Ñ X ˆS Y satisfying πX ˝ ψ = ψX

and πY ˝ ψ = ψY.

These properties may naturally be expressed through commutative diagrams,identical to the ones used for sets in the previous section, and the notions of aCartesian diagram and Cartesian squares are carried over to any category.

The two arrows πX ˝ ψ and πY ˝ ψ that determine the arrow ψ : Z Ñ XˆS Y,are called Components

(Komponentene)the components of φ, and the notation ψ = (ψX, ψY) is sometimes used.

If ψX : X1 Ñ X and ψY : Y1 Ñ Y are two arrows over S, there is a unique arrowdenoted ψX ˆ ψY from X1 ˆS Y1 to XˆS Y whose components are ψX ˝ πX1 andψY ˝ πY1 .

If the fibre product exists, it is unique up to a unique isomorphism, as is truefor solutions to any universal problem. However, it is a good exercise to checkthis in detail in this specific situation.Exercise 7.2 Show that if the fibre product exists in the category C, it is uniqueup to a unique isomorphism. M

It is not so hard to come up with examples of categories where fibre productsdo not exist. For instance, consider the funny category C where the objects aresubsets X of the integers with an even number of elements, and the morphismsgiven by inclusions Y Ă X. In this category, the fibre product of Y Ă X andZ Ă X over X would be YX Z Ă X. However, YX Z does not necessarily havean even number of elements!

What is of course much more disappointing, is that fibre products fail toexist in our good old categories like manifolds or affine varieties. This shows yetanother reason why we need to make the transition from varieties to schemes.Exercise 7.3

fibre products of schemes 139

a) Give an example showing that the fibre product does not always exist in thecategory of manifolds;

b) Give an example showing that the fibre product does not always exist in thecategory of affine varieties.

M

Exercise 7.4 Let C be a category and X, Y and S three object from S. Convinceyourself that the simple product of two objects in C/S equals their fibre productin C. Show that one formally may add a final object ˚ to any category C andthat the fibre product Xˆ˚ Y exists in the extended category if and only if thesimple product XˆY exists in C, and in case they exist, they are equal. M

7.2 Fibre products of schemes

It is a fundamental property of schemes that their fibre products exist uncon-ditionally, and most of this chapter is devoted to a proof of this. It consists ofa rather long sequence of reductions basically to the affine case — where thespectrum of the tensor product provides a product — and the glueing techniquesdeveloped in Chapter 4. We shall prove

Theorem 7.1 (Existence of fibre products) Let X Ñ S and Y Ñ S betwo schemes over the scheme S. Then their fibre product XˆS Y exists.

The projections onto X and Y will frequently be denoted by respectively πX

and πY. We will see several examples later which show that the underlyingset of a product can be very different from the product of the underlying setsof X and Y. However, the ‘scheme-valued points’ behave well; that is, for anyS-scheme T Ñ S, there is a canonical isomorphism of sets of T-points

HomSch/S(T, XˆS Y) » HomSch/S(T, X)ˆHomSch/S(T, Y).

This is just another way of formulating the universal property of a product.

Products of affine schemesThe category AffSch of affine schemes is, more or less by definition, equivalentto the category of rings, and in the category of rings we have the tensor product.The tensor product enjoys a universal property dual to the one of the fibreproduct. To be precise, assume A1 and A2 are B-algebras, i.e. we have two mapsof rings αi

A1 A2

Bα2α1


There are two maps βi : Ai Ñ A1bB A2 sending a1 P A1 to a1b 1 and a2 to 1b a2,respectively. These are both ring homomorphisms, since aa1b 1 = (ab 1)(a1b 1)respectively 1b aa1 = (1b a)(1b a1), and they fit into the commutative diagram

A1bB A2

A1 A2

B.

β1 β2

α2α1

(7.3)

because α1(b)b 1 = 1b α2(b) by the definition of the tensor product A1bB A2

(this is the significance of the tensor product being taken over B; one can moveelements in B from one side of the b-glyph to the other).

Moreover, the tensor product is universal in this respect. Indeed, assume thatγi : Ai Ñ C are B-algebra homomorphisms, i.e. γ1 ˝ α1 = γ2 ˝ α2; or phraseddifferently, they fit into the commutative diagram analogous to (7.3) with theβi’s replaced by the γi’s. The association a1b a2 Ñ γ1(a1)γ(a2) is B-bilinear andhence extends to a B-algebra homomorphism γ : A1bB A2 Ñ C, which obviouslyhas the property γ ˝ βi = γi. Expressed diagrammatically, the universal propertyappears as

A1bB A2

A1 C A2

B

β1

γ1 γ2

β2

α2α1

(7.4)

Transforming all this into geometry by applying the Spec-functor, we arrivea the diagram

Spec(A1bB A2)

Spec A1 Spec A2

Spec B,

π1 π2

(7.5)

and the affine scheme Spec(A1bB A2) enjoys the property of being universalamong affine schemes sitting in a diagram like (7.5). Hence Spec(A1bB A2)

equipped with the two projections π1 and π2 serves as the fibre product in thecategory AffSch of affine schemes. One even has the stronger statement; it is thefibre product in the larger category Sch of schemes.


Proposition 7.2 Given morphisms φi : Spec Ai Ñ Spec B for i = 1, 2. Thenthe spectrum Spec(A1bB A2) with the two projection π1 and π2 defined asabove, is the fibre product of the Spec Ai’s in the category of schemes.

Unravelled, the conclusion of the proposition reads: if Z is any scheme andψi : Z Ñ Spec Ai are morphisms with φ1 ˝ ψ1 = φ2 ˝ ψ2, there exists a uniquemorphism ψ : Z Ñ Spec(A1bB A2) such that πi ˝ ψ = ψi for i = 1, 2.Proof: We know that the proposition is true whenever Z is an affine scheme;so the subtle point is that Z may not necessarily be affine. For short, we letX = Spec(A1bB A2).

The proof is just an application of the gluing lemma for morphisms. Onecovers Z by open affines Uα and covers the intersections Uαβ = Uα XUβ by openaffine subsets Uαβγ as well. By the affine case of the proposition, for each Uα weget a map ψα : Uα Ñ X, such that πi ˝ ψα = ψi|Uα . By the uniqueness part of theaffine case, these maps must coincide on the open affines Uαβγ, and therefore onthe intersections Uαβ. They can thus be patched together to a map ψ : Z Ñ X,which is unique since the ψα’s are unique. o

A preliminary lemmaRecall that any open subset U of a scheme Y has a canonically defined schemestructure as an open subscheme (the structure sheaf is defined as OY|U). Hence,if f is any morphism f : X Ñ Y, the inverse image f´1(U) is in a natural wayan open subscheme of X. The following lemma will turn out to be useful:

Lemma 7.3 If X ˆS Y exists and UĎX is an open subscheme, then U ˆS Y existsand is canonically isomorphic to the open subscheme π´1

X (U) with the two restrictionsπY|π´1

X (U) and πX|π´1X (U) as projections.

Proof: The situation is displayed in the following diagram

Y

Z π´1X (U) XˆS Y

U X,

ψY

ψUπX

πY

and we are to verify that π´1X (U) together with the restriction of the two pro-

jections satisfy the universal property. If Z is a scheme and ψU : Z Ñ U andψY : Z Ñ Y are two morphisms over S, we may consider ψU as a map into X,and therefore they induce a map of schemes ψ : Z Ñ XˆS Y with ψX = πX ˝ ψ

and ψY = πY ˝ ψ. Clearly πX ˝ ψ = ψU takes values in U and therefore ψ takes


values in π´1X (U). It follows immediately that ψ is unique (see the Exercise

below), and we are through. o

When identifying π´1X (U) with UˆS Y, the inclusion map π´1

X (U)ĎXˆS Y willcorrespond to the map ιˆ idY where ι : U Ñ X is the inclusion, so anticipatingthe notion of base change, one may reformulate the lemma by saying that openimmersions stay open immersions under base change (see Proposition 7.7 onpage 149).Exercise 7.5 Assume that UĎX is an open subscheme and let ι : U Ñ X be theinclusion map. Let φ and ψ be two maps of schemes from a scheme Z to U andassume that ι ˝ φ = ι ˝ ψ. Then φ = ψ. M

The basic patchingThe following proposition will lay at the ground for all the gluing necessary forthe construction of a fibre product:

Proposition 7.4 Let φX : X Ñ S and φY : Y Ñ S be two maps of schemes,and assume that there is an open covering tUiuiPI of X such that Ui ˆS Y existfor all i P I. Then XˆS Y exists. The products Ui ˆS Y form an open coveringof XˆS Y and projections restrict to projections.

Proof: The tactics will be to patch together the different schemes Ui ˆS Y,and see that the result, indeed, is a product X ˆS Y. We begin with someuseful notation: let Uij = Ui XUj be the intersections of the Ui and Uj, and letπi : Ui ˆS Y Ñ Ui denote the projections. By Lemma 7.3 there are isomorphismsθji : π´1

i (Uij)Ñ UijˆS Y, and the gluing functions we shall use are τji = θ´1ij ˝ θji,

which identify π´1i (Uij) with π´1

j (Uij). Here is the picture:

Ui ˆS Y Ě π´1i (Uij) Uij ˆS Y π´1

j (Uji)ĎUj ˆS Y.θji

»

θ´1ij

»

The gluing maps τij satisfy the gluing conditions because they are compositionsof maps that satisfy them, and we shall see that the scheme resulting from thegluing process will serve as the product XˆS Y.

The two projections are essential parts of the product and must not beforgotten: The projection onto Y stays invariable there since Y is never touchedduring the construction. The projection onto X is obtained by gluing theprojections πi along the π´1

i (Uij). Under the identification of π´1i (Uij) with

the product Uij ˆS Y the projection πij onto Uij corresponds to the restrictionπi|π´1

i (Uij)by Lemma 7.3, and this means that the equalities πi|π´1

i (Uij)= πij ˝ θji

hold true. Saying that the two restrictions πi|π´1i (Uij)

and πj|π´1j (Uij)

become

equal after gluing, is to say that the equality

πi|π´1i (Uij)

= πj|π´1j (Uij)

˝ τji


holds (remember that in the gluing process points x and τji(x) are identified),but this is the case since

πj|π´1j (Uij)

˝ τji = πij ˝ θij ˝ τji = πij ˝ θij ˝ θ´1ij ˝ θji = πij ˝ θji = πi|π´1

i (Uij).

Consequently we may glue the πi’s together to obtain the projection πX.It is a matter of easy verifications that the glued scheme with the two

projections has the required universal property. o

It is worth while commenting that the product X ˆS Y is not defined asa particular scheme, it is just an isomorphism class of schemes — productsare only defined up to isomorphism (but surely a unique one). So speakingabout the product is an abuse of language (justified by there being a uniqueisomorphism respecting projections between any two ‘products’). In the proofabove, while both π´1

i (Uij) and π´1j (Uij) represent products Uij ˆS Y, they are

however not equal, merely canonically isomorphic. In the construction we couldhave used any of them or any non-specified representative of the isomorphismclass (as we in fact did, since this makes the situation appear more symmetric ini and j).

An immediate consequence of the Proposition 7.4 is that fibre products existover an affine base S.

Lemma 7.5 Assume that S is affine, then XˆS Y exists.

Proof: First if Y as well is affine, we are done. Indeed, cover X by open affinesets Ui. Then Ui ˆS Y exists by the affine case, and we are in the position toapply Proposition 7.4 above. We then cover Y by affine open sets Vi. As we justverified, the products XˆS Vi all exist, and applying Proposition 7.4 once morewe may conclude that XˆS Y exists. o

The final reductionLet tSiu be an open affine covering of S and let Ui = φ´1

X (Si) and Vi = φ´1Y (Si).

By Lemma 7.5 the products Ui ˆSi Vi all exist. Using the following lemma and,for the third time, the gluing statement in Proposition 7.4, we are through witha proof of the existence of fibre products (Theorem 7.1 on page 139).

Lemma 7.6 With current notation, we have the equality Ui ˆSi Vi = Ui ˆS Y. That is,Ui ˆS Y exists and the projections are πUi and πY|Vi .

Proof: We contend that Ui ˆSi Vi satisfies the universal product property of


Ui ˆS Y. The key diagram is

Z

Ui Y Vi

S

ψ ψ1

φX|UiφY

where ψ and ψ1 are two given maps. If one follows the left path in the diagram,one ends up in Si, and hence the same must hold following the right path.But then, Vi being equal to the inverse image φ´1

Y (Si), it follows that ψ1 factorsthrough Vi, and by the universal property of Ui ˆSi Vi there is a morphismZ Ñ Ui ˆSi Vi with the requested properties. o

NotationIf S = Spec A one often writes XÂ Y in short for XˆSpec A Y. If S = Spec Z, onewrites X ˆY. In case Y = Spec B the shorthand notation XbA B is frequentlyseen as well; it avoids writing Spec twice.

As in any category, diagrams arising from fibre products are frequentlycalled Cartesian diagrams

(Kartesiskediagrammer)

Cartesian diagram; that is, the diagram

Z X

Y S

πX

πY φX

φY

is said to be a Cartesian diagram if there is an isomorphism Z » XˆS Y withπX and πY corresponding to the two projections.

Exercise 7.6 (Basic properties of the fibre product.) Let X, Y and Z be threeˇ

schemes over the scheme S. Show the following basic properties of the fibreproduct; there are canonical isomorphisms over S all compatible with all actualprojections and all unique:

a) (Reflectivity) XˆS S » X;

b) (Symmetry) XˆS Y » YˆS X;

c) (Associativity) (XˆS Y)ˆS Z » XˆS (YˆS Z).

Assume moreover that T is a scheme over S and that Y is as well a scheme overT; show that there is a unique S-isomorphism compatible with all projections

d) (Transitivity) XˆS TˆT Y » XˆS Y,

where X ˆS T is a scheme over T via the projection onto T and Y is a schemeover S via the map T Ñ S. M

examples 145

Exercise 7.7 (Functoriality.) Show by using the universal property that giventwo S-morphisms φ : X1 Ñ X and ψ : Y1 Ñ Y there is a unique morphismφˆ ψ : X1 ˆS Y1 Ñ XˆS Y such that the two squares in the diagram commute.

X1 X1 ˆS Y1 Y1

X XˆS Y Y.

φ φˆψ

πX1 πY1

ψ

πX πY

Be precise about what uniqueness means. If φ1 : X2 Ñ X1 and ψ1

: Y2 Ñ Y1 isanother pair of morphisms, show that (φˆ ψ) ˝ (φ1 ˆ ψ1) = (φ ˝ φ1)ˆ (ψ ˝ ψ1),with equality unregarded of which representatives of the products are used(recall that the product is merely defined up to (unique) isomorphism). M

7.3 Examples

Varieties versus SchemesIn the important case that X and Y are integral schemes of finite type overthe algebraically closed field k the product of the two as varieties coincideswith their product as schemes over k, with the usual interpretation that thevariety associated to the scheme X is the set of closed points X(k) with inducedtopology.

The product Xˆk Y will be a variety (i.e. an integral scheme of finite typeover k) and the closed points of the product Xˆk Y will be the direct product ofthe closed points in X and Y.

It is not obvious that Abk B is an integral domain when A and B are, andin fact, in general, even if k is a field, it is by no means true. But it holds truewhenever A and B are of finite type over k and k is an algebraically closed field.The standard reference for this is Zariski and Samuel’s book Commutative algebraI which is the Old Covenant for algebraists. It is also implicit in Hartshorne’sbook ([?]), exercise 3.15 b) on page 22.

However that the tensor product Abk B is of finite type over k when A andB are, is straightforward. If u1, . . . , um generate A over k and v1, . . . , vm generateB over k the products uib 1 and 1b vj generate Abk B.

Non-algebraically closed fieldsThe situation is more subtle when one works over fields that are not algebraicallyclosed. To illustrate some of the phenomena that can occur, we study a few basicexamples. They all show different aspects of the product of spectra of two fields;i.e. the spectrum of a tensor product of fields.

examples 146

Examples(7.7) A simple but illustrative example is the product Spec CˆSpec R Spec C. Thisscheme has two distinct closed points, and it is not integral — it is not evenconnected!

The example also shows that the underlying set of the fibre product is notnecessarily equal to the fibre product of the underlying sets, although this wastrue for varieties over an algebraically closed field. In the present case the threeschemes involved all have just one element and their fibre product has just onepoint. So we issue warnings: The product of integral schemes is in general notnecessarily integral! The underlying set of the fibre product is not always thefibre product of the underlying sets.

The tensor product CbR C is in fact isomorphic to the direct product CˆC

of two copies of the complex field C; indeed, we compute using that C =

R[t]/(t2 + 1) and find

CbR C = R[t]/(t2 + 1)bR C = C[t]/(t2 + 1) = C[t]/(t´ i)(t + i) = CˆC,

where for the last equation we use the Chinese remainder theorem and that therings C[t]/(t˘ i) both are isomorphic to C.(7.8) This previous little example can easily be generalized: assume that L isa simple, separable field extension of K of degree d; that is, L = K(α) wherethe minimal polynomial f (t) of α over K is separable and of degree d. Let Ωbe a field extension of K in which the polynomial f (t) splits completely, e.g. anormal extension of L (e.g. any algebraically closed field containing K) – then byan argument completely analogous to the one above, one finds the equality

LbK Ω = Ωˆ . . .ˆΩ,

where the product has d factors. Consequently Spec L ˆSpec K Spec Ω has anunderlying set with d points, even if the three sets of departure all are primespectra of fields and thus singletons.

One may push this further and construct examples where the fibre productSpec KˆSpec K Spec Ω is not even Noetherian and has infinitly many points!(7.9) Let k be a field and x and y two variables. Consider the tensor productA = k(x)bk k(y). We can regard this as a localization of k[x, y] where we inverteverything in the multiplicative set S = tp(x)q(y)|p(x), q(y) ‰ 0u. Let us showthat A has infinitely many maximal ideals. Suppose that m Ă A is a maximalideal; it has the form S´1p for some prime ideal p Ă k[x, y] which is maximalamong the primes that do not intersect S. In this case we must have pX k[x] = 0,since otherwise there would be a non-zero p(x) P pX S. Similarly pX k[y] = 0,which implies that p has height at most 1. Hence, either p = (0), or p = ( f ) forsome irreducible polynomial f P k[x, y] not a product of a polynomial from k[x]and one from k[y]. If follows that A has dimension 1, and A has infinitely manymaximal ideals – in fact uncountably many if e.g. k = C.

examples 147

This example shows how strange the fibre product really is – Spec A is aninfinite set, even though it is the fibre product of two schemes with one-pointunderlying sets. We will see more examples like this in the end of this chapter.(7.10) In this example we let L P C[x, y] be a linear polynomial that is not real,for example L = x + iy + 1, and we introduce the real algebra A = R[x, y]/(LL).The product LL of L and its complex conjugate is a real irreducible quadric;which in the concrete example is (x + 1)2 + y2. The prime spectrum Spec A istherefore an integral scheme. However, the fibre product Spec AˆR Spec C isnot irreducible being the union of the two conjugate lines L = 0 and L = 0 inSpec C[x, y].

The scheme Spec A has just one real point, namely the point (´1, 0) (i.e.corresponding to the maximal ideal (x + 1, y)). The C-points however, areplentiful. They are C-points of A2

R(C), which all are the orbits t(a, b), (a, b)uof the complex conjugation with (a, b) non-real, and form the subset of thoset(a, b), (a, b)u such that L(a, b) = 0.(7.11) Another example along same lines as Example 2 shows that the fibreproduct XˆS Y is not necessarily reduced even if both X and Y are; the pointbeing to use an inseparable polynomial f (t) instead of a separable one in 2. Let kbe a non-perfect field whose characteristic is p, which means there is an elementa P k not being a p-th power. Let L be the field extension L = k(b) where bp = a.That is, L = k[t]/(tp ´ a); this is a field since tp ´ a is an irreducible polynomialover k. However, upon being tenzored by itself over k, it takes the shape

Lbk L = L[t]/(tp ´ a) = L[t]/(tp ´ bp) = L[t]/ ((t´ b)p) ,

which is not reduced, the non-zero element t´ b being nilpotent. So we issue athird warning: the fibre product of integral schemes is not in general reduced!

K

Exercises(7.8) With the assumptions of the example above, check the statement thatLbK Ω » Ωˆ . . .ˆΩ, the product having d factors.(7.9) Assume that A is an algebra over the field k having a countable sette1, e2, . . . , ei, . . . u of mutually orthogonal idempotents, i.e. eiej = 0 if i ‰ jand eiei = 1, and assume that ei A » k. Assume also that every element is a finitelinear combination of the ei’s. Show that the ideal Ij generated by the ei’s withi ‰ j is a maximal ideal.(7.10) Let k be a field and let X and Y be schemes over k. Show that the k-pointsof Xˆk Y are in bijection with the usual Cartesian product of sets X(k)ˆY(k).(7.11) Let A = R[x, y]/(x2 + xy + 1). Show that Spec A is irreducible and thatSpec AˆSpec R Spec C is isomorphic to the disjoint union of two affine lines A1

C.(7.12) Let a P Q be a rational number that is not a square and for each i P N let

base change 148

Li = Q(ηi) where η2ii = a. Then the Li’s form a tower of fields

L1 Ă ¨ ¨ ¨ Ă Li Ă Li+1 Ă ¨ ¨ ¨ Ă C,

and each Li+1 is a quadratic extension of Li. Let L be the union of the Li’s. Showthat the algebras LibQ C form a tower

L1bQ C Ă ¨ ¨ ¨ Ă LibQ C Ă Li+1bQ C Ă ¨ ¨ ¨ Ă CbQ C

and that LbQ C =Ť

i LibQ C. Show that each Li+1bQ C is isomorphic toLibQ Cˆ LibQ C as a C-algebra and conclude that LbQ C as an C-algebrais isomorphic to direct sum of countably many copies of C’s. Conclude thatSpec LˆSpec Q Spec C is not even a Noetherian scheme.

M

7.4 Base change

The fibre product is in constant use in algebraic geometry, and it is an aston-ishingly versatile and flexible instrument. We shall comment on some of themost frequently encountered applications, and we begin with the notion of basechange.

In its simplest and earliest appearances base change is just extending thefield over which one works; e.g. in Galois theory, or even in the theory of realpolynomials, when studying an equation with coefficients in a field k one oftenfinds it fruitful to study the equation over a bigger field K. Generalizing this toextensions of algebras over which one works, and then to schemes, one arrivesnaturally at the fibre product.

Let X be a scheme over S and T Ñ S be a map. Considering T Ñ S as achange of base schemes one frequently writes XT = XˆS T and says that XT isobtained from X by Base change

(Basisforandring)base change or frequently that XT is the pullback of X along

T Ñ S. This is a functorial construction since if φ : X Ñ Y is a morphism overS, there is induced a morphism φT = φˆ idT from XT to YT over T, and oneeasily checks that φT ˝ πT coincides with the natural projection map XT Ñ T (orin other words, the outer rectangle in the diagram in the margin is Cartesian). XT X

YT Y

T S

φT φ

πT

πY

IfP is a property of morphisms, one says that P is stable under base change if forany T over S, the map φT has the property P whenever φ has it. Examples 4

and 5 showed that neither being irreducible nor being reduced are propertiesstable under base change.

On the other hand, one way of phrasing Lemma 7.3 on page 141 is to saythat being an open immersion is stable under base change. The same applies toclosed immersions:

base change 149

Proposition 7.12 (Pullbacks of open and closed immersions)Assume given a Cartesian diagram of schemes

ZY Z

Y X

φY φ

If the morphism φ : Z Ñ X is a closed immersion, then the morphism φY : ZY Ñ

Y is one as well. If φ is an open immersion, then φY is also an open immersion.

Proof: Only the statement about closed immersions needs a proof. Assumefirst that X and Y are both affine, say X = Spec A and Y = Spec B. FromX = Spec A it follows that Z = Spec A/a for some ideal a (Proposition 3.18 onpage 85), and therefore ZY = ZˆX Y = Spec A/abA B = Spec B/aB. Hence ZY

is a closed subscheme of Y.The issue is local on Y (Exercise 3.3 on page 85), so assume that UĎY is

an open affine that maps into an open affine VĎX (one may cover Y by suchby first covering X by affine opens and subsequently covering each of theirinverse images in Y by affine opens). Then by Lemma 7.6 on page 143 one hasφ´1(V)ˆX Y = φ´1(V)ˆV U, and by the affine case this is a closed subschemeof U. o

Exercises(7.13) Finite type and base change. Show that being of finite type (respectivelybeing finite or being locally of finite type) is a property stable under base change.Show that the product of two morphisms of finite type (respectively of finite or

locally of finite type) is of finite type (respectively of finite or locally of finitetype).(7.14) Flat base change. In a base change staging, with X a scheme over S andˇ

T Ñ S a morphism, it is often crucial to be able to compare functions on theschemes X and XT, and for so-called flat base changes the situation is optimal.Assume X is a scheme over Spec A and let B be an A-algebra. Show that thereis a natural B-algebra homomorphism Γ(X,OX)bA B Ñ Γ(XB,OXB) which isan isomorphism whenever B is a flat A-algebra.(7.15) Let p and q be two different primes. Show the following identities:

a) Spec Fp ˆSpec Z Spec Fq = H;

b) Spec Z(p) ˆSpec Z Spec Z(p) = Spec Z(p);

c) Spec Z(p) ˆSpec Z Spec Z(q) = Spec Q.

M

scheme theoretic fibres 150

7.5 Scheme theoretic fibres

In most parts of mathematics, when one studies a map of some sort, it is veryuseful to understand the fibres of the map. This is also true in the theory ofschemes.

Suppose that φ : X Ñ Y is a morphism of schemes and that y P Y is a point.On the level of topological spaces, we are interested in the preimage φ´1(y), andwe aim at giving a scheme theoretic definition of the fibre φ´1(y). Having thefibre product at our disposal, nothing is more natural than defining the fibre to bethe fibre product Spec k(y)ˆY X, where Spec k(y)Ñ Y is the map correspondingto the point y; recall that the field k(y) is defined as k(y) = OY,y/my and thatthe ‘point-map’ Spec k(y) Ñ Y is the composition Spec k(y) Ñ SpecOY,y Ñ Y.It is common to write Xy for the scheme theoretic fibre and reserve the notationφ´1(y) for the topological fibre. Thus the scheme theoretic fibre of φ over y fitsinto the Cartesian diagram

Xy = XˆY Spec k(y) X

Spec k(y) Y.

φ

Note that the fibre Xy enjoys the property that for every morphism ψ : Z Ñ Xthe composition φ ˝ ψ factors through Spec k(y)Ñ Y if and only if ψ itself ‘takesvalues’ (read factors through) the fibre Xy.

As the next lemma will show, the underlying topological space of Xy is thetopological fibre φ´1(y), but it will be endowed with an additional and canonicalscheme structure. In many cases, the fibre will not be reduced, and this is mostlya good thing since it makes certain continuity results true.

Proposition 7.13 Let X and Y be schemes and φ : X Ñ Y a morphism. Lety P Y be a point. Then

i) The inclusion Xy Ñ X of the scheme theoretic fibre is a homeomor-phism onto the topological fibre φ´1(y);

ii) If X = Spec B and Y = Spec A, it holds that Xy = Spec(B/pyB)py ;iii) If X = Spec B and Y = Spec A and y is a closed point, one has

Xy = Spec B/myB.

Proof: We begin with the two affine cases ii) and iii), but even in the generalcase we may obviously assume that Y is affine, say Y = Spec A.

The map φ between two affine schemes is induced by a map of rings α : A ÑB. Let p = pyĎ A be the prime ideal corresponding to y P Y. We have theequalities

t q P Spec B | pĎ α´1(q) u = t q P Spec B | pBĎ q u = V(pB).


In the particular case that p is a maximal ideal, the inclusion pĎ α´1(q) must bean equality, and the sets above describe the fibre set-theoretically:

φ´1(p) = t qĎ B | q Ě pB u = V(pB).

The closed subset V(pB) of Spec B with induced topology from Spec B is canon-ically homeomorphic to Spec(B/pB). Thus we have a homeomorphism betweenSpec B/pB and the topological fibre φ´1(p). On the other hand, by standardequalities between tensor products one has

B/pB = AbA A/pA = BbA k(y),

and so the scheme theoretical fibre φ´1(y), which is given by Xy = X ˆY

Spec k(y) = Spec BbA k(y), is in a canonical way homeomorphic to the topolog-ical fibre. This proves iii).

If p is not a maximal ideal, the set Spec B/pB can certainly be bigger than thefibre, the extra prime ideals being those q for which α´1(q) is strictly bigger thanp. When we localize in the multiplicative system S = AzpĎ A, these superfluousprime ideals become non-proper since they all contain elements of the form α(s)with s P S. It follows that the points in the fibre φ´1(y) correspond exactly tothe primes in the localized ring (B/pB)p. Standard formulas for tensor productsgive on the other hand the equality

(B/pB)p = BbA Ap/pAp = BbA k(y).

The Zariski topology on the spectrum Spec(B/pB)p (which is the topology it hasas a scheme-fibre) coincides with the one induced from the Zariski topology ofSpec(B/pB) (which is the topology as topological fibre), and hence ii) is proven.

In the general case, one considers the diagram (7.6) below where U is openand affine in X. The two small squares are Cartesian, the lower by the definitionof the fibre and the upper one after Lemma 7.3 on page 141, and because thefibre product is transitive (Exercise 7.6 on page 144), it follows that the largersquare is Cartesian as well; and so Xy XU is the fibre over y of the map U Ñ Y.

Xy XU U

Xy X

Spec k(y) Y

(7.6)

By the affine case, the two topologies we examine agree on Xy XU, and as Ucan be any open affine, they share a basis and must be equal. o

Example 7.14 We take a look at a simple but classic example: the map

φ : X = Spec k[x, y]/(x´ y2)Ñ Spec k[x]


induced by the injection B = k[x] Ñ k[x, y]/(x´ y2) = A. Geometrically onewould say this is just the projection of the ‘horizontal’ parabola onto the x-axis.

If a P k, computing the fibre over ma = (x´ a) yields that the fibre Xa equalsthe spectrum of the ring

k[x, y]/(x´ y2)bk[x] k(a) » k[y]/(y2 ´ a).

where k(a) denotes the field k(a) = k[x]/(x´ a) (which of course is just a copyof k), and where we use the isomorphism A/abA M » M/aM for an ideal a inA and an A-module M.

Several cases can occur, apart from the characteristic two case, which isspecial.

i) In case a has a square root in k, say b2 = a, the polynomial y2 ´ afactors as (y´ b)(y + b), and the fibre becomes the product

Spec k[y]/(y´ b)ˆ Spec k[y]/(y + b),

which is the disjoint union of two copies of Spec k.

ii) If a does not have a square root in k, the fibre equals Spec k(?

a) wherek(?

a) is a quadratic field extension of k. The fibre is a singleton, butwith multiplicity two, the multiplicity materializing as the degree ofthe field extension k Ă k(

?a).

iii) The final case appears when a = 0. The fibre is not reduced, butequals Spec k[y]/(y2). The fibre is again a singleton with multiplicitytwo, the multiplicity this time materializes as the length of OXa .

We also notice that the generic fibre of φ is the quadratic extension k(x)(?

x)of the function field k(x). In all cases the ‘number of points’ in the fibre whencounted with the proper multiplicity, equals two, and this illustrates one of thepermanence properties alluded to above.

When k is not algebraically closed there are also points P in Spec k[x] whosemaximal ideals mP = ( f (x)) are generated by a non-linear polynomial irre-ducible over k. The analyses above goes through word by word with the solechange that k is replaced by the extension K = k(P) = k[x]/( f (x)) of k. When ahas a square root in K, the fibre XP becomes the disjoint union of two copies ofSpec K, and when not, it will equal Spec K(

?a).

Over perfect fields k of characteristic two the picture is completely different.Then a is a square, say a = b2 and as (y2 ´ b2) = (y´ b)2 none of the fibresare reduced (they equal Spec k[y]/(y´ b)2), except the generic fibre which isk(x)(

?x). K

Example 7.15 A similar example can be obtained from the map

f : Spec B Ñ Spec A,

where B = Spec k[x, y, z]/(xy ´ z) and A = k[z] and f is induced from theinclusion k[z]Ñ k[x, y, z]/(xy´ z). As before, we assume k algebraically closed,


pick a closed point a P Spec A, and consider the fibre

Xa = Spec (BbA k(a)) = Spec k[x, y]/(xy´ a)

Again there are two cases. If a ‰ 0, then xy´ a is an irreducible polynomial,and so Xa is an integral scheme. This is intuitive, since it corresponds to thehyperbola V(xy´ a) in A2

k . If a = 0, we are left with X0 = Spec k[x, y]/(xy),which is not irreducible; it has two components corresponding to V(x) andV(y). X0 is reduced however.

For good measure, we also consider the fibre over the generic point η ofSpec A. This corresponds to

k[x, y, z]/(xy´ z)bk[z] k(z) = k(z)[x, y]/(xy´ z),

which is an integral domain. Hence Xη is integral. K

Exercises(7.16) Compute the fibre product Spec(Z/2)ˆSpec Z Spec(Z/3). Explain yourˇ

answer geometrically.(7.17) Let X = Spec B and Y = Spec A. Let φ : X Ñ Y be a map such thatφ7 : A Ñ B makes B into a free A-module of rank n. Prove that for each pointy P Y it holds that

ÿ

xPφ´1(y)

[k(x) : k(y)] lengthOXy,x = n.

(7.18) Let p and q be two different prime numbers and consider the morphismφ : A1

k Ñ A2k induced from the map k[x, y] Ñ k[t] which is defined by the

assignments x ÞÑ tp and y ÞÑ tq. Determine all scheme theoretic fibres of φ.(7.19) Let k an algebraically closed field. Consider the k-algebra A given as A =

k[x, y, z]/(xy, xz, yz) and let X = Spec A. Consider the map φ : X Ñ A1 dual tothe k-algebra homomorphism k[t]Ñ A that sends t to x + y + z. Determine allscheme theoretic fibres of φ. Hint: Heuristics: X(C) is the union of the axes inC3 and the map sends points on X(C) to the sum of their coordinates.(7.20) A du Val singularity. Let k be an algebraically closed field. Let n ě

2 be an integer and consider the two rings A = k[x, y, z]/(xy ´ zn+1) andB = k[u, v, z]/(uv ´ zn´1). Let X = Spec A and Y = Spec B. Show that theassignments x ÞÑ uz, y ÞÑ vz and z ÞÑ z induce a morphism φ : Y Ñ X. Provethat φ is birational and determine its scheme theoretic fibres over closed points.(7.21) Another du Val singularity. Let k be an algebraically closed field. LetA = k[x, y, z]/(x2 + y3 + z5) and let X = Spec A (this is the famous du Valsingularity E8). Furthermore, let B = k[u, v, z]/(u2 + u3z + z3) and Y = Spec B.Show that the assignments x ÞÑ uz, y ÞÑ vz and z ÞÑ z induce a morphism


φ : Y Ñ X. Prove that φ is birational and determine its scheme theoretic fibresover closed points.(7.22) Describe the scheme theoretic fibres in all points of the following mor-phisms.

a) f : Spec C[x, y]/(xy´ 1)Ñ Spec C[x];b) f : Spec C[x, y]/(x2 ´ y2)Ñ Spec C[x];c) f : Spec C[x, y]/(xy)Ñ Spec C[x];d) f : Spec Z[x, y]/(xy2 ´m)Ñ Spec Z, where m is a non-zero integer.

(7.23) Determine all the scheme theoretic fibres of the morphism Spec Z[(1 +?

5)/2]Ñ Spec Z[?

5] induced by the natural inclusion Z[?

5]ĎZ[(1+?

5)/2].M

Chapter 8

Separated schemes

The topology on schemes behaves very differently from the usual Euclideantopology. In particular, schemes are not Hausdorff, except in trivial cases – theopen sets in the Zariski topology are simply too large. Still we would like tofind an analogous property that can serve as a satisfactory substitute for thisproperty. The route we take is to impose that the diagonal should be closed;closed in the Zariski topology of the product, of course.

By the immense freedom we have for gluing schemes together there are lotsof non-separated schemes in the world of schemes. On the other hand, thenon-separated schemes are a bit strange, and one doesn’t frequently encounternon-separated schemes in practice There are

counterexamples:‘moduli spaces’ and‘quotient spaces’ areclasses of schemeswhich are sometimesnon-separated.

. In fact, the first edition of EGA reservedthe word preschema for what we today call schemes and schema for a separatedscheme.

More importantly, some very nice and advantageous properties hold only forseparated schemes, and this legitimates the notion. For instance, in a separatedscheme, the intersection of two affine subsets is again affine (this is a propertywhich will be important later on).

Of course, one needs good criteria to be sure we have a large class ofseparated schemes. We will soon see that all affine schemes are separated, andwe will see in Chapter 9 that the same is true for projective schemes also.

We begin with defining the diagonal and giving some of its properties, theproofs are of a quite formal nature and also work in any category where thestatements are meaningful.

8.1 The diagonal

Let X/S be a scheme over S. There is a canonical map ∆X/S : X Ñ X ˆS Xof schemes over S called the diagonal map or the diagonal morphism. The twocomponent maps of ∆X/S are both equal to the identity idX; that is, the definingproperties of ∆X/S are πi ˝∆X/S = idX for i = 1, 2 where the πi’s denote the twoprojections.

the diagonal 156

In the case that X and S are affine schemes, the diagonal has a simple andnatural interpretation in terms of algebras; it corresponds to the most naturalmap, namely the multiplication map:

µ : AbB A Ñ A.

The multiplication map sends ab a1 to the product aa1, and then extends toAbB A by linearity. The projections correspond to the two algebra homomor-phisms ιi : A Ñ AbB A that send a to ab 1 respectively to 1b a. Clearly it holdsthat µ ˝ ιi = idA, and on the level of schemes this translates into the definingrelations for the diagonal map. Moreover, µ is clearly surjective, so we haveestablished the following:

Proposition 8.1 If X is an affine scheme over the affine scheme S, then thediagonal ∆X/S : X Ñ XˆS X is a closed immersion.

The conclusion here is not generally true for schemes, and shortly we shallgive counterexamples. However, from the proposition we just proved, it followsreadily that the image ∆X/S(X) is always locally closed, i.e. the diagonal is locallya closed immersion:

Proposition 8.2 The diagonal ∆X/S is locally a closed immersion.

Proof: Begin with covering S by open affine subsets and subsequently covereach of their inverse images in X by open affines as well. In this way one obtainsa covering of X by affine open subsets Ui whose images in S are containedin affine open subsets Si. The products Ui ˆSi Ui = Ui ˆS Ui are open andaffine, and their union is an open subset containing the image of the diagonal.By Proposition 8.1 above the diagonal restricts to a closed immersion of Ui inUi ˆSi Ui. o

Exercises(8.1) In the setting of the proof of Proposition 8.2 above, show that ∆X/S|Ui =

∆Ui/S.(8.2) Let T Ñ S be a morphism and let X and Y be two schemes over T. Showˇ

that there is a Cartesian diagram

XˆT X XˆS X

T TˆS T,

ι

fˆ f∆T/S

and conclude that the natural inclusion ι : XˆT Y Ñ XˆS Y is a locally closedimmersion.(8.3) Pullback of diagonals. Let X Ñ S and T Ñ S be morphisms betweenˇ

schemes, and as usual, let XT = XˆS T. Show that the diagonal ∆X/S pulls back

separated schemes 157

to the diagonal ∆XT/T; in other words, that there is a canonical Cartesian square

XT XT ˆT XT

X XˆS X.

∆XT /T

∆X/S

(8.4) The diagonal and monomorphisms. In a general category the classical def-ˇ

inition of injective maps is not meaningful and is replaced by the notion ofmonomorphisms, which reads as follows: a monomorphism in the category C

is an arrow f : X Ñ Y in C such that for any two arrows g, g1 : Z Ñ X in C anequality f ˝ g = f ˝ g1 implies g = g1. The dual concept is that of epimorphisms:if g, g1 : Y Ñ Z are two arrows in C so that g1 ˝ f = g ˝ f , then g = g1.

In any category were fibre products exist, one has notion of the graph ofan arrow f : X Ñ Y over S, namely the arrow Γ f : X Ñ X ˆS Y with definingproperties πX ˝ Γ f = idX and πY ˝ Γ f = f .

a) Show that the diagonal ∆X/S of any scheme X over S is a monomorphism.As is the graph Γ f : X Ñ X ˆS Y of any morphism f : X Ñ Y betweenschemes over S.

b) Show that the diagonal of a monomorphism is the identity. In precise termsthis means that the following diagram is Cartesian:

X X

X S.

idX

idX

f

f

c) Conclude that ‘the diagonal of the diagonal’ is the identity.

M

8.2 Separated schemes

We have now come to the definition of the property that will play the role of theHausdorff property for schemes.

Definition 8.3 One says that the scheme X/S is separated over S, or thatthe structure map X Ñ S is separated, if the diagonal map ∆X/S : X Ñ XˆS Xis a closed immersion. One says for short that X is separated if it is separatedover Spec Z.


Examples(8.4) Any morphism Spec B Ñ Spec A of affine schemes is separated (by Propo-sition 8.1).(8.5) The simplest example of a scheme that is not separated is obtained bygluing the prime spectrum of a discrete valuation ring to itself along the genericpoint.

To give more details let R be a dvr with fraction field K. Then Spec R = tx, ηu

where x is the closed point corresponding to the maximal ideal, and η is thegeneric point corresponding to the zero ideal. The generic point η is an openpoint (the complement of tηu is the closed point x) and corresponding tothe open immersion Spec K Ñ Spec R. By the gluing lemma for schemes(Proposition 4.3 on page 96), we may glue one copy of Spec R to another copy ofSpec R by identifying the generic points; that is, the open subschemes Spec K inthe two copies.

In this manner we construct a scheme ZR together with two open immersionsιi : Spec R Ñ ZR. They send the generic point η to the same point, which is anopen point in ZR, but they differ on the closed point x.

It follows that the diagonal is not closed. If π : Spec R Ñ XˆX is the morphisminduced by ι1 and ι2, the preimage π´1(∆X(X)) of the diagonal is exactly theset of points in Spec R, where the morphisms ιi agree. But this set has exactlyone point, the open point, which is not closed. (See also Proposition 8.8 below).(8.6) The affine line X with two origins constructed on page 109 in Chapter 5

is not separated over S = Spec k. It was constructed as the union of twoaffine lines Ui = Spec k[u] glued together along their common open subsetU12 = Spec k[u, u´1]. Hence there are two open immersions Spec k[u] Ñ Xwhich agree on U12, which is not closed.

It is also instructive to examine the diagonal in detail. Denote the two originsby O1 and O2. Then the scheme Xˆk X is glued together by four affine charts


U1 Û1, U1 Û2, U2 Û1, and U2 Û2, all isomorphic to A2k = Spec k[x, y].

Here the image ∆X(X) intersects U1 Û1 along the diagonal V(x´ y) Ă A2k »

U1 Û1. On the other hand, in U1 Û2, the intersection U1 Û2 X ∆(X) doesnot contain the origin (O1, O2). It follows that ∆X(X) is not closed. In fact, thefibre product Xˆk X contains four origins

(O1, O1), (O1, O2), (O2, O1), (O2, O2).

The image of the diagonal morphism only contains the two origins (O1, O1) and(O2, O2), while the closure of ∆X(X) contains all four.

K

Example 8.7 Monomorphisms and locally closed immersions. Any monomorphismis separated because the diagonal is just the identity morphism (Exercise 8.4).In particular every graph (including the diagonal) is separated. Locally closedimmersions are monomorphisms (we give a proof below) and are thus separated.In particular this applies to both open and closed immersions; they are allseparated.

Proposition 8.8 Locally closed immersions are monomorphisms, hence theyare separated.

Proof: This holds true basically because surjective ring maps are epimorphismsin the category Rings. Assume a locally closed immersion ι : X Ñ Y is given,and let fi : Z Ñ X for i = 1, 2 be two morphisms such that ι ˝ f1 = ι ˝ f2. Coverthe image ι(X) by affines Ui = Spec Ai such that the restrictions ι|Vi are closedimmersions where Vi = ι´1(Ui). Then Vi is affine and the restriction ι|Vi isinduced by a surjective ring homomorphism α = ι7(Ui) : Ai Ñ Ai/ai.

By assumption f 71 ˝ ι7 = f 72 ˝ ι7 so we have f 71(Vi) ˝ α = f 72(Vi) ˝ α. The mapα being surjective it ensues that f 71(Vi) = f 72(Vi), and we deduce, citing thefundamental theorem about maps into affines (Theorem 4.6 on page 98), that f1

and f2 coincide on the open subsets f´11 (Vi) (which equal f´1

2 (Vi)). These coverZ and we can conclude that f1 = f2. o

K

Some of the most basic properties of separated morphisms are listed in thenext proposition:


Proposition 8.9 The following hold true:

i) (Immersions) Locally closed immersions are separated, in particularopen and closed immersions are;

ii) (Composition) Let f : T Ñ S and g : X Ñ T be morphisms. If both fand g are separated, the composition g ˝ f is separated as well. If X isseparated over S, it is separated over T;

iii) (Base change) Being separated is a property stable under base change:if f : X Ñ S is separated and T Ñ S is any morphism, then fT : XˆS

T Ñ T is separated:

Proof: The first statement was proven above. To prove statement ii), let thetwo separated morphism be f : X Ñ T and g : T Ñ S. The crux of the proof isthat the following diagram is Cartesian (see Exersice 8.2):

XˆT X XˆS X

T TˆS T.

ι

fˆ f∆T/S

(8.1)

Note that ∆X/S = ι ˝ ∆X/T. Assume first that T Ñ S is separated, then ∆T/S isa closed immersion, hence ι will be one as well (Proposition 7.7 on page 149).It follows that ∆X/S = ι ˝ ∆X/T is a closed immersion (composition of closedimmersions are closed immersions), and so X is separated over S. For theconverse statement, assume that X is separated over S. Then the compositionι ˝ ∆X/T being equal to ∆X/S is a closed immersion, hence ∆X/T is closed.

When proving statement iii), it suffices to cite Exercise 8.3 on page 156, thatdiagonals pull back to diagonals, and again Proposition 7.7, that pullbacks ofclosed immersions are closed immersions. o

We introduce separatedness mostly because they give good formal properties.In some sense the schemes category is still a little bit "too large", and separatedschemes have properties that make them closer to varieties. Here is one of theseproperties:

Proposition 8.10 Assume that X is a separated scheme over an affine schemeS = Spec A, and assume that U and V are two affine open subsets of X.Then the intersection U X V is also affine, and the natural product mapΓ(U,OU)bA Γ(V,OV)Ñ Γ(U XV,OX) is surjective.

Proof: The product U ˆS V is an open and affine subset of X ˆS X, andU XV = ∆X/S(X)X (U ˆS V). So if the diagonal is closed, U XV is a closedsubscheme of the affine scheme U ˆS V, hence affine (Proposition 3.18). By theconstruction of the fibre product of affine schemes one has

Γ(U ˆS V,OUˆSV) = Γ(U,OU)bA Γ(V,OV),


and as U XV is a closed subscheme of U ˆS V, the restriction map

Γ(U ˆS V,OUˆSV)Ñ Γ(U XV,OUXV)

is surjective, as we wanted to show. o

Conversely, we have

Proposition 8.11 Let X be a scheme over Spec A, and let U = tUiuiPI be anaffine cover of X such that

i) all intersections Ui XUj are affine;ii) Γ(Ui,OX)bA Γ(Uj,OX) Ñ Γ(Ui XUj,OX) is surjective for each

i, j P I.

Then X is separated over S.

Proof: Let π1, π2 : XˆS X Ñ X be the two projections and let ∆ : X Ñ XˆS Xdenote the diagonal morphism ∆X/S. Let Ui = Spec Bi and Uj = Spec Bj be twoopen sets in the covering U . We have

∆´1(π´11 (Ui)X π´1

2 (Uj)) = ∆´1(π´11 (Ui))X ∆´1(π´1

2 (Uj)) = Ui XUj, (8.2)

Also, from the universal property of the fibre product it ensues that π´11 (Ui)X

π´1(Uj) = Ui ˆS Uj Ă X ˆS X, and from this we deduce that ∆ is a closedimmersion if each restriction

∆ij : Ui XUj Ñ Ui ˆS Uj

of ∆ is a closed immersion. But this follows from the assumptions: by i)the intersection Ui XUj is affine, say Ui XUj = Spec Cij, and by ii) the ringhomomorphism Bi bA Bj Ñ Cij is surjective. Hence ∆ij is a closed immersionfor each i, j, and the proof is complete. o

Example 8.12 The above provides us with a convenient criterion to check thata scheme is separated, given an affine covering. For instance, let us show thatthe projective line P1

k is separated. P1k is covered by the two affine subsets U1 =

Spec k[x] and U2 = Spec k[x´1], which have affine intersection Spec k[x, x´1]. Toconclude, we need only check that the map

k[x]bk k[x´1]Ñ k[x, x´1]

is surjective, and it is. K

Example 8.13 Here is a non-separated scheme where two affine open sets havenon-affine intersection. We glue two copies of the affine plane A2

k together alongthe complement U12 = A2

k ´ V(x, y) of the origin. If U1 and U2 denote thetwo open immersions of the affine plane, then U1 XU2 = U12, but the open setU12 is not affine (see the example in Section 5.2 on page 104). In this example,


the multiplication map in the proposition coincides with k[x, y] b k[x, y] ÑΓ(U12,OU12), which is surjective. K

Another useful property is that morphisms into separated schemes aredetermined on open dense sets, at least when the source is reduced:

Proposition 8.14 Let X and Y be two schemes over S and f1, f2 : Y Ñ X twomorphisms over S. Assume that Y is a reduced scheme and X is separated overS. Moreover, assume there is an open immersion ι : U Ñ Y with dense imagesuch that f1 ˝ ι = f2 ˝ ι. Then f1 = f2.

Proof: We may assume that Y is affine, say Y = Spec A. The morphismf : Y Ñ XˆS X whose two components are f1 and f2 enters in the diagram

U Y XˆS X

E X,

ι

j

f

η ∆X/S

where the right square is Cartesian. We have assumed that f1 ˝ ι = f2 ˝ ι, sof ˝ ι factors through ∆X/S and, according to the universal property of pullbacks,the map ι factors through E. Now, pullbacks of closed immersions are closedimmersions, so that the image η(E) is closed, and by Proposition 3.18 on page 85

it is shaped like Spec A/a for some ideal a. The image η(E) contains the denseset ι(U) and therefore is equal to Y. Thus

?a = (0), and I is contained in

the nilradical of A which is zero as A is reduced. Hence η is an isomorphism.Consequently, f factors through the diagonal and f1 = f2. o

Example 8.15 The above proposition fails when X is not separated. For instance,if X is the affine line with two origins, then there are two morphisms ιj : A1

k Ñ Xfor j = 1, 2 which agree on a dense open set, but they are not equal. K

Example 8.16 Likewise, it may fail when the scheme Y is not reduced: letY = Spec k[x, y]/(y2, xy) and consider the two maps f j : Y Ñ Spec k[u], j = 1, 2defined by u ÞÑ x and u ÞÑ x + y respectively. These agree over the distinguishedopen set D(x), but they are different. K

Varieties versus schemes againWith the notion of separatedness we can finally state the definition of a variety:

Definition 8.17 A variety X is an integral, separated scheme of finite typeover an algebraically closed field.

This definition should be compared with the definition from Chapter 4.There we defined a variety to be a scheme in the image of the functor

Var/k Ñ Sch/k,


which associates a k-variety V to a scheme Vs over k. As varieties satisfy theHausdorff axiom, it is immediate that the corresponding scheme Vs is separated.Thus the two notions agree.

From now on a ‘variety’ will always refer to a scheme satisfying the requestsin Definition 8.9. Basically, any theorem from the ‘classical setting’ regardingvarieties carries over to varieties in the new sense. This is justified by thefollowing theorem:

Theorem 8.18 The functor V Ñ Vs is fully faithful and gives and equivalencebetween the category of varieties Var/k and the subcategory of Sch/k of schemessatisfying Definition 8.9.

Exercises(8.5) Show that X Ñ S is separated if and only if the image of the diagonal map∆X/S is a closed subset of XˆS X.(8.6) Show that ZR ˆ ZR with ZR as in Example 2 above is obtained by gluingfour copies of Spec R together along their generic points. Show that the diagonalis open and not closed.(8.7) The graph of a morphism. A morphism φ : X Ñ Y over S has a graph Γφ : X ÑXˆS Y; it is the pullback of the diagonal ∆Y/S under the morphism φˆ idY : XˆY Ñ YˆS Y. Show that the graph is a closed immersion when Y is separated.(8.8) Closed immersions. Let f : X Ñ Y and g : Y Ñ Z be morphisms of schemes.

a) Assume that g is separated. Show that if the composition g ˝ f is a closedimmersion, then f is a closed immersion. Hint: Consider the diagram

X XˆZ Y Y

X Z

Γ f

g

g˝ f

where the square is Cartesian and Γ f is the graph of f .

b) Show by an example that in general f is not necessarily a closed immersioneven if g ˝ f is. Hint: For one of the copies of A1, say U1, in the affine line Xwith two origins constructed on page 109 in Chapter 5, exhibit a morphismX Ñ A1 that restricts to the identity on U1.

(8.9) Let T be a scheme and X and Y two schemes over T with structure mapsˇ

φX : X Ñ T and φY : Y Ñ T. Let T Ñ S be a morphism.

a) Show there is a Cartesian diagram

XˆT Y XˆS Y

T TˆS T

ι

φXˆφY

∆T/S


where ι is the natural inclusion as in Exercises 8.2.

b) Show that ι is separated and that ι is a closed immersion if T is separatedover S.

(8.10) A morphism φ : X Ñ Y is said to be affine if for some cover tUiu of Yof open affine sets, the inverse images φ´1(Ui) are affine. Show that affinemorphisms are separated.(8.11) Let R and S be two dvr’s with the same fraction field, and denote by mR

and mS the two maximal ideals. Assume that R and S different in the sensethat mR X S Ę mS and mS X R Ę mR. Let Z be the scheme obtained by gluingSpec R and Spec S together along the generic points. Show that Z is affine, moreprecisely, show that Z is isomorphic to Spec (RX S).(8.12) The Hausdorff axiom. Let Y be a separated scheme over S and let f , g : X ÑY be two S-morphisms from X to Y. Show that the set Z Ă X of points x P X sothat f (x) = g(x), is closed in X.(8.13) Equalizers. Let X and Y be schemes over S and f1 and f2 two morphismsfrom Y to X. Let f : Y Ñ XˆS X be the morphism whose components are thefi’s; that is, fi = πi ˝ f (as usual, the πi’s are the two projections). The pullbackf´1∆X/S is called the Equalizer (ekvilisator)equalizer of the fi’s, and we shall denote it by η : E Ñ Y. Inother words, the diagram below is Cartesian:

E Y

X XˆS X.

η

f∆X/S

a) Show that a morphism g : Z Ñ Y satisfies f1 ˝ g = f2 ˝ g if and only if gfactors via η;

b) Show that X is separated if and only if all equalizers of maps into X areclosed.

(8.14) Let X = Spec C and S = Spec R. Recall that the product XˆS X consistsof two (closed) points. Which one is the diagonal? Can you find another R-algebra A so that if Y = Spec A it holds that YˆS Y » XˆS X and the diagonalis the other point?(8.15) Let A be a B-algebra. Show that the kernel of the multiplication mapµ : AbB A Ñ A is generated by the elements of shape ab 1´ 1b a. Hint: Itholds true that

ř

i aib bi =ř

i(aib 1´ 1b ai) ¨ 1b bi +ř

i 1b aibi.(8.16) Prove a partial converse to statement ii) in Proposition 8.5:ˇ

a) if f and g are two composable morphisms of schemes and f ˝ g and f areseparated, prove that then g is separable. Hint: Show that ι in the diagram(8.1) always is separated, then resort to Exercise 8.8 above.

b) Show by way of an example that the third alternative is not true; i.e. exhibitf and g so that f ˝ g and g are separated but f is not.


(8.17) Show that if a scheme X is separated (over Z), then for every scheme Yˇ

and every morphism f : X Ñ Y, the morphism f is separated.M

Chapter 9

Projective schemes

The projective varieties are fundamental in the theory of varieties, not justbecause they are interesting objects of study, but also because they in manyaspects are easier to handle than non-projective ones. In the scheme-world thereis a construction extending the notion of projective varieties vastly; from anypositively graded ring R one constructs a scheme Proj R called the projectivespectrum. The construction is somewhat parallel to that of the prime spectrumof a ring, but the two differ significantly in many respects. For instance, andperhaps most strikingly, Proj R does not depend functorially on R in the sensethat maps between graded rings do not always give maps between the projectiveschemes. Moreover, different R’s may yield isomorphic projective spectra.

Before we begin the construction, we include a motivating section recallingsome features of the projective spaces over the complex numbers.

9.1 Motivation

Let us recall the usual construction of complex projective space: as a topologicalspace CPn is the quotient space

CPn =(

Cn+1 ´ 0)

/C˚

by the action of C˚ on Cn+1 that scales the coordinates. Of course the orbitsof C˚ in Cn+1zt0u are just the lines through the origin, which is the traditional‘variety-way’ of thinking about CPn.

We can translate this into algebra as follows: for each polynomial function fon Cn+1 and each λ P C˚ a complex number, we get a new polynomial functionf λ by defining f λ(x) = f (λx), and this gives an action of C˚ on the polynomialring C[x0, . . . , xn]. Now, the actions of C˚ are well-understood: any time C˚ actson a complex vector space V (in a not too exotic manner), we can decomposethat vector space into the direct sum

V =ÿ

dPZ

Vd

motivation 167

of eigenspaces; that is, each Vd is the subspace of vectors v with C˚ acting asvλ = λdv. Thus for any C-algebra R on which C˚ acts through algebra homo-morphisms — which means that ( f g)λ = f λgλ — the algebra R decomposes asR =

À

dPZ Rd as a vector space over C, and since the action is through algebrahomomorphisms, one readily verifies that Rd ¨ Rd1 ĎRd+d1 . In other words, Ris a graded algebra. In the illustrative case at the top, the induced grading onC[x0, . . . , xn] is just the usual one.

Leaving the realm of complex manifolds and entering the world of schemes,we want to take the quotient of An+1

C´ 0 = Spec C[x0, . . . , xn]´V(x0, . . . , xn) by

this action. We write PnC for the corresponding quotient space equipped with

the quotient topology. The notation PnC, rather than CPn, is used to emphasize

that the quotient is taken with respect to the Zariski topology, and not the usualtopology.

One may try to put a scheme structure on PnC by looking for reasonable

open covers. Note that the open subsets of PnC correspond to C˚-invariant open

subsets of An+1C

´ 0. It is not too hard to see that D( f ) Ă An+1C

is C˚-invariantif and only if f is a homogeneous polynomial, and in case it is, we shall writeD+( f ) Ă Pn

C for the open subset corresponding to D( f ) Ă An+1C

´ 0.To define a structure sheaf on Pn

C we must figure out what the spaces ofsections OPn

C(D+( f )) should be. While it is true that D( f ), being an affine

scheme, has a structure sheaf whose global sections is C[x0, . . . , xn, f´1], we haveto take more care in deciding which sections to take, to make things compatiblewith the C˚-action: a function on D+( f ) should be a function on D( f ) that isinvariant under the action of C˚. That is, we should have gλ = g, which meansprecisely that g has degree zero. Thus we define

OPnC(D+( f )) = C[x0, . . . , xn, f´1]0,

where the subscript means that we take the degree 0 part.We can generalize the above construction for any affine C-scheme with an

action of C˚. Such a scheme corresponds to a graded C-algebra R. To make areasonably good quotient it is necessary to remove the locus in Spec R that isfixed by C˚, and it is not too hard to prove the following:

Lemma 9.1 The fixed locus of C˚ acting on Spec R is V(R+), where R+ denotes theideal generated by elements of positive degree.

We then proceed to consider the quotient space P of Spec R´V(R+) by C˚.Again, the C˚-invariant distinguished open subsets in Spec R of the form D( f )where f is homogeneous, constitute a basis for the topology on Spec R´V(R+).These descend to open subsets D+( f ) Ă P = (Spec R´V(R+)) /C˚, whichform a basis for the quotient topology. Finally, whenever f is homogeneous,OSpec R(D( f )) has a natural grading, and we may define a B-sheaf on P by

basic remarks on graded rings 168

setting OP(D+( f )) = OSpec R(D( f ))0, and of course, we must check that we geta scheme P.

Beside of inducing a grading on R, the action of C˚ plays very little role here.Realizing this, we can in fact build a scheme P from any graded ring R: Weconstruct the topological space of P from the set of homogeneous prime ideals ofR (with the induced Zariski topology), and define a structure sheaf on it by theformula like the one above. This is essentially the ‘Proj’-construction.

9.2 Basic remarks on graded rings

We begin the story about projective schemes with a short recap of the basics ofgraded rings and modules (see also CA Section ??, p. ?? and Section ??, p. ??).In the literature one can find a diversity of definitions of what a graded ringshould be, of more or less general flavour (one can even allow the degree totake values in any monoid), but we shall adhere to the very simple convention:A graded ring R is a ring with a decomposition

R =à

nPN0

Rn = R0 ‘ R1 ‘ ¨ ¨ ¨

as an abelian group such that Rm ¨ Rn Ď Rm+n for each m, n ě 0. Note that R0 isa subring of R and that each of the Rn’s is an R0-module. The elements in Rn

are said to be homogeneous of degree n, and one writes deg x = n when x P Rn.(Note that 0 has no well-defined degree, but is considered homogeneous of anydegree.)

Every non-zero element x P R can be expressed uniquely as a sum x =ř

nPN0xn with xn P Rn. The non-zero terms in the sum are called the homogeneous

components of x.A homomorphism φ : R Ñ S between two graded rings R and S is homo-

geneous of degree d if φ(Rn)Ď Sdn. In fact, one may allow d to be a rationalnumber, but except in a few rather exotic cases, it will be a natural number. Thegraded rings together with the homogeneous maps form a category GrRings.That two graded rings are isomorphic means that they are isomorphic as ringsand that the gradings are the same except for a possible scaling of the degrees.

An R-module M is graded if it has a similar decomposition M =À

nPZ Mn

as an abelian group such that Rm ¨ Mn Ď Mm+n for all. A map of graded R-modules is an R-linear map φ : M Ñ N such that φ(Mn) Ă Nn for all n P Z.Note that contrary to what we required for maps between graded rings, degreesare preserved.

As usual, a non-zero element x P M is homogeneous of degree n if it lies inMn. Just like ring elements, any member x P M may be expressed in a uniqueway as x =

ř

nPN0xn with each xn in Mn, and the non-zero terms are called the

homogeneous components of x.

basic remarks on graded rings 169

An ideal a Ă R is homogeneous if the homogeneous components of eachelement in a belongs to a. This is the case if and only if a is generated byhomogeneous elements. It is readily verified that intersections, sums andproducts of homogeneous ideals are homogeneous.

We will write R+ for the sumÀ

ną0 Rn; this is naturally a homogeneous idealof R, which we call the irrelevant ideal. If R = k[x0, . . . , xn] with the standardgrading, then R+ = (x0, . . . , xn).Example 9.2 Veronese rings. Common examples of graded rings are the so-called

Veronese rings(Veronese-ringer)

Veronese rings rings associated to a graded ring R. For any natural number d, welet let R(d) denote the subring of R given by

À

ně0 Rnd. K

LocalizationOccasionally we will meet graded rings having elements of negative degree;they are defined as above except that they decompose as

R =à

nPZ

Rn.

These are sometimes called Z-graded rings. One way such rings appear isas localizations of graded rings. Indeed, if TĎR is a multiplicative system allwhose elements are homogeneous, one may define a grading on T´1R by lettingdeg g/t = deg g´ deg t for t P T and g a homogeneous element from R. Inother words, one puts

(T´1R)n = t f /t P T´1R | f P Rn, t P T and deg f ´ deg t = n u.

Then, as is easily verified, the localized ring T´1R decomposes as the directsum as T´1R =

À

nPZ(T´1R)n, which makes it a Z-graded ring. The same

construction also works very well for graded modules, so that T´1M is a gradedmodule whose homogeneous elements are of shape xt´1 with x homogeneousand deg xt´1 = deg x´ deg t.

One example of multiplicative sets of the graded sort, is the sets T(p) consist-ing of all homogeneous elements in R not lying in a given homogeneous primeideal p. Another example is the set S of non-negative powers of a homogeneouselement f .

Some algebraic geometry-texts use the notations M(p) and M( f ) for thedegree zero part of the localizations Mp and M f respectively. We will howevernot adopt this notation (e.g. writing things like ‘k[x](x)’ gets a little bit confusingand one ends up with monstrous notation like ‘k[x]((x))’).Example 9.3 For the polynomial ring R = A[x0, . . . , xn] equipped with thestandard grading, the degree 0 part of the localization Rxj is generated by themonomials x0x´1

j , . . . , xnx´1j , so

(Rxj)0 = A[x0x´1j , . . . , xnxj

´1].

K

the proj construction 170

Exercises(9.1) Let R be a graded ring and p a homogeneous prime ideal. Show thatˇ

(Rp)0 is a local ring whose maximal ideal is given as q = t f g´1 | f P p, g PT(p), deg f = deg g u.(9.2) Let R be a graded ring and p a homogeneous ideal in R. Show that p isˇ

prime if and only if x ¨ y P p implies x P p or y P p for all homogeneous elementsx and y.(9.3) Let R and S be graded rings and φ : R Ñ S a homomorphism of gradedrings. Show that the inverse image φ´1(p) of an ideal pĎ S is homogeneouswhenever p is.

M

9.3 The Proj construction

Motivated by the discussion in the introduction, we make the following defini-tion:

Definition 9.4 Let R be a graded ring. We denote by Proj R the set of homoge-neous prime ideals of R that do not contain the irrelevant ideal R+. It is calledthe projective spectrum of R.

One endows Proj R with a topology by letting for each homogeneous ideal a,

V(a) = t p P Proj R | p Ě a u,

and just like in the case of Spec R, these sets comply with the axioms forclosed sets of a topology. This topology is called the Zariski topology on Proj R.Indeed, the following identities hold true; the conditions that the primes arehomogeneous and not contain the irrelevant ideal, do not disturb the proofswhich remain mutatis mutandis the same as for the closed sets in Spec R:

i) V(ř

ai) =Ş

V(ai);

ii) V(ab) = V(aX b) = V(a)YV(b);

iii) V(?a) = V(a),

where a, b and the ai’s are homogeneous ideals. The key point is that sums,products and radicals persist being homogeneous when the involved ideals are.

The reason behind the name ‘the irrelevant ideal’ is that R+ does not playany role when it comes to forming closed sets in Proj R, neither do ideals whoseradical equals R+. This is made clear by the following lemma. Note that bydefinition we have V(R+) = H.

Lemma 9.5 For any homogeneous ideal a it holds that V(a) = V(aX R+). In fact, ifI is an ideal such that

?I = R+, it holds that V(a) = V(aX I).


Proof: Since V(R+) = H, condition iii) above implies that V(I) = H, andcondition ii) then gives V(aX I) = V(a)YV(I) = V(a). o

Thus, when constructing the closed sets V(a), it suffices to work with idealscontained in the irrelevant ideal. In fact, we can take a lying in any prescribedpower of the irrelevant ideal.

Incidentally, we will not get more closed sets if we allow all ideals a and notjust the homogeneous ones: any ideal a has a ‘homogenization’ associated withit, which is the ideal generated by all homogeneous components of membersof a, and which gives the same closed subset V(a) of Proj R — a homogenousprime ideal contains a if and only if all homogenous components of elementsin a lie in it. In other words, the Zariski topology on Proj R is nothing but thetopology induced on Proj R from the Zariski topology on Spec R by means ofthe inclusion Proj R Ď Spec R.

Distinguished open subsetsAs in the affine case, there are some distinguished open sets. For each f P Rwhich is homogeneous of positive degree, we let D+( f ) be the collection ofhomogeneous ideals in (not containing the irrelevant ideal R+) that do notcontain f , or in other words, D+( f ) = D( f )X Proj R. These are open sets withrespect to the Zariski topology on Proj R; the complement of D+( f ) equals theclosed set V( f ).

The next result is important in understanding the local structure of Proj R.In particular, it will be essential when we define the scheme structure on Proj R.

Proposition 9.6 We have D+( f ) X D+(g) = D+( f g). Also, the D+( f )form a basis for the topology on Proj R when f runs through the homogeneouselements of R of positive degree.

Proof: The first part is evident by the definition of a prime ideal. The secondfollows as in the affine case: V(a) is the intersection of the V(( f ))’s for thehomogeneous f P aX R+, so Proj R´V(a) is the union of these D+( f ). Henceevery open set is a union of sets of the form D+( f ). o

Exercise 9.4 Let R be a graded ring and f and t fiuiPI homogenous elementsfrom R. Show that the distinguished open sets D+( fi) cover D+( f ) if and onlyif a power of f lies in the ideal ( fi|i P I). M

Dehomogenization and homogenizationIn the affine case, there is a canonical homeomorphism between D( f ) andSpec R f which associates a prime p P D( f ) with the prime ideal pR f . In perfectanalogy with this, associating the degree zero part of pR f with p P D+( f ) givesa homeomorphism between D+( f ) and Spec (R f )0.


Example 9.7 To illustrate this correspondence in a simple example, whichwill hopefully clarify what’s going on in the general case, let us consider thering R = k[x, y, z], and the distinguished open set D+(z). The monomialsof degree zero in Rz are products of powers of xz´1 and yz´1, so we have(Rz)0 = k[xz´1, yz´1]. Consider then a principal ideal a = ( f ) in R generatedby a homogeneous polynomial f of degree d. Because z is invertible in Rz, andbecause of the identity

f (xz´1, yz´1, 1) = z´d f (x, y, z),

the ideal aRz becomes aRz = (z´d f )Rz; moreover, since z´d f is of degree zero itlies in (Rz)0, and consequently it holds true that

(aRz

)0 = aRz X R0 = (z´d f )R0.

So when we pass to (Rz)0, the generator f is replaced by the dehomogenizedpolynomial z´d f .

There is also a straightforward way of making a polynomial g in the ringk[xz´1, yz´1] homogeneous: one simply gives g a factor zd with d being thedegree of g. This will almost all the time be an inverse to the dehomogenizationprocess above; there is just one fallacy: any factor of f which is a power of z,disappears when f is dehomogenized, and there is no means of recovering itknowing only z´d f . K

The general set up of the isomorphism D+( f ) » Spec (R f )0 follows thepattern in the example. Basically one dehomogenizes elements of the ideals withrespect to f (and homogenizes to get them back), but expressed in a necessarilygeneral formalism.

Proposition 9.8 Let R be a graded ring and let f P R be homogeneous ofdegree d. The canonical map φ : D+( f )Ñ Spec(R f )0 that is defined by

φ(p) = pR f X (R f )0,

has the following three properties:

i) φ is a homeomorphism;ii) For any homogeneous element g P R such that D+(g) Ď D+( f ),

letting u = gd f´deg g P (R f )0, we have φ(D+(g)) = D(u);iii) If a Ă R is a homogeneous ideal, then φ(V(a)XD+( f )) = V(aR f X

(R f )0).

Proof: Note that φ is just the restriction to Proj RX Spec R f of the canonicalmap Spec R f Ñ Spec(R f )0 induced by the inclusion map (R f )0 Ă R f . Thereforeit is continuous.

To prove i), that φ is a homeomorphism, we begin by construct the inversemap ψ : Spec(R f )0 Ñ D+( f ). Given a prime ideal p P Spec(R f )0, let us defineψ(p) to be the direct sum ψ(p) =

À

ně0 ψ(p)n where

ψ(p)n = tx P Rn|xd ¨ fń P pu.


One may think of ψ(p)n as consisting of the homogenous elements of degree nthat end up in p when dehomogenized (recall that d = deg f ).

The first thing to check is that ψ(p) is a homogeneous prime ideal. Once weknow it is an ideal, it will be homogeneous by definition, and it is nearly trivialto see it is closed under multiplication by elements from R. The tricky part isactually to prove it is an additive subgroup. To that end, assume we are giventwo elements x, y P ψ(p)n. By definition we have xd ¨ fń P p and yd ¨ fń P p, soby the Binomial theorem (x + y)2d ¨ f´2n P p; indeed, each term in the expandedsum contains either xd ¨ fń or yd ¨ fń. Therefore it holds that (x + y)d ¨ fń P p,and consequently ψ(p) is closed under addition.

To show that ψ(p) is prime, it suffices to verify the defining property forpairs of homogeneous elements, so we assume that x P Rm and y P Rn aretwo elements such that (xy)d f´(m+n) P p. Since f R p, it holds that (xy)d P p,and hence either x P p or y P p. Therefore we have either xd f´m P ψ(p)m oryd fń P ψ(p)n; in other words, ψ(p) is prime.

That the maps φ and ψ are mutually inverse follows almost by construction.Indeed, to verify pĎψ(pR f X (R f )0), note that elements x P pn = pX Rn satisfyxd fń P pR f X (R f )0. Conversely, if x P ψ(pR f X (R f )0)n, then xd fń P pR f X

(R f )0, and there is an N ą 0 so that f Nxd P p. As p is prime, and f R p, we havexd P p, and hence x P p. This proves that ψ ˝ φ = 0 and proving φ ˝ ψ = id isdone similarly. Thus φ is bijective. That it is a homeomorphism follows from itbeing continuous and open — openness ensues from statement ii), which wenow proceed to prove.

Proof of ii): Let g P R be an element with D+(g) Ă D+( f ). Then forp P D+( f ), the following series of equivalences holds true since pR f is a primeideal:

p P D+(g)ô gr f´s R pR f for some r, s ą 0

ô gr f´s R pR f for all r, s ą 0

ô gdeg f f´deg g R pR f X (R f )0 = φ(p)

Hence φ(D+(g)) = D(u).Proof of iii): This follows essentially by the definition of φ. Let p P V(a)X

D+( f ), so that aĎ p and f R p. Then aR f X (R f )0Ď pR f X (R f )0, which givesthe inclusion ‘Ď1. Conversely, given a prime ideal p Ď (R f )0 such that aR f X

(R f )0Ď p, its preimage p1 = pX R will be a homogeneous prime ideal in R notcontaining f , and so φ(p1) = p1R f X (R f )0 Ě aR f X (R f )0. This completes theproof. o

There is an analogue homogenization-dehomogenization process for mod-ules, which we shall need later on when dealing with coherent sheaves onProj R:


Proposition 9.9 Let R be a graded ring and M a graded R-module. Letf and g be two homogeneous elements such that D+(g)ĎD+( f ). If we letu = gd f´deg g P (R f )0, there is a canonical homomorphism (M f )0 Ñ (Mg)0

which induces an isomorphism ((M f )0)u » (Mg)0.

Proof: Write gk = a f . The localization map M f Ñ Mg is given by x f´m ÞÑ

amxg´mk, where x P M. This induces a map (M f )0 Ñ (Mg)0 because deg x +

m(deg a´ k deg g) = deg x´m deg f . The element u acts as an invertible ele-ment on (Rg)0, so the map (M f )0 Ñ (Mg)0 factors via a map

ρ : ((M f )0)u Ñ (Mg)0.

We claim that this is an isomorphism.ρ surjective: Explicitly, we have

ρ(

x fńu´m) = f tmńxg´dm,

where t = deg g. Take any element y ¨ g´l P (Mg)0 where deg y = tl. Choose mlarge so that dm ě l. Define x = gdm´ly ¨ f´(tmń) P (M f )0. We have deg x = nd,and hence x fń ¨ u´m P ((M f )0)u is an element that maps to the given y ¨ g´l .

ρ is injective: If x fń P (M f )0 maps to 0 in (Mg)0, then there is an l ą 0 sothat gldanx = 0 P M. Multiplying up by powers of a and f , we get a relationof the form g(l+n)dx = 0 P M, and hence u(l+n)dx = 0 P (M f )0. But thenx fń = 0 P ((M f )0)u. o

Proj R as a schemeWe shall now give Proj R a scheme structure, and the first step will be to makeit a locally ringed space. In other words, we need to define the structure sheaf —a sheaf of rings OProj R— and while doing this, relying on Proposition 9.5, weshall see that the locally ringed space is locally affine. So (Proj R,OProj R) willbe a scheme. The order of the day is: restrict OSpec R to Proj R and ‘take degreezero parts’.

To carry out the order of the day, we let B be the basis for the topology onProj R consisting of the distinguished open subsets. For each D+( f ), we set

O(D+( f )) = (R f )0.

The localization maps R f Ñ Rg are all homogenous of degree zero so that (R f )0

maps into (Rg)0, and we may use the maps (R f )0 Ñ (Rg)0 as restriction maps.This gives us a B-presheaf O. We proceed to establish that O is a B-sheaf: IftD+( fi)u is a covering of D+( f ) (with the fi’s homogeneous), the distinguishedopen subsets D( fi) of Spec R will cover D( f ), and consequently the standardsequence

0 R fś

i R fi

ś

i,j R fi f jα β

(9.1)


will be exact simply because OSpec R is a sheaf. Singling out pieces of degreezero is an exact operation and applied to (9.1) yields the exact sequence

0 (R f )0ś

i(R fi)0ś

i,j(R fi f j)0α β

which exactly says that O is a B-sheaf. We denote the unique sheaf extensionof O by OProj R. Notice that the formula OProj R(D+( f )) = (R f )0 is still valid.

According to Proposition 9.5 on page 172 there is a canonical homeo-morphism D+( f ) » Spec(R f )0 which sends D+(g) Ă D+( f ) to the subsetD(u) Ď Spec(R f )0 with u = f deg gg´deg f . Because u is of degree zero, it holdsthat (Rg)0 »

((R f )0

)u. This means that O restricts to the B-sheaf induced

by the structure sheaf on Spec(R f )0, and so OProj R restricts to OSpec (R f )0. The

locally ringed space (Proj R,OProj R) is therefore locally affine; in other words itis a scheme.

Definition 9.10 For a graded ring R, we call the scheme (Proj R,OProj R) theprojective spectrum of R.

The projective spectrum Proj R is in a natural way a scheme over Spec R0:the structure map π : Spec R Ñ Spec R0 restricts to a continuous map on Proj R,which turns out to be a morphism. For this to be true, it suffices that itsrestriction to D+( f ) be a morphism for each homogeneous f . But under theidentification φ : D+( f ) » Spec (R f )0 from Proposition 9.5 this restriction turnsinto the composition π|D+( f ) ˝ φ´1, and one check that this coincides with thestructure map Spec (R f )0 Ñ Spec R0 (that comes from the map R0 Ñ (R f )0); inother words, it holds true that

φ(p)X R0 =(pR f X (R f )0

)X R0 = pX R0.

Indeed, one inclusion is obvious, and if for some x P p it holds that y = fńx PR0, we find y P p since x = f ny lies there, but f does not.

Among the most prominent varieties are the projective spaces, and they haveanalogues over any ring, in fact over any base scheme.

Definition 9.11 We define the projective n-space to be the scheme

Pn = Proj Z[x0, . . . , xn].

More generally, for a ring A, the projective n-space over A is the scheme

PnA = Proj A[x0, . . . , xn].

There is also a projective n-space PnS over any scheme S. It is defined as

PnS = Pn ˆZ S; it may be obtained by gluing together the projective schemes Pn

A(which are open subsets in Pn

S), one for each affine open set U = Spec A Ă S.


Examples(9.12) Let A be a ring and let R = A[t] with the grading given by deg t = 1and deg a = 0 for all a P A. Then the structure map gives an isomorphismProj R » Spec A.(9.13) The projective line P1

A once more. Let us study the case of a polynomialring in R = A[s, t] where s and t have degree one and see that P1

A = Proj R tiesup with the version of P1

A as defined in Chapter 5 (in Section 5.3 on page 105);indeed, we shall see that the new P1

A is glued together from affine schemes inprecisely the same manner as is the old P1

A.Note that Proj R is covered by the distinguished open sets D+(s) and D+(t)

(since s and t generate the irrelevant ideal). Write for simplicity U = D+(s) »Spec (Rs)0 and V = D+(t) » Spec (Rt)0. Then Proj R is glued together fromU, V along U XV = D+(st) » Spec(Rst)0.

Note first that the degree zero part of Rs » A[s, s´1, t] equals A[s´1t], andsymmetrically it holds that (Rt)0 = A[st´1]. The intersection D+(st) is thedegree zero part of Rst which is given as (Rst)0 = A[s´1t, st´1]. In otherwords, if we write u = s´1t, it holds true that (Rs)0 = A[u], (Rt)0 = A[u´1]

and that (Rst)0 = A[u, u´1] = A[u]u. Hence U » Spec A[u] = A1A and V »

Spec A[u´1] » A1A, and they are patch together along Spec A[u, u´1], exactly as

in the gluing scheme used to construct the old P1A in Section 5.3.

(9.14) Projective n-space. In the same vein, we can show that PnA = Proj R where

R = A[x0, . . . , xn] coincides with the previous construction of PnA via gluing.

The case when A = k is a field is the most interesting. In this case Pnk is

a scheme whose closed k-points Pn(k) coincides with the variety of projectiven-space.

Since Pnk is covered by n + 1 copies of An

k , Pnk is reduced and it is also

irreducible, since Spec R ´ V(R+) is. Thus Pnk is an integral scheme. Then,

since Ank is a dense open subset, we have k(Pn

k ) = k(Ank ) = k(X1, . . . , Xn). In

particular, this has transcendence degree n over k, so that Pnk has dimension n

by Theorem 6.10. More intrinsically, we may also write

k(Pn) =

"

g(x0, . . . , xn)

h(x0. . . . , xn)

ˇ

ˇ g, h homogeneous of the same degree*

.

(9.15) Proj of a one dimensional ring. Let R = k[x, y]/(xy) with the naturalgrading. Speaking geometrically, Spec R is the union of the x- and y-axes, andso Spec R´V(x, y) is the union of the axes with the origin excluded, and weexpect Proj R to consist of two points. On the other hand, R has, apart from theirrelevant ideal R+ = (x, y), just two homogeneous prime ideals, (x) and (y), soProj R indeed has just two points.

K


Some basic properties of Proj RA few fundamental properties are as follows:

Proposition 9.16 (Properties of Proj R) Let R be a graded ring.

i) Proj R is separated;ii) If R is Noetherian, then Proj R is Noetherian; in particular, Proj R is

quasi-compact;iii) If R is finitely generated over R0, then Proj R is of finite type over

Spec R0;iv) If R is an integral domain, then Proj R is integral.

Proof: We use the fact that X is covered by the affine open sets D+( f ) where fruns over the homogeneous elements of R+. These open sets are clearly affine(Proposition 9.5), and so is their intersection: D+( f )XD+(g) = D+( f g). Thusto prove that Proj R is separated, we need only check condition ii), namely that(R f )0 b (Rg)0 Ñ (R f g)0 is surjective for any f , g P R+, which it is.

The remaining properties can all be checked on an affine covering, althoughin ii) and iii) it must be finite. In our case Proj R is covered by the affinesSpec(R f )0, which are Noetherian (respectively of finite type or integral) providedR is Noetherian (respectively finitely generated or an integral domain) and inboth cases that R is Noetherian or of finite type over R0, the irrelevant ideal isfinitely generated, and so Proj R is covered by finitely many D+( f )’s; (let f runthrough a set of generators of R+). o

Example 9.17 When the ring R is not Noetherian, it may very well happen thatProj R is not quasi-compact (!), which is in stark contrast with the case of affineschemes: the prime spectrum Spec A is always quasi-compact whatever the ringA is. For an explicit example we may take R = k[x1, x2, . . . ] to be a polynomialring in infinitely many variables. Then R+ = (x1, x2, . . . ), and Proj R is coveredby the open sets D+(xi), but there is clearly no finite sub-cover: if I is any finitesubset of N, the family tD+(xi) | i P I u does not cover Proj R as its union doesnot contain the prime ideal (xi|i P I). Compare this with the quasi-affine schemeSpec R´ tR+u, which neither is quasi-compact.

This situation is somewhat counterintuitive, given the usual heuristic thatcomplex projective varieties (i.e. closed subsets of the compact space CPn) arecompact, whereas affine varieties (e.g. An or A1 ´ 0) are not. The explanationis that the usual notions of ‘compactness’ do not behave so well in the Zariskitopology; there are other notions like ‘properness’ which better capture theproperties we want. K

functoriality 178

Exercises(9.5) Prove that for a graded ring R, and homogeneous elements f , g P R, thenatural map (R f )0 b (Rg)0 Ñ (R f g)0 is surjective.(9.6) If R is an integral domain, show that the function field of X = Proj R isˇ

given as

k(X) =!g

h| g, h have the same degree

)

Ď K(R) (9.2)

.(9.7) Show that Proj R is empty if and only if every element in R+ is nilpotent.(9.8) Give examples of a non-noetherian graded ring R such that Proj R isˇ

Noetherian, of R that is not of finite type over a field k, but Proj R is, and R whichis not an integral domain, but whose projective spectrum Proj R is integral.Hint: The irrelevant ideal is irrelevant.

M

9.4 Functoriality

Unlike the case of affine schemes, the proj-construction is not entirly functorial.A graded ring homomorphism φ : R Ñ S does not induce a morphism betweenthe projective spectra Proj S and Proj R. The reason is that some primes in Proj Smay pullback to R to contain the irrelevant ideal R+. However, as we will seeshortly, this is the only obstruction to defining a morphism, and discarding thebadly behaved primes, we find and open set where a morphism can be defined.

Given a homomorphism φ : R Ñ S of graded rings we introduce the setG(φ) Ď Proj S of homogeneous prime ideals p in S that do not contain φ(R+);or equivalently, they do not contain φ(R+)S = R+S. In particular, these primeideals have their inverse images φ´1(p) in Proj R, and the assignment p ÞÑ φ´1(p)

sets up a mapF : G(φ)Ñ Proj R.

The set G(φ) is an open subset of Proj S being the complement of V(R+S) andhas the canonical induced scheme structure as an open subscheme of Proj S.Giving it that structure, we have:

Proposition 9.18 Let φ : R Ñ S be a homomorphism of graded rings. Thenthe map F : G(φ)Ñ Proj R is a morphism of schemes.

Proof: First of all, the map F is continuous because the Zariski topologieson Proj R and Proj S are induced from those of Spec S and Spec R, and F is therestriction of the map between the two Spec ’s induced by φ.

As usual, it suffices to check that the restriction of F will be a morphism oneach distinguished open subset, and that these agree on intersections. To beprecise, we consider distinguished open subsets D+( f ) and D+(φ( f )); whenf runs through R+ the former cover Proj R and the latter G(φ). Note that

functoriality 179

F´1(D+( f )) equals D+(φ( f )) — which is contained in the set G(φ) becauseG(φ) = V(R+S)c.

We rely on the canonical isomorphisms between D+( f ) and Spec(R f )0 andbetween D+(φ( f )) and Spec (S f )0 established in Proposition 9.5. The naturalmap (R f )0 Ñ (Sφ( f ))0 induced by φ gives a morphism D+(φ( f )) Ñ D+( f ),whose underlying topological map clearly equals F|D+(φ( f )), just follow a homo-geneous prime ideal in S the two ways to (R f )0 around the diagram

(Sφ( f )) S f S

(R f )0 R f R.

That these morphisms match up over intersections, is a matter of easy verifica-tion: indeed, the map f : (R f g)Ñ Sφ( f g) induced by φ, equals the localization ofboth the maps R f Ñ Sφ( f ) and Rg Ñ Sφ(g) induced by φ. o

Example 9.19 Projection from a point. To illustrate why restriction to the openset G(φ) is necessary, we consider the case where R = k[x0, x1], S = k[x0, x1, x2]

and φ is the inclusion map. Note that the prime ideal a = (x0, x1) defines anelement in Proj S, but its restriction to R is the whole irrelevant ideal of R. Infact, G(φ) = Proj S´V(a), and the map

ψ : P2k ´V(a)Ñ P1

k

is nothing but the projection from the point (0 : 0 : 1) which sends a point withhomogeneous coordinates (x0 : x1 : x2) to the one with coordinates (x0 : x1). Itis a good exercise to prove that there can be no morphisms Pm

k Ñ Pnk for m ą n

in general. (See Section 16). K

Example 9.20 Consider the map φ : k[u, v]Ñ k[x, y] of graded k-algebras definedby the two assignments u ÞÑ x2 and v ÞÑ y2. The exceptional set G(φ) isempty, because if a prime ideal pĎ k[x, y] contain s (u, v)k[x, y] = (x2, y2), it willcertainly contain (x, y). Hence the map φ gives rise to a morphism P1

k Ñ P1k . Its

action on k-points is (a0 : a1) ÞÑ (a20 : a2

1). K

Exercise 9.9 Let A be a ring and let R = A[x0, . . . , xn]. Define a morphism ofˇ

schemesπ : An+1

A ´V(R+)Ñ PnA

that generalizes the usual quotient construction of CPn. M

Exercise 9.10 In this exercise A denotes a ring. Consider the homomorphismof graded rings φ : A[x0, x1, x2]Ñ A[x0, x1, x2] defined by the three assignmentsxi Ñ xjxk where the indices satisfy ti, j, ku = t1, 2, 3u. Determine the open setG(φ) in the two cases

a) A = k is a field;

b) A is the ring of integers.

M

functoriality 180

Closed immersionsA large number of examples of morphisms between projective spectra as con-structed above are the ones associated with graded quotient homomorphismφ : R Ñ R/a, where a Ă R is a homogeneous ideal. In this case φ(R+) = (R/a)+so G(φ) is the entire spectrum Proj R/a, and the corresponding map ι is definedeverywhere. Thus we obtain a morphism

ι : Proj R/aÑ Proj R,

whose image is V(a). We contend that ι is a closed immersion, and notice thatone may verify this on an open cover of Proj R. So let f P R be a homogeneouselement. We know that ι´1(D+( f )) = D+(φ( f )), and that the restriction of ι toι´1(D+( f )) may be identified with the morphism

Spec ((R/a)φ( f ))0 Ñ Spec (R f )0

induced by the degree zero part of the localization R f Ñ (R/a) f of φ. But thisis obviously surjective, hence ι|ι´1(D+( f )) is a closed immersion.

We will prove in Chapter 12 that, in fact, any closed immersion arises in thisway, under some mild assumptions on R.Example 9.21 Homogeneous coordinates. The simplest conceivable closedimmersion is that of a closed point in Pn

k . At least if k is algebraically closed,such points a are given by their homogeneous coordinates a = (a0 : ¨ ¨ ¨ : an), themaximal ideal corresponding to a is generated by the minors of the matrix(

x0 x1 . . . xn

a0 a1 . . . an

). (9.3)

Indeed, the vanishing of those minors describes vectors in kn+1 dependent on(a0, . . . , an); or in other words, points lying on the line through (a0, . . . , an).

There is an analogue of this for projective spaces PnA over an arbitrary ring

A, that to an n-tuple a = (a0, . . . , an) of elements from A gives an A-point ofPn

A; that is, a section of the structure map π : PnA Ñ Spec A. The appropriate

necessary condition on the ai’s (generalizing the condition that not all ai arezero) is that the ai’s generate the unit ideal in A. Moreover, two such tuples givethe same section if and only if they are proportional by a unit from A.

Let a be the ideal in A[x0, . . . , xn] generated by the minors of the matrix (9.3);in other words

a = (aixj ´ ajxi |0 ď i, j ď n).

We claim that π induces an isomorphism between V(a) and Spec A; its in-verse will then be a closed embedding ιa : Spec A Ñ Pn

A. The open distin-guished sets D(ai) cover Spec A, and it will suffice to see that the restrictionπ|π´1(D(ai))

: V(a)X π´1(D(ai)) Ñ D(ai) is an isomorphism for each i. So re-placing Spec A by D(ai), we may well assume that one of the ai’s, say a0, is

projective schemes 181

invertible. Since a0xi ´ aix0 belongs to a, we deduce that xi ´ aia´10 x0 P a, and

hence A[x0, . . . , xn]/a = A[x0]. By Example 4, it follows that the structure maprestricts to an isomorphism on V(a). Clearly a simultaneous scaling does notchange aia´1

0 , and if aia´10 = a1ia

1´10 , it holds that a1i = a10a´1

0 ai.It is not true in general that all maps Spec A to Pn are of the ‘homogeneous

coordinate form’ (a1 : ¨ ¨ ¨ : an), but if A is local (e.g. a field) it holds true.

Lemma 9.22 Assume that A is a local ring. Then every section Spec A Ñ PnA of the

structure map is given by (a1 : ¨ ¨ ¨ : an) where at least one ai is a unit. Another suchtuple (a11 : ¨ ¨ ¨ : a1n) gives the same map if and only if a1i = αai for a unit α P A.

One must remember that the lemma is relative to a fixed sequence of variablesx0, . . . , xn.Proof: Assume that a section f : Spec A Ñ Pn

A of the structure map be given.Then the image of the closed point lies in D+(xν) for some ν, and so f factorsthrough D+(xν). This means that f 7 is a map from A[xνx´1

ν , . . . , xnx´1ν ] to A;

the image ai = f 7(xix´1ν ) are elements in A and (a0 : ¨ ¨ ¨ : 1 : ¨ ¨ ¨ : an) will be the

appropriate homogeneous coordinates giving the map f (where the ‘one’ is inthe ν-th slot); indeed, with a as above, the section f factors through V(a), andas the structure map of V(a) is an isomorphism, f will be an isomorphism ontoV(a) o

K

9.5 Projective schemes

Let S be a scheme and let X be a scheme over S. We say that X is projective overS, or that the structure morphism f : X Ñ S is Projective morphisms

(projektive morfier)projective, if f : X Ñ S factors

as f = π ˝ ι where ι : X Ñ PnS is a closed immersion and π : Pn

S Ñ S is theprojection. X is Quasi-projective

morphisms(kvasiprojektivemorfier)

quasi-projective over S if X Ñ S factors via an open immersionX Ñ X and a projective S-morphism X Ñ S.1

The primary examples is of course X = PnA Ñ Spec A for a ring A. More

generally, if X = Proj R where R is a graded R0-algebra generated in degree oneby finitely many elements and S = Spec R0, then X is projective over S, sincein this case, we can define the projective immersion ι by taking a surjectionR0[x0, . . . , xn]Ñ R, which upon taking Proj, gives a closed immersion X Ñ Pn

R0.

Note that projectivity is a relative notion: it is the morphism X Ñ S whichis projective, not X itself. For instance, P1

k[t] is projective over Spec k[t], but it isnot over Spec k. Intuitively, it is the fibers of X Ñ S which are projective; in theexample, the (scheme-theoretic) fiber over s P S = Spec k[t] equals the projectiveline P1

k(s) over k(s). Still, if we are working in the category of schemes over, say,

1There exist slightly different definitions in the literature, see https://stacks.math.columbia.edu/tag/01VW

https://stacks.math.columbia.edu/tag/01VW

https://stacks.math.columbia.edu/tag/01VW

the veronese embedding 182

a field k or Z, we still refer to a scheme X being ‘projective’ when it is projectiveover the base scheme.Example 9.23 For A = C[t], the scheme X = Proj A[x, y, z]/(zy2 ´ x3 ´ txz2) isprojective over A1

C = Spec A. The fiber of X Ñ A1C over any closed point a P A1

C

is an integral projective subscheme of dimension one: V(zy2 ´ x3 ´ axz2) Ă P2C.

K

9.6 The Veronese embedding

Let R be a graded ring and let d be a positive integer. In Example 1 we definedthe Veronese ring R(d) associated to a graded ring R as

À

n Rdn. I this sectionwe aim at showing that the inclusion φ : R(d) Ñ R induces an isomorphism

vd : Proj R Ñ Proj R(d).

The first step will naturally be to show that vd is a morphism. This is truebecause in this case G(φ) = Proj R since any prime p such that p Ě R+ X R(d)

must also contain all of R+; indeed, if a P R+, note that ad P R+ X R(d) and soa P p as well! The map vd is called the The Veronese

embedding (Veronese-embeddingen)

Veronese embedding, or the d-uple embeddingof Proj R.

Proposition 9.24 The Veronese embedding vd is an isomorphism.

Proof: There are many things to check here, so we will sketch the proof, andleave the remaining verifications for the reader.

First we note that vd is injective: if p, q P Proj R are two prime ideals suchthat pX R(d) = qX R(d), then for a homogeneous element x P R it holds true that

x P pô xd P pô xd P q,ô x P q

and hence p = q.To show that vd is surjective, let q P Proj R(d) and define the homogeneous

ideal in R by

p =8à

n=0

!

x P Rn|xd P q)

.

It is easy to see that multiplication by homogenous elements from R leaves p

invariant, and that if a product of homogeneous elements lies in p, one of thefactors does. Using the little trick from Proposition 9.5 considering (x + y)n+m

with x and y homogeneous of degree n and m, one infers it is additively closedas well. So p is a prime ideal. That pX R(d) = q follows immediately from p

being prime. So we conclude that vd is bijective.The proof then proceeds to show that the maps vd and its inverse are open,

hence they are homeomorphisms. Now, f d P R(d) for each homogeneous f P R+,so for each prime p P Proj R it holds true that f P p if and only if f d P pX R(d).

more intricate examples 183

This shows that vd maps D+( f ) bijectively onto D+( f d). For the different f , thedistinguished open subsets D+( f ) cover Proj R, and the D+( f d) cover Proj R(d),and it ensues that vd is a homeomorphism.

Finally, one checks that vd induces isomorphisms between the two distin-guished open sets D+( f ) and D+( f d); this is a consequence of the identificationin Proposition 9.5 on page 172 and the undemanding equality (R(d)

f d )0 = (R f )0.o

Example 9.25 Classic Veronese varieties. These varieties are named after the ItalianMathematician Guiseppe Veronese, who by many is considered the founderof algebraic geometry of higher dimensional varieties. He considered mapsPn Ñ PN given by a basis for the homogenous part of the polynomial ringR = k[x0, . . . , xn] of degree d (so N is the dimension of that space). For instance,the map P2 Ñ P5 that sends a point with homogeneous coordinates (x0 : ¨ ¨ ¨ : x2)

to (x20 : x0x1 : x0x2 : x2

1 : x1x2 : x22) is one of the sort whose image is the famous

Veronese surface.In Proj-terminology a Veronese embedding corresponds to the ring homo-

morphism k[t0, . . . , tN ] Ñ k[x0, . . . , xn] = R that sends each variable ti to acorresponding basis elements. The image of this map is precisely the Veronesering R(d), and thus it induces, according to Section 9.4 on page 180, a closedimmersion of Proj Rd into Proj R. As a footnote, let us note that this explainsthe a priori mysterious qualifier ‘embedding’ in the name ‘Veronese embedding’above. K

Remark on rings generated in degree one We will frequently assume thatthe ring R is generated in degree one, that is, R is generated as an R0-algebra byR1. The reason for this will become clear in the next section. Intuitively, it isbecause we want Proj R to be covered by the ‘affine coordinate charts’ D+(x)where x should have degree 1.

We remark that this assumption is in fact not very restrictive: Any projectivespectrum of a finitely generated ring is isomorphic to the Proj of a ring generatedin degree 1. This is because of the basic algebraic fact that if R is finitelygenerated, then some subring R(d) will have all of its generators in one degree,and since Proj R(d) » Proj R, we don’t change the Proj by replacing R with R(d).

Exercise 9.11 Let x and y be two points in Pnk . Prove there is an open affine

UĎPnk containing both x and y. M

9.7 More intricate examples

The construction of the projective spectra is definitely more subtle than theconstruction of the prime spectra. As we allude to in the introductory remark,one of the vast differences is that unlike the harmonious relation between ringsand prime spectra, which is one–to–one, a huge number of different graded


rings give rise to isomorphic Proj’s. Already the Veronese embeddings furnishan infinity of examples: for any natural number d, the two schemes Proj R andProj R(d) are isomorphic (see Section 9.6). The first following example givesexamples of a different kind. Our second example illustrates the versatility ofthe Proj-construction and its usefulness to making ‘quotient by k˚-actions’ otherthan ‘schemy-versions’ of the classical one from projective geometry, a taste ofthe so-called weighted projective spaces.Example 9.26 The weigthed projective spaces P(p, q). Let k be a field and p and qtwo relatively prime natural numbers and let d = pq. Consider the polynomialring R = k[x, y], but endow it with the non-standard grading giving x degreep and y degree q. We claim that Proj R » P1

k , or more specifically that R(d) isisomorphic to the polynomial ring A = k[u, v] graded in the non-standard butinnocuous way that deg u = deg v = d. Clearly Proj A » P1

k .Observe that a homogeneous element in R(d) is a linear combination of

monomials xαyβ with pα + qβ = dn; hence q divides α and p divides β and soα1 + β1 = n with α = qα1 and β = pβ1. There is a homomorphism of gradedk-algebras A = k[u, v] Ñ R(d) that sends u Ñ xq and v Ñ yp. This is injectivesince xq and yp are algebraically independent when x and y are, so to see it isan isomorphism, it will suffice to check it is surjective on each homogeneouscomponent: now, as we just saw, (R(d))dn has a basis consisting of the monomialsxqαypβ with α + β = n; and for the same α’s and β’s the monomials uαvβ form abasis for An. K

Example 9.27 The weighted projective space P(1, 1, p). This is another examplealong the same lines. Again we begin with a polynomial ring R = k[x, y, z]endowed with a slightly exotic grading; we put deg x = deg y = 1 and deg z = pfor some natural number p. Then Proj k[x, y, z] is a so-called weighted projectivespace and one often sees it denoted by P(1, 1, p).

As the reader might guess, both this example and the previous one arespecial cases of the general construct P(p1, . . . , pr) = Proj k[x1, . . . , xr] where wegive k[x1, . . . , xr] a grading by setting deg xi = pi.

The scheme X = Proj R has a covering of the three open affines D+(x), D+(y)and D+(z). Both D+(x) and D+(y) are isomorphic to A2

k ; it is a straightforwardexercise to verify that (Rx)0 = k[yx´1, zx´p] and (Ry)0 = k[xy´1, zy´p], andthat these are polynomial rings. However, the third distinguished open affineD+(z) is not isomorphic to A2

k . In fact, it has a singularity! Clearly xpíyiz´1,for 0 ď i ď p, are homogeneous elements of degree zero in (Rz)0, and it isalmost trivial that they generate (Rz)0, so that (Rz)0 = k[xpz´1, . . . , ypz´1]. Onerecognizes this ring as an isomorphic copy of the pth Veronese ring A(p) of thepolynomial ring A = k[u, v]. And anticipating parts of the story, this is the coneover a so-called projective normal curve of degree p, whose apex is a singularpoint. K

Example 9.28 The Blow-up as a Proj. Consider the ring A = k[x, y] and the ideal


I = (x, y). We can form a new graded ring by introducing a new formal variablet and setting

R =à

kě0Iktk

where I0 = A. In R, the new variable t has degree 1, and the other variablesx and y have degree 0. One may think about R as the subring of A[t] ofpolynomials shaped like

ř

ν avtν where the coefficient aν belongs to Iν.The map p ÞÑ pX A, induces a morphism

π : Proj R Ñ Spec A = A2k .

The irrelevant ideal R+ is generated by xt and yt so that Proj R is glued togetherby the two open affine subschemes Spec(Rxt)0 and Spec(Ryt)0.

Note that there is a map of graded rings φ : A[u, v]Ñ R, where both u and vare of degree one, which is given by the assignments

u ÞÑ xt

v ÞÑ yt.

This is surjective since I is generated by x and y. Note also that the kernelcontains the element xv´ yu. In fact, by Exercise 9.12 below, we have

Lemma 9.29 R » A[u, v]/(xv´ yu).

From this description we see that Proj R is covered by the two distinguishedopen sets D+(u) = Spec(Rv)0 and D+(v) = Spec(Rv)0. Here

(Ru)0 » (A[u, v]u/(xv´ yu))0 = k[x, vu´1]

and(Rv)0 » (A[u, v]v/(xv´ yu))0 = k[y, uv´1].

These are glued along Spec(Ruv)0 » (A[u, v]uv/(xv´ yu))0, and one finds

(A[u, v]uv/(xv´ yu))0 = k[x, y, uv´1, vu´1]/(x ¨ vu´1 ´ y) » k[x, uv´1, vu´1]

In particular, we see that Proj R coincides with the previous blow-up construction.K

Exercise 9.12 Prove Lemma 9.13. Mˇ

Exercise 9.13 (The weighted projective space P(1, 1, p).) Let R be as in theˇ

Example 15 above, and let A = k[x, y, w] with the usual grading. Furthermore,let α : R Ñ A be the homomorphism that sends z to wp and neither touches xnor y. Show that α is homogeneous of degree zero and induces a morphismπ : P2

k Ñ X. Describe the fibres of π over closed points in case k is algebraicallyclosed.

M

Exercise 9.14 Let R = k[x, y, z] be the polynomial ring given the gradingdeg x = 1, deg y = 2 and deg z = 3, and let X = Proj R (also denoted P(1, 2, 3)).


The aim of the exercise is to describe the three covering distinguished sub-schemes D+(x), D+(y) and D+(z).

a) Show that (Rx)0 = k[yx´2, zx´3] and that D+(x) » A2k .

b) Show that (Ry)0 » k[x2y´1, z2y´6, xzy´2]. Show that the graded algebrahomomorphism k[u, v, w]Ñ (Ry)0 given by the assignments x ÞÑ yx´2, v ÞÑz2y´6 and w ÞÑ xzy´2 induces an isomorphism k[u, v, w]/(w2 ´ uv) » (Ry)0.Hence D+(y) is a hypersurface in A3

k ; the so-called ‘cone over a quadric’.Show it is not isomorphic to A2

k (check the local ring at the origin).

c) Show that Rz = k[x3z´1, y3z´2, xyz´1] and that the map k[u, v, w] Ñ (Rz)0

defined by the assignments x ÞÑ x3z´1, v ÞÑ y3z´2 and w ÞÑ xyz´1 inducesan isomorphism k[u, v, w]/(w3´ uv) » (Rz)0. Show that it is not isomorphicto A2.

d) Show that the map R+ Ñ k[U, V, W] sending x ÞÑ U, y Ñ V2 and z ÞÑ W3

induces a map P2k Ñ Proj R, and describe the fibres over closed points.

M

Chapter 10

Sheaves of modules

OX-modulesŤ

Quasi-coherent sheavesŤ

Coherent sheaves

When you study commutative algebra may be you are primarily interested inthe rings and ideals, but probably you start turning your interest towards themodules pretty quickly; they are an important part of the world of rings, andto get the results one wants, one can hardly do without them. The categoryModA of A-modules is a fundamental invariant of a ring A; and in fact, itmay be is the principal object of study in commutative algebra. There is alsoan analogue viewpoint for schemes for which the so called quasi-coherent OX-modules form an important attribute, if not a decisive part of the structure.They constitute a category QCohX with many properties paralleling those of thecategory ModA. In fact, in case the scheme X is affine, i.e. X = Spec A, the twocategories ModA and QCohX are equivalent as we will prove shortly. Imposingfiniteness conditions on the OX-modules, one arrives at the category CohX ofthe so-called coherent OX-modules, which in the Noetherian case parallel thefinitely generated A-modules.

We start out the chapter by describing the much broader concept of an OX-module. The theory here is presented for schemes, but the concept is meaningfulfor any ringed space.

In the literature one finds several different approaches to the quasi-coherentsheaves. We follow EGA ([?]) and Hartshorne ([?]) and introduce the quasi-coherent modules first for affine schemes. If X = Spec A one defines an OX-module ĂM associated with each A-module M, and in the general case thesemodules serve as the local models for the quasi-coherent ones.

There is a notion of quasi-coherent OX-modules on a general locally ringedspace. These modules are important in some other branches of mathematics (e.g.complex function theory), but we will concentrate our efforts on quasi-coherentmodules on schemes.

sheaves of modules 188

10.1 Sheaves of modules

A module over a ring is an additive abelian group equipped with a multiplica-tive action of A. Loosely speaking, we can multiply members of the moduleby elements from the ring, and of course, the usual set of axioms must besatisfied. In a similar way, if X is a scheme, an OX-module is an abelian sheaf Fwhose sections can be multiplied by sections of OX, the multiplicator and themultiplicand of course being sections over the same open subset.

More formally, we define an OX-modules(OX-moduler)

OX-module as a sheaf F equipped with multipli-cation maps F (U)ÔX(U)Ñ F (U), one for each open subset U of X, makingthe space of sections F (U) into a OX(U)-module and this in a way compatiblewith all restrictions. In other words, for every pair of open subsets VĎU, thenatural diagram below is required to commute

F (U)ÔX(U) F (U)

F (V)ÔX(V) F (V).

where vertical arrows represent restrictions maps and horizontal ones multipli-cation maps.

Maps, or Homomorphisms ofOX-modules(homomorfier avOX-moduler)

homomorphisms of OX-modules are simply maps α : F Ñ G betweenOX-modules considered as abelian sheaves that respect the multiplication bysections of OX. That is, for each open U the map αU : F (U) Ñ G(U) is anOX(U)-module homomorphism. Thus we obtain a category of OX-modules,which we denote by ModX.

The category ModX is an additive category: the sum of two OX-homomor-phisms as maps of abelian sheaves is again an OX-homomorphism. So forall F and G the set HomOX (F ,G) of OX-homomorphisms from F to G is anabelian group, and one verifies readily that the compositions maps are bilinear.Moreover, the direct sum of two OX-modules as abelian sheaves has an obviousOX-structure with multiplication being defined componentwise. In fact, thisdefinition works for arbitrary direct sums (or coproducts as they also are called).For any family tFiuiPI of OX-modules the direct sum

À

iPI Fi is an OX-module(see Exercise 10.3 below).

The notions of kernels, cokernels and images of OX-module homomorphismsnow appear naturally. Each of these corresponding abelian constructions arecompatible with the multiplication by sections of OX, and therefore they haveOX-module structures. The respective defining universal properties (in thecategory of OX-modules) come for free, and one easily checks that this makesModX an abelian category.Example 10.1 Ideal sheaves are important examples of OX-modules. Formally,a sheaf I is an ideal sheaf if I (U) Ă OX(U) for each open set U Ă X.


For a concrete example, consider A2k = Spec k[x, y], let p be the origin, and

define for U Ă A2, the presheaf

I (U) = t f P OX(U) | f (p) = 0u .

Then it is readily checked that I is a sheaf, and hence an ideal sheaf, becauseeach I(U) is an ideal. K

Example 10.2 In the same manner, the quotient sheaf OX/I is an OX-module.This is is a priori not completely obvious, because there is a sheafificationinvolved in forming the quotient sheaf (however, see Exercise 10.2). K

Example 10.3 If F is a sheaf obtained by gluing together sheaves Fi defined ona covering U = tUiu, and each Fi is an OUi -module, then F is an OX-module.(To see this, use the explicit construction of F in Chapter 4). K

Example 10.4 Write P1 for the projective line over a field k, and considerthe sheaves OX(a) from Section 5.8. That is, OP1(a) is the sheaf obtained bygluing OU0 to OU1 using the isomorphism OU1 |U0XU1 Ñ OU0 |U0XU1 on U0XU1 =

Spec k[x, x´1] given by multiplication by xa. Then OP1(a) is an OX-module. Assuch it is a very special; it is ‘locally free’ in the sense that it restricts to thestructure sheaf on the opens in an open covering (however, OX(a) » OX onlyfor a = 0). K

The OX-modules (or at least some of them) play a fundamental role in thetheory of schemes, and shortly we shall see a long series of examples. Thesewill all be so called quasi-coherent sheaves. The examples we now describeare a bunch of wild examples, intended to show that OX-modules without anyrestrictive hypothesis are very general and often unmanageable objects.

Example 10.5 (Modules on spectra of dvr’s). Modules on the prime spectrum ofa discrete valuation ring R are particularly easy to describe. Recall that thescheme X = Spec R has only two non-empty open sets, the whole space X itselfand the singleton tηu consisting of the generic point. The singleton tηu is theunderlying set of the open subscheme Spec K, where K denotes the fraction fieldof R.

We claim that to give an OX-module is equivalent to giving an R-module M,a K-vector space N and a R-module homomorphism ρ : M Ñ N.

Indeed, given an OX-module F , we get the R-module M = F (X), andN = F (tηu) which is a vector space over K = OX(tηu). The homomorphismρ is just the restriction map F (X)Ñ F (tηu). Conversely, given the data M, Nand ρ, we can define a presheaf F by setting F (X) = M and F (tηu) = N andas ρ is a map ρ : F (X) Ñ F (tηu), we can use it as the restriction map. If wealso set F (H) = 0, we have a presheaf F which satisfies the two sheaf axioms.Furthermore, since M and N are modules over OX(X) = R and OX(tηu) = Krespectively, this makes F into an OX-module.


Note that the restriction map can be just any R-module homomorphismM Ñ N. In particular, it can be the zero homomorphism, and in that case Mand N can be completely arbitrary modules. Again, this illustrates the versatilityof general OX-modules. K

Example 10.6 (Godement sheaves again). Recall the Godement construction fromSection 1.5 on page 29 in Chapter 1. Given any collection of abelian groupstAxuxPX indexed by the points x of X, we defined a sheaf A whose sectionsover an open subset U was

ś

xPU Ax, and whose restriction maps to smalleropen subsets were just the projections onto the corresponding smaller products.Requiring that each Ax be a module over the stalk OX,x, makes A into an OX-module; indeed, the space of sections Γ(U,A) =

ś

xPU Ax is automatically anOX(U)-module, the multiplication being defined componentwise with the helpof the stalk maps OX(U)Ñ OX,x. Clearly this module structures is compatiblewith the projections, and thus makes A into an OX-module. K

Exercises(10.1) Assume that F and G are OX-modules and that α : F Ñ G is a mapbetween them. Show that the kernel, cokernel and image of α as a map ofabelian sheaves indeed are OX-modules, and that they respectively are thekernel, cokernel and image of α in the category of OX-modules as well. Showthat a complex of OX-modules is exact if and only it is exact as a complex ofabelian sheaves.(10.2) Suppose that F is a presheaf of OX-modules (i.e. a presheaf satisfyingthe usual OX-module axioms). Show that the sheafification F+ naturally is anOX-module.(10.3) Show that the category ModX has arbitrary products and coproducts, byshowing that the products and coproducts in the category of abelian sheavesAbShX are OX-modules and are the products, respectively the coproducts, inthe category ModX.(10.4) Assume that p1, . . . , pr is a set of primes, and let Z(pi) as usual denote thelocalization at the prime ideal (pi). Let X be the scheme obtained by gluing theschemes Xi = Spec Z(pi) together along their common open subschemes Spec Q.Describe the OX-modules on X.(10.5) Let A =

ś

1ďiďn Ki be the product of finitely many fields and let X =

Spec A. Describe the category ModX.M

Tensor products and Hom’sFor two OX-modules F and G we may also define the tensor product, denotedby F bOX G. As in many other cases, the tensor product F bOX G is defined byfirst describing a presheaf which subsequently is sheafified. The sections of the


presheaf, temporarily denoted by F b1OXG, are defined in the natural way by

(F b1OXG)(U) = F (U)bOX(U) G(U). (10.1)

Example 10.7 Let us continue Example 4 on page 189. We claim that for eacha, b P Z, there is an isomorphism of OX-modules

OP1(a)bOP1(b) » OP1(a + b). (10.2)

Indeed, we have OP1(a)|Ui = OUi and OP1(b)|Ui = OUi for each i = 0, 1. Whenwe identify OUi bOUi

OUi = OUi , we see that the tensor product on the left-handside is obtained by gluing the OUi ’s using the isomorphism xa ¨ xb = xa+b, whichis exactly the sheaf on the right-hand side.

By the way, this example also shows why the sheafification is neccessary inthe defiition of the tensor product. Indeed, the presheaf OP1(´1)b1 OP1(1) hashas value

OP1(´1)(P1)bOP1 (P1) OP1(1)(P1) = 0bk k2 = 0

on the open set U = P1. On the other hand, (10.2) gives an isomorphism ofsheaves OP1(´1)bOP1(1) » OX, and OP1(P1) = k. K

There is also a The sheaf ofOX-homomorphisms(knippet avOX-homomorfier)

sheaf of OX-homomorphisms between F and G. Recall fromExample 9 on page 20 the sheaf Hom(F ,G) of homomorphisms between theabelian sheaves F and G whose sections over an open set U is the groupHom(F |U ,G|U) of homomorphisms between the restrictions F |U and G|U . In-side this group one has the subgroup of the maps being OX-homomorphisms,and these subgroups, for different open sets U, are respected by the restrictionmaps. So they form the sections of a presheaf; it turns out to be a sheaf, andthat is the sheaf HomOX (F ,G) of OX-homomorphisms from F to G.

Exercises(10.6) Let F and G be two OX-modules on the scheme X. Show that the tensorproduct F bOX G enjoys a universal property with respect to bilinear mapsanalogous to the one enjoyed by the tensor product of modules over a ring A.(10.7) Find an example of two OX-modules F and G so that the presheaf (10.1)ˇ

is not a sheaf.(10.8) Let F and G be two OX-modules on the scheme X. Show that the stalk(F bOX G)x at the point x P X is naturally isomorphic to the tensor productFxbOX,x Gx of the stalks Fx and Gx. Show that the tensor product is right exactin the category of OX-modules.(10.9) Show that the sheaf-hom HomOX (F ,G) of two OX-modules as definedabove on page 191 is a sheaf. Show that HomOX (F ,G) is right exact in thesecond variable and left exact in the first.(10.10) Adjunction between Hom and b. Show that for any three OX-modules F ,


G and H there is a natural isomorphism

HomOX (F , HomOX (G,H)) » HomOX (F bOX G,H),

which is functorial in all three variables. Hint: This is easier than it looks —reduce to the usual tensor product for modules over rings.(10.11) With the notation of Example 7, show that

HomOP1 (OP1(a),OP1(b)) » OP1(b´ a).

Conclude that there are no non-trivial maps of sheaves OP1(a)Ñ OP1(b) whena ą b.

M

Pushforward and Pullback of OX-modulesIn Chapter 1 we introduced two functors between the categories AbShX andAbShY associated with a continuous map f : X Ñ Y between topological spaces:the pushforward functor f˚ and the inverse image functor f´1. In this section weparallel these two constructions when f is a morphism of schemes to obtainfunctors f˚ and f ˚ between ModX and ModY. They form an adjoint pair offunctors.

Pushforward

Let f : X Ñ Y be a morphism of schemes. If F is an abelian sheaf on X, recallthat the pushforward f˚F was defined to be the sheaf on Y whose sections overan open U is f˚F (U) = F ( f´1U); in particular, we have f˚OX(U) = OX( f´1U).When F is an OX-module, it is then clear that each f˚F (U) is a module overf˚OX(U) in a canonical way, and hence we may use the map of sheaves of ringsf # : OY Ñ f˚OX to equip f˚F with a natural OY-module structure.

Definition 10.8 The above OY-module f˚F is called the Direct images ofOX-modules (direktebilder av OX-moduler)

direct image or thePushforward(frempuff)

pushforward of F under f .

This construction is clearly functorial in the sheaf F , and so we obtain afunctor f˚ : ModX Ñ ModY. That the pushforward f˚ is left exact follows easilyfrom Lemma 1.24 on page 43 in Chapter 1. It is also functorial in the morphismf in the sense that ( f ˝ g)˚ = f˚ ˝ g˚ when f and g are composable morphismof schemes; indeed, this follows from the two equalities ( f ˝ g)´1 = g´1 ˝ f´1

and ( f ˝ g)# = g# ˝ f #.

Pullback

If G is an OY-module, we also have a ‘pullback’ f ˚G, which is an OX-module,but this construction is a little bit more involved. Recall that we in Chapter 1

defined (Definition 1.27 on page 44) the inverse image f´1G of a sheaf G by


sheafifying the presheaf that assigns to an open subset UĎX the inverse limitlimÐÝ f (U)ĎV G(V) of all G(V) where V contains f (U). When G is an OY-module,

this sheaf is naturally an f´1OY-module (because f´1OY is expressed as ananalogue inverse limit), and we can make f´1G into an OX-module using themap f´1OY Ñ OX (that makes OX an f´1OY-algebra). Finally we take thetensor product and define:

f ˚G = OX b f´1OYf´1G.

The assignment G ÞÑ f ˚G is functorial, so we get a functor f ˚ : ModOY Ñ ModOX .The above OX-module is called the Pullbacks

(tilbaketrekninger)pullback of G under f .

Example 10.9 We have f ˚OY = OX. Indeed, the tensor product identityA bA B = B induces a natural isomorphism f´1OY b f´1OY

OX = OX (aftersheafifying the tensor product presheaf!). K

Proposition 10.10 For a point x P X we have the following expression for thestalk of the pullback

( f ˚G)x = G f (x) bOY, f (x)OX,x.

Proof: This follows from the facts that taking stalks commutes with sheafifica-tion and tensor products, and ( f´1G)x = G f (x). o

Adjoint properties of f˚ and f ˚

At first sight, the definition of the pullback might seem a bit out of the blue.It is defined from f´1G, tensorizing with OX over f´1OY to rig it into beinga OX-module. However, as in the case of the inverse image functor f´1, theimportant point is what the sheaf does, rather than how it is explicitly defined.In the present case, the pullback is the adjoint of a functor which is easy tounderstand, namely f˚:

Proposition 10.11 The functors f˚ and f ˚ between the categories ModOX andModOY are adjoint. In other words, if F P ModOX and G P ModOY , there is afunctorial isomorphism

HomOX ( f ˚G,F ) » HomOY(G, f˚F ).

Proof: See Exercise 10.15. o

In particular, applying this to the two maps id f˚G and id f˚F provide us withthe canonical maps

η : G Ñ f˚ f ˚G, ν : f ˚ f˚F Ñ F .


Previously we already saw that f˚ is a left-exact functor, and this impliesthat f ˚ is right-exact by the general property of adjoint functors that adjoints toleft exact functors are right exact.

Corollary 10.12 f ˚ is right exact and f˚ is left-exact.

At this point, the sheaf f ˚G is not so useful, since it’s complication definitiondoes not allow for actual computations. However, we will see that when G is aquasicoherent sheaf, we have a concrete formula for it.

Pullback of sectionsWe can also pull back sections. If G is an OY-module and s P G(V), then we geta section f ˚(s) = η(s) P Γ( f´1(V), f ˚G) by the map η : G Ñ f˚ f ˚G.Example 10.13 If f : X Ñ Y is a morphism of affine varieties over k, andG = OY, then the above pullback coincides with the ‘usual’ pullback of regularfunctions. More precisely, if V Ă Y is an open set, and g : V Ñ k is a regularfunction, then f ˚(g) is an element of OX( f´1V), and can thus be regarded as aregular function on f´1(V); and f ˚(g) is of course nothing but the compositiong ˝ f : f´1(V)Ñ k. K

Exercises(10.12) Find examples of morphisms f : X Ñ Y such thatˇ

a) f˚OX fi OY

b) f´1OY fi OX

(10.13)ˇ

a) Show that the natural map f´1OY Ñ OX is an isomorphism if and only iffor each x P X, the stalk map OY, f (x) Ñ OX,x is.

b) Let f : X Ñ Y be a finite morphism of noetherian schemes. Show thatf´1OY Ñ OX is an isomorphism if and only if for each x P X, there is anopen set U Ă X containing x and an open set V Ă Y of f (x) such that finduces an isomorphism U Ñ V.

(10.14) Prove that applying f ˚ commutes with taking tensor products of sheaves,ˇ

i.e. f ˚(G bH) » f ˚G bOX f ˚H for any two OX-modules G and H. Does thesame hold for f˚?(10.15) Prove Proposition 10.3 above.ˇ

(10.16) Write out a proof that f ˚ is right-exact.(10.17) Let ι : U Ñ X be the an open immersion. Show that for every OX-moduleF on U it holds true that ι˚ι˚F = F .

M

quasi-coherent sheaves 195

10.2 Quasi-coherent sheaves

In the Cambridge Dictionary, we find the following for the word ‘coherence’:the situation when the parts of something fit together in a natural or reasonable way.

In our setting, the coherence of a sheaf should mean that there are some strongrelations between the sections over different open sets, at least over sets fromsome sufficiently large collection of open sets. In our context the open affinesubsets stand out as obvious candidates to form such a collection, and indeed, aquasi-coherent sheaf on the scheme X will turn out1 to have a coherence propertydescribed as follows: In general for any OX-module F and any pair VĎU ofaffine open sets in X there is a canonical map

F (U)bOX(U) OV(V)Ñ F (V) (10.3)

induced by the map sending sb f to ρUV(s) ¨ f , where ρUV denotes the restrictionmaps of F . The salient point is that when F is a quasi-coherent sheaf, this map isan isomorphism. In fact the converse holds true as well; that is, the map in (10.3)being an isomorphism for all pairs VĎU of open affine sets is equivalent to Fbeing quasi-coherent.

Quasi-coherent sheaves on affine schemesIn this section we shall work over an affine scheme X = Spec A. For each A-module M we intend to define an OX-module ĂM, whose construction completelyparallels what we did when constructing the structure sheaf OX on X = Spec A.Letting B again be the base for the Zariski topology on X consisting of thedistinguished open subsets, we define a B-presheaf ĂM by letting sections overD( f ) be given by

ĂM(D( f )) = M f ,

and letting restriction maps be the canonical localization maps: when D(g) ĎD( f ), it holds true that gn = a f for some a and some n, and there is the canonicallocalization map M f Ñ Mg sending m f´r to armgńr. The same proof as for OX

(Proposition 3.3 on page 72) shows that this is actually a B-sheaf, and hencegives rise to a unique sheaf on X, which we continue to denote by ĂM.

Definition 10.14 An OX-module F on X = Spec A is said to be Quasi-coherentOX-modules(kvasikoherentOX-moduler)

quasi-coherent if it is isomorphic to an OX-module of the form ĂM for some A-moduleM.

M f N f

Mg Ng

α f

αg

The tilde-construction is functorial in M. For any A-module homomorphismα : M Ñ N there is an obvious way of obtaining an OX-module homomorphism

1One might have used this coherence property as the definition, but because of obscure reasons wechoose another definition.


rα : ĂM Ñ rN; indeed, the maps α f : M f Ñ N f are OX(D( f ))-linear homomor-phisms compatible with localization maps, and thus induce a map between ĂMand rN. Clearly one has Ćφ ˝ ψ = rφ ˝ rψ, and the ‘tilde-operation’ is therefore acovariant functor from A-modules to OX-modules.

The three main properties of the sheaf ĂM are listed in the proposition thatfollows.

Proposition 10.15 Let A be a ring and M and A-module. The sheaf ĂM onSpec A has the following three properties.

i) Stalks: let x P Spec A be a point whose corresponding prime ideal isp, then the stalk ĂMx of M at x P X is

ĂMx = Mp = MbA Ap;

ii) Sections over distinguished open sets: if f P A, one has

Γ(D( f ), ĂM) = M f = MbA A f

in particular it holds true that Γ(X, ĂM) = M;iii) Sections over arbitrary open sets: for any open subset U of Spec A

covered by the distinguished sets tD( fi)uiPI , there is an exact sequence

0 Γ(U, ĂM)ś

i M fi

ś

i,j M fi f jα β

where α and β are the natural maps as defined in (1.1) on page 17.

Proof: These properties are completely analogous to the statements in Proposi-tion 3.6 on page 74 in Chapter 3 about the structure sheaf OX, and the proofsare mutatis mutandis the same. The first property follows since the stalks ĂMx

and the localizations Mp are direct limits of the same modules over the sameinductive system (indexed by the distinguished open subsets D( f ) containingx), the second ensues from the way we defined ĂM, and the third is just thegeneral exact sequence expressing the space of sections of a sheaf over an openset in terms of the space of sections over members of an open covering and theirintersections. o

The tildes enjoy a certain universal property among the OX-modules onX = Spec A. Assume that an OX-module F is given on X, and let M = F (X)

denote the global sections of F . There is a natural map of OX-modules

β : ĂM Ñ F .

As usual, it suffices to tell what this map does to sections over the distinguishedopens U = D( f ). As F is an OX-module, multiplication by f´1 in the spaceof sections F (D( f )) makes sense since OX(D( f )) = A f . Hence we may map


the section m fń P M f of ĂM(D( f )) to the section of F over D( f ) obtainedby multiplying the restriction of m to D( f ) by fń; i.e. we send m fń tofń ¨m|D( f ). This is clearly a map of A f -modules, and everything works wellwith the restriction map, so we get a well defined map β. For later reference, westate this observation as a lemma:

Lemma 10.16 Given an OX-module F on the affine scheme X = Spec A. Then thereis a unique OX-module homomorphism

β : ČF (X)Ñ F

that induces the identity on the spaces of global sections. Moreover, it is natural in thesense that if α : F Ñ G is a map of OX-module inducing the map a : F (X) Ñ G(X)

on global sections, one has βG ˝ ra = α ˝ βF .

Exercise 10.18 Check that the map β in the lemma is a well defined map (thereare choices involved in the definition). M

Lemma 10.17 In the canonical identification of the distinguished open subset D( f )with Spec A f , the OX-module ĂM restricts to ĄM f .

Proof: As Γ(D( f ), ĂM) = M f , there is a map β f : ĄM f Ñ ĂM|D( f ) that ondistinguished open subsets D(g)ĎD( f ) induces an isomorphism between thetwo spaces of sections, both being equal to the localization Mg. o

Categorical propertiesWe now survey a few of the most remarkable properties of the ‘tilde-functor’.

Lemma 10.18 For any two A-modules M and N, the association φ Ñ rφ gives abijection HomA(M, N) » HomOX (

ĂM, rN) whose inverse is α ÞÑ α(X).

Proof: That rφ = α when φ = α(X) may be checked on distinguished open setswhere it boils down to the definition of rφ and the fact that α commutes with thelocalization maps. That Γ(X, rφ) = φ follows directly from the definition of rφ. o

Thus the tilde functor is fully faithful, and establishes an equivalence betweenModA and a subcategory of ModX. It is a strict subcategory; most OX-modulesare not quasi-coherent.

Lemma 10.19 The tilde-functor is exact.

Proof: Assume given an exact sequence of A-modules:

0 // M1 // M // M2 // 0.

That the induced sequence of OX-modules

0 // ĂM1 // ĂM //ĄM2 // 0


is exact is a direct consequence of the three following facts. The stalk of a tilde-module ĂM at the point x with corresponding prime ideal p is Mp, localization isan exact functor, and finally, a sequence of abelian sheaves is exact if and only ifthe sequence of stalks at every point is exact. o

Tensor products and Hom’s

The next lemma says that the tilde-functor takes the tensor product MbA Nof two A-modules to the tensor product ĂMbOX

rN of the two correspondingOX-modules.

Lemma 10.20 There is a canonical isomorphism ČMbA N » ĂMbOXrN.

Proof: As usual, let B be the basis for the Zariski topology consisting ofdistinguished open sets. The tensor product ĂMbOX

rN is the sheaf associatedto the presheaf T given as U ÞÑ ĂM(U)bOX(U)

rN(U). Over the distinguishedopen set U = D( f ) the sections of ČMbA N equals (NbA M) f , so there is a mapof B-presheaves T Ñ ČMbA N coming from the assignment m/ f a b n/ f b to(mb n)/ f a+b, which in fact induces an isomorphism M f bA f N f » (MbA N) f .

After sheafifying and extending the B-sheaves, we get a the desired map ofsheaves

ĂMbOXrN Ñ ČMbA N.

This is an isomorphism since it is an isomorphism over every distinguishedopen set. o

Recall that an A-module M is said to have finite presentation if there is anexact sequence of the form

Am Ñ An Ñ M Ñ 0

If M is of finite presentation, the tilde-functor sends the A-module of homomor-phisms HomA(M, N) to the sheaf of homomorphisms HomOX (

ĂM, rN). However,this is not true if M is not of finite presentation, the only lacking desirableproperty of the functor Ą(´). There is always a canonical map

HomA(M, N) f Ñ HomA f (M f , N f ), (10.4)

but without some finiteness condition (like being of finite presentation) on M, itis not necessarily an isomorphism.Example 10.21 Even in the simplest case of an infinitely generated free moduleM =

À

iPI Aei that map is not surjective. An element in HomA(M, N) f isof the form α fń where α : M Ñ N is A-linear, as opposed to elements inHomA f (M f , N f ) which are given by their values mi fńi on the free generators


ei, and the salient point is that the ni’s may tend to infinity in which case we canfind no n that works for all i’s. K

Exercise 10.19 Show that the map (10.4) above is an isomorphism when M isof finite presentation. Hint: First observe that this holds when M = A. Thenuse a presentation of M to reduce to that case. M

If M is an A-module of finite presentation, the global sections of the sheaf-hom HomOX (

ĂM, rN) equals HomA(M, N), and there is a map

HomA(M, N)rÑ HomOX (ĂM, rN). (10.5)

Moreover, in that case, the maps in (10.4) are isomorphisms, and the map in(10.5) above induces isomorphisms between the spaces of sections of the twosides over each distinguished open subset. One concludes that the map is anisomorphism of sheaves, and thus one has HomA(M, N)r» HomOX (

ĂM, rN).

Lemma 10.22 If M is of finite presentation, then there is a canonical isomorphism ofsheaves HomA(M, N)r» HomOX (

ĂM, rN).

Exercise 10.20 Show that the tilde-functor is additive; i.e. takes direct sums ofmodules to the direct sum and sums of maps to the corresponding sums. M

Example 10.23 Assume that A is an integral domain and that K is the fieldof fractions of A. Show that the OSpec A-module rK is a constant sheaf in thestrong sense; that is, Γ(U, rK) = K for any non-empty open UĎX, and that therestriction maps all equal the identity idK. K

Pushforward

Suppose we are given a morphism f : X Ñ Y between the two affine schemesX = Spec B and Y = Spec A. We let φ = f 7(X) : A Ñ B be the ring mapcorresponding to f . If M is an B-module, can one describe the sheaf f˚ĂM on Y?The answer is not only yes, but the description is very simple. The B-module Mcan be considered as a A-module via the map φ : A Ñ B, and we denote thisA-module by MA. Clearly this is a functorial construction in M. In this settingone has

Proposition 10.24 f˚ĂM = ĄMA.

Proof: Let a P A. The crucial observation is that f´1(D(a)) = D(φ(a)) (indeed,a prime ideal p Ă A satisfies a P φ´1(p) if and only if φ(a) P p) from which itfollows that

Γ(D(a), f˚(ĂM)) = Γ(D(φ(a)), ĂM) = Mφ(a).

Note that an element a P A acts on MA as multiplication by φ(a): this meansthat the module on the right is isomorphic to (MA)a = Γ(D(a), ĄMA). Thus thereis an isomorphism of B-sheaves f˚ĂM » ĄMA, and we are done. o

Example 10.25 Consider the morphism f : A1k = Spec k[x] Ñ A1

k = Spec k[y]induced by the ring map k[y] Ñ k[x] given by y ÞÑ xn where n ą 0. What is


f˚OA1? Well, as a k[y]-module, k[x] is isomorphic to

k[y][x]/(y´ xn) » k[y]‘ k[y]x‘ ¨ ¨ ¨ ‘ k[y]xn´1.

Hence by Proposition10.13, we find that f˚OA1k» On

A1k. K

Pullback

Recall the notion of pullback of a sheaf along a morphism f : X Ñ Y. Thisis a relatively complicated operation since it involves taking a direct limit, atensor product, and finally a sheafification. The next result tells us that when Xand Y are affine and G is a sheaf of the form ĂM on Y, there is a much simplerdescription of the pullback f ˚G which will allow us to do local computationsmore easily.

Proposition 10.26 Let f : Spec B Ñ Spec A be a morphism induced by aring map φ : A Ñ B, and let M be an A-module. Then

f ˚(ĂM) = ČMbA B. (10.6)

Proof: First, note that the proposition holds in the special case when M is afree module, i.e. M = AI (here the index set I is allowed to be infinite); this issimply because f ˚OY = OX and f ˚ commutes with taking arbitrary direct sums.To prove it in general, we pick a presentation of M of the form

AJ AI M 0.γ

(10.7)

Applying the tilde functor followed by f ˚ we get a sequence

f ˚(ĂAJ) f ˚(ĂAI) f ˚(ĂM) 0f˚(γ)

which is exact since both the tilde functor and f ˚ are right-exact. On theother hand, in a similar way, first tensorizing the sequence (10.7) by B andsubsequently applying the tilde functor, we obtain the exact sequence (the tensorproduct is right-exact and the tilde functor exact)

ČAJ bA B ČAI bA B ČMbA B 0.γbid

Comparing the two sequences using that the proposition holds for free modules,finishes the proof. o

Example 10.27 Consider again the map f : A1k Ñ A1

k from Example 12. Letus compute f ˚rI where I is the ideal I = (y ´ a) Ă k[y]. Explicitly, this isgiven by the tilde of the k[x]-module (y ´ a)k[y] bk[y] k[x], a submodule ofk[y]bk[y] k[x], which is mapped to the ideal (xn ´ a) under the isomorphismk[y]bk[y] k[x] » k[x]. Hence f ˚rI is the ideal sheaf of a subscheme of A1

k , the n-throots of a when k is algebraically closed. K

The adjoint property


Recall that we defined, for a morphism f : X Ñ Y, two natural maps

ν : f ˚ f˚F Ñ F η : G Ñ f˚ f ˚G

where F P ModX and G P ModY, and where ν and η are maps respectively fromModX and ModY. In the case the two schemes are affine and the sheaves arequasi-coherent sheaves we may understand these maps in the following way.Let the schemes be X = Spec B and Y = Spec A and the two sheaves F = ĂMand G = rN with M a module over B and N one over A. By what we saw above,it holds true that f˚ĂM = ĂMA, and so we have

f ˚ f˚ĂM = ČMA bA B.

The point is that since the tensor product is over the ring A, we cannot moveB over to the left hand side, but we do have a natural map of B-modulesMA bA B Ñ M, which is given by mb b ÞÑ bm, and the tilde of this map will bethe map ν : ČMA bA B Ñ ĂM.

To explain the other map η, note that f ˚ rN = N bA B. This yields

f˚ f ˚ rN = Č(N bA B)A,

and consequently η is induced by the map

N Ñ (N bA B)A

given by n ÞÑ nbA 1.Exercise 10.21 Let as in the paragraph f : Spec B Ñ Spec A be a morphism ofˇ

affine schemes.

i) It is instructive to verify the adjoint property of f˚ and f ˚ for sheavesof the form ĂM. If F = ĂM for a B-module M and G = rN for anA-module N, show, using the adjoint properties of Hom and b thatHomX( f ˚G,F ) = HomY(G, f˚F ).

ii) By the Yoneda Lemma, an A-module P is completely determined bythe functor HomA(P,´). Use this fact and the computation in i) togive a new proof that f ˚ĂM = ČMbA B.

M

Exercise 10.22 Let X = Spec A and let f P A be an element. Denote byι : D( f )Ñ X be the inclusion map. Describe the stalk of ι˚OD( f ) at every pointx R D( f ). M

Quasi-coherent sheaves on general schemesHaving established the class of sheaves on affine schemes, which will serve aslocal models for the quasi-coherent sheaves on general schemes, we are nowready to take on the general definition.


Definition 10.28 Let X be a scheme and F an OX-module. One says that Fis a Quasi-coherent

sheaves(kvasikoherenteknipper)

quasi-coherent OX-module, or quasi-coherent sheaf for short, if there isan open affine covering tUiuiPI of X, say Ui = Spec Ai, and modules Mi overAi such that F |Ui »

ĂMi for each i.

Phrased in slightly different manner, an OX-module F is quasi-coherent if itis locally of tilde-type. In particular, the modules ĂM on affine schemes Spec A areall quasi-coherent. Quasi-coherence is a local property: if F is quasi-coherent ina neighbourhood of every point, it will be quasi-coherent. The full subcategoryof ModX whose objects are the quasi-coherent sheaves is denoted QCohX.

The restriction of a quasi-coherent sheaf F to any open set UĎX is quasi-coherent. Indeed, it will suffice to verify this for affine schemes, as then therestriction F |UXUi will be quasi-coherent. So let X be affine. By Lemma 10.8 therestriction of a sheaf of tilde-type to a distinguished open set is of tilde-type. Asany open U in an affine scheme is the union of distinguished open subsets, itfollows that F |U is quasi-coherent.

For an OX-module F to be quasi-coherent, we require that F be locally oftilde-type for just one open affine cover. However, it turns out that this willhold for any open affine cover, or equivalently, that F |U is of tilde-type for anyopen affine subset UĎX. This is a much stronger than the requirement in thedefinition, and it is somewhat difficult to prove. As a first corollary, we arriveat the a priori not obvious conclusion that the modules of the form ĂM are theonly quasi-coherent OX-modules on an affine scheme. We shall also see thatquasi-coherent modules enjoy the coherence property (10.3) on page 195 thatwas the point of departure for our discussion.

We start by showing a lemma that establishes the coherence property (10.3)in a very particular case, i.e. for sections over distinguished open sets of a quasi-coherent OX-module on an affine scheme X = Spec A. For any distinguishedopen set D( f )ĎX it holds true that OX(D( f )) = A f , and consequently for anyOX-module there is a canonical map F (X)bA A f Ñ Γ(D( f ),F ) which sendssb a fń to a fń ¨ s|D( f ). This turns out to be an isomorphism whenever F isquasi-coherent:

Lemma 10.29 Suppose that X = Spec A is an affine scheme and that F is a quasi-coherent OX-module. Let D( f )ĎX be a distinguished open set. Then the followinghold:

i) F (D( f )) » F (X) f ;ii) Let s P F (X) be a global section of F and assume that s|D( f ) = 0, then

sufficiently large powers of f kill s; that is, for sufficiently large integers none has f ns = 0;

iii) Let s P F (D( f )) be a section. Then for a sufficiently large n the section f nsextends to a global section of F . That is, there exists an n and a global section


t P F (X) such that t|D( f ) = f ns.

The naive intuition here is that on D( f ) we allow regular functions such as fń,and these extend, of course, to regular functions on all of X after multiplicationby some power of f .Proof: The first statement is by the definition of localization equivalent to thetwo others, so it will suffice to prove those.

To prove ii), we observe that since the sheaf F is quasi-coherent by hypothesis,and since the affine scheme X = Spec A is quasi-compact, there is a finite openaffine covering of X by distinguished sets D(gi) such that F |D(gi) »

ĂMi for someAgi -modules Mi. The given section s of F restricts to sections si of F |D(gi) overD(gi); that is, to elements si of Mi.

Further restricting F to the intersections D( f ) X D(gi) = D( f gi) yieldsequalities F |D( f gi) = (ĂMi) f , and by hypothesis, the section s restricts to zero inΓ(D( f gi),F ) = (Mi) f . This means that the localization map sends si to zero in(Mi) f . Hence si is killed by some power of f , and since there is only finitelymany gi’s, there is an n with f nsi = 0 for all i; that is, ( f ns)|D(gi) = 0 for all i. Bythe locality axiom, it follows that f ns = 0.

Attacking iii), assume a section s P Γ(D( f ),F ) is given. Our task is toshow that f ns extends to a global section of F for some n. Each restrictions|D( f gi) P Γ(D( f gi),F ) = (Mi) f is of the form fńsi with si P Mi = Γ(D(gi),F ),and by the usual finiteness argument, n can be chosen uniformly for all i.This means that the different si = f ns and sj = f ns mach on the intersectionsD( f )XD(gi)XD(gj), and by the second part of the lemma applied to Spec Agi gj ,one has f N(si´ sj) = 0 on D(gi)XD(gj) for a sufficiently large integer N. Hencethe different f Nsi’s patch together to give the desired global section t of F . o

Theorem 10.30 Let X be a scheme and F an OX-module. Then F is quasi-coherent if and only if for all open affine subsets UĎX = Spec A, the restrictionF |U is isomorphic to an OX-module of the form ĂM for an A-module M.

Proof: As quasi-coherence is conserved when restricting OX-modules toopen sets, we may surely assume that X itself is affine; say X = Spec A. LetM = F (X). We saw in Lemma 10.7 on page 197 that there is a natural mapβ : ĂM Ñ F that on distinguished open sets sends m fń to fńm|D( f ). OnD( f ), this map is injective: fńm|D( f ) = 0, then for some n, f nm = 0 by thefundamental Lemma 10.16 (i), and so m fń in the localization M f . Likewise,given s P F (D( f )), we choose t P F (X) so that t|D( f ) = f ns as in LemmaLemma 10.16 (ii); this has the property that fńt|D( f ) = s, so the map is alsosurjective. Hence the two sheaves are isomorphic. o

Applying the theorem to affine schemes yields the important fact that anyquasi-coherent sheaf (in the sense of Definition 10.15) F on an affine scheme


X = Spec A is of the form ĂM for an A-module M.

Theorem 10.31 Assume that X = Spec A. The tilde-functor M ÞÑ ĂM is anequivalence of the categories ModA and QCohX with the global section functoras inverse.

When speaking about mutually inverse functors one should be very careful;in most cases such a statement is an abuse of language. Two functors Fand G are mutually inverses when there are natural transformations, bothbeing an isomorphism, between the compositions F ˝ G and G ˝ F and theappropriate identity functors. In the present case one really has an equalityΓ(X, ĂM) = M, so that Γ ˝ Ą(´) = idModA . On the other hand, the naturaltransformation ČΓ(X,F ) Ñ F from Lemma 10.7 on page 197 gives just anisomorphism of functors.

Theorem 10.17 has the important corollary that the global section is an exactfunctor:

Corollary 10.32 Let X = Spec A be an affine scheme. Then the global sectionfunctor Γ(X,´) : QCohX Ñ ModA is exact.

Proof: The inverse to any exact equivalence of categories is exact. o

Example 10.33 Quasi-coherent modules on P1. Consider the projective line P1k

over k. It comes equipped with the usual affine open covering U0 = Spec k[x]and U1 = Spec k[x´1] which are glued together along Spec k[x, x´1]. A quasi-coherent sheaf on P1

k is given by a triple (M0, M1, τ), where

i) M0 is a module over OX(U0) = k[x];ii) M1 is a module over OX(U1) = k[x´1];

iii) τ is an isomorphism of modules over k[x, x´1]:

τ : M1 bk[x´1] k[x, x´1]Ñ M0 bk[x] k[x, x´1].

Concrete examples are the sheaves OP1k(n) from Section 5.8; where the data are

M0 = k[x], M1 = k[x´1] and τ : k[x, x´1]Ñ k[x, x´1] is the multiplication by xn.K

Finally, we show that this notion of ‘quasi-coherent’ coincides with the onealluded to in the introduction of Section 10.2.

Theorem 10.34 Let X be a scheme and let F be an OX-module on X. Then Fis quasi-coherent if and only if for any pair VĎU open affine subsets the naturalmap

F (U)bOX(U) OX(V)Ñ F (V) (10.8)

is an isomorphism.


Proof: By Theorem 10.17 it suffices to treat the case that X is affine, sayX = Spec A. Assume first that the maps (10.8) are isomorphisms. We maytake V = D( f ) and U = X and M = F (X). Then from (10.8) it followsthat Γ(D( f ),F ) = M f which shows that the canonical map β : ĂM Ñ F is anisomorphism over all distinguished open subsets, and therefore an isomorphism.Hence F is quasi-coherent.

To argue for the reverse implication, we may again assume X = Spec A,U = X and V = Spec B. So suppose that F is quasi-coherent; that is, F = ĂMfor some A-module M. Let ι : V Ñ X denote the inclusion map. We haveι˚ĂM = ĂM|U » ČMbA B. Taking global sections, this isomorphism turns intoexactly the map (10.8), so we get our desired isomorphism. o

Example 10.35 Quasi-coherent sheaves on spectra of dvr’s. The example of andiscrete valuation ring is always useful to consider, and we continue exploringExample 5 above. Consider the OX- module F given by the data M, N, ρ.We claim that F is quasi-coherent if and only if ρ b K : M bR K Ñ N is anisomorphism (of K-vector spaces).

If F is quasi-coherent, then every point has a neighbourhood on which F isthe tilde of some module. The only neighbourhood of the unique closed pointis X itself, and so F = ĂM. Therefore, N = F (U) = M(0) = MbR K and ρ is anisomorphism. Conversely, if ρb K : MbR K Ñ K is an isomorphism, then F isgiven by F (X) = M and F (tηu) = MbR N, and so F » ĂM is quasi-coherent.

K

Example 10.36 Another nice consequence of the equivalence in Theorem 10.18

is that any purely categorical construction commutes with the tilde-functor —any universal property that holds in ModX holds as well in QCohX. For instance,if tMiuiPI is a directed system if modules, it will be true that (lim

ÝÑMi)

r is thedirect limit lim

ÝÑĂMi in the category QCohX; and in fact in ModX as well:

Proposition 10.37 Assume that X = Spec A and that tMiuiPI is a directedsystem of modules. Then (lim

ÝÑMi)

r is the direct limit in ModX of the systemtĂMiu.

Proof: That limit in QCohX is also the limit in the larger category ModX followsleisurely by Lemma 10.7 on page 197; details are left to the students. o

K

Exercise 10.23 (Direct limits of quasi-coherent sheaves.) In general, if F is a directedsystem of OX-modules, one defines the direct limit lim

ÝÑFi by sheafifying the

presheaf that sends U to limÝÑ

Fi(U).

i) Show that limÝÑ

Fi is the direct limit in the category ModX; that is, it hasthe required universal property;

coherent sheaves 206

ii) Show that forming the direct limit commutes with restriction; i.e. forevery open UĎX it holds that lim

ÝÑ(Fi|U) = (lim

ÝÑFi)|U ;

iii) Show that if all the Fi’s are quasi-coherent, then the direct limit limÝÑ

Fi

is quasi-coherent.

M

10.3 Coherent sheaves

Let A be a ring. One may formulate several finiteness conditions for an A-module M, of course, M may be finitely generated, and is said to be of Modules of finite

presentation(endeligpresentertemoduler )

of finitepresentation if for some integers n and m there is an exact sequence

An Am M 0.

This is a convenient finiteness condition for modules; it means that the modulecan be written as Am/N where N is the image of a map An Ñ Am. In otherwords, M is the cokernel of a map between two free modules of finite rank.Example 10.38 If I = ( f1, . . . , fm) is a finitely generated ideal of A, then A/I isfinitely presented; there is an exact sequence of the form Am Ñ A Ñ A/I Ñ 0.

K

Example 10.39 Let A = k[x, y, z, w]. The matrix defining the twisted cubic

m =

(x y zy z w

)(10.9)

defines a map ν : A3 Ñ A2, and the cokernel Coker ν has finite presentation. K

A more restrictive finiteness condition is the following one:

Definition 10.40 One says that M is Coherent modules(koherente moduler)

coherent if the following two require-ments are fulfilled:

o M itself is finitely generated;

o Every finitely generated submodule of M is finitely presented.

The second statement is equivalent to the kernel of every A-linear mapAm Ñ M being finitely generated.

This ‘coherence’ condition originated in the theory of analytic functions andanalytic geometry where coherent non-Noetherian rings are frequent (e.g. thering of analytic functions on an open set in C).

When A is Noetherian, which is frequently the case in algebraic geometry,the three conditions being coherent, being finitely generated and being of finitepresentation coincide. The key point is that every submodule NĎM of a finitelygenerated module M over a Noetherian ring A is also finitely generated. For

coherent sheaves 207

instance, to show that a finitely generated module M over a Noetherian ringis coherent, we just consider a map α : Am Ñ M. Since A is Noetherian, everysubmodule of Am is finitely generated, in particular the kernel Ker α will be.

With this notion under our belt, we are ready for the definition of a ‘coherent’OX-module:

Definition 10.41 On a scheme X a quasi-coherent OX-module F is Coherent OX-modules(koherenteOX-moduler)

coherentif there is a covering of X by open affine sets Ui = Spec Ai such that F |Ui »

ĂMi

with the Mi’s being coherent Ai-modules. Moreover, F is Finitely presentedsheaves (endeligpresenterte knipper)

finitely presented ifeach Mi is finitely presented as an Ai-module and of

Sheaves of finite type(knipper av endeligtype)

finite type if each Mi isfinitely generated over Ai.

So if X is Noetherian (or locally Noetherian), the condition that M be coher-ent is equivalent to the a priori weaker condition that M is finitely generated.

The definition above is the one that appears in ega and in the Stacks project.However, it differs slightly from the notation used in other texts (e.g. Harts-horne’s book). In that book, an OX-module F is said to be coherent if thereis a covering by open affines Ui = Spec Ai such that F |Ui =

ĂMi with the Mi’sbeing finitely generated Ai-modules. When X is (locally) Noetherian, these twodefinitions coincide, but they are far from being equivalent in general.

Henri Cartan(1904 – 2008)

The notion of a coherent sheaf was actually introduced by Henri Cartan inthe theory of holomorphic functions of several variables around 1944. In 1950

Kiyoshi Oka proved that the sheaf OCn of analytic functions on C n is coherent,and this is, in fact, a very difficult theorem (although it seems like a trivialstatement if the other definition of ‘coherence’ is used); already for n = 1 it isnon-trivial and hinges on convergence properties of Weierstrass’s products

One benefit of using coherent modules rather than finitely generated onesis that the category of coherent modules is an abelian category, even in thenon-Noetherian setting. However, a problem is that coherence is very difficult tocheck in general, and actually, for some schemes, even affine ones, the structuresheaf OX is not coherent!Example 10.42 (A ring that is not coherent). The following is an almost tautologicalexample of a monogenic module that is not coherent. Let R = k[x, y, ti, ui|i P N]

and a = (tix ´ uiy|i P N). Then the R-module A = R/a is not coherent: theideal (x, y) is finitely generated, but the relations are not. Indeed, map the freemodule Re‘ R f with basis e, f into A by sending e Ñ x and f Ñ y. The kernelhas generators uie1 + tie2 for i P N and is not finitely generated; its image in Runder e.g. the first projection equals the ideal (ui|i P N) which for sure is notfinitely generated. K

categorical and functorial properties 208

10.4 Categorical and Functorial properties

There is an important strengthening of Corollary 10.19 on page 204 to the casethat merely the leftmost sheaf is quasi-coherent and without other restrictionson the two others than being OX-modules (and, in fact, the not yet established‘white magic of cohomology’ will show that even this is not necessary). Theresult is a special case of a very central result that the so-called cohomologygroups These groups are

constructed to copewith the global sectionfunctor not beingexact. They will beextensively treatedlater on starting inChapter 13; see alsoAppendix B

Hi(X,F ) vanish for i ą 0 when F is quasi-coherent on an affine X; butwe find in worth while to anticipate the general result in order to complete thepicture of quasi-coherent modules, and it will also be required in the proof ofProposition 10.26 below.

Proposition 10.43 Let X be an affine scheme and F , G and H three OX-modules. Assume they live in the short exact sequence

0 F G H 0.β

If F is quasi-coherent, the sequence of global sections

0 F (X) G(X) H(X) 0

is exact.

We begin by stating a lemma:

Lemma 10.44 Let σ P H(X) be a section. If D( f ) is a distinguished open so thatσ|D(g) can be lifted to a section s of G(D( f )), then for sufficiently large integers n, thesection f nσ may be lifted to G(X)

Given the lemma, the proposition follows by a standard ‘partition of unityargument’: chose a finite covering tD( fi)u of X = Spec A so that each restrictionσ|D( fi) lifts to a section of G(D( fi)). According to the lemma there is an n sothat each f n

i σ|D( fi) lifts to a section τi P G(X). Since the D( f ni )’s cover X, we

may write 1 =ř

i ai f n, and thenř

i aiτi lifts σ.Proof of the lemma: The proof is a ‘patching’-proof with two steps using afinite open affine cover tD(gi)u of X: firstly, we extend the sections f ns|D( f gi)

with n ąą 0 to sections ti of G(D(gi)), secondly we patch them together, whichrequires a new power of f as factor.

Chose the cover tD(gi)u of X by finitely many distinguished open affines,all so small that σ|D(gi) extends to a section si P G(D(gi)). Over D(gi f ) the twosections si|D(gi f )) and s|D(gi f ) both lift σ|D(gi f ), and hence their difference belongsto the space F (D(gi f )). By Lemma 10.16 on page 202 and the fact that thecovering tD(gi)u is finite, for n sufficiently large there are sections ri P F (D(gi))

such that ri|D(gi f ) = f ns|D(gi f ) ´ f nsi|D(gi f ). Then ti = ri + f nsi are sections ofG(D(gi)) that restrict to f ns|D(gi f ).


Now, we would want to glue the ti’s together to a global section of G, andthis can be done at least after giving each ti a factor a high power of f : letUij = D(gi)X D(gj) = D(gigj) and consider ti|Uij ´ tj|Uij P G(Uij). It maps tozero in D(gigj f ), and by Lemma 10.16 is therefore killed by a high power f nij

of f ; and as usual, we may use the same exponent, m say, for every pair i, j. Itfollows that the sections f mti of G(D(gi)) coincide on intersections D(gi)XD(gj)

and they can thus be glued together to a section t of G, and this maps to f n+mσ

since each f mti maps to f n+mσ|D(gi). o

The category of quasi-coherent sheaves has several nice properties:

Proposition 10.45 Suppose that α : F Ñ G is a map of quasi-coherent sheaveson the scheme X.

i) The kernel, cokernel and the image of α are all quasi-coherent.ii) The category QCohX is closed under extensions; that is, if

0 F G H 0 (10.10)

is a short exact sequence of OX-modules with the two outer sheavesF and H being quasi-coherent, the middle sheaf G is quasi-coherentas well.

Proof: If α : F Ñ G is a map of quasi-coherent OX-modules, on any openaffine subsets U = Spec A of X it may be described as α|U = ra where a : M Ñ Nis a A-module homomorphism and M and N are A-modules with F |U = ĂMand G|U = rN. Since the tilde-functor is exact, one has Ker α|U = (Ker a)r.Moreover, by the same reasoning, it holds true that Coker α|U = (Coker a)randIm α|U = (Im a)r. Suppose now that an extension like (10.10) is given with Fand H quasi-coherent, the two other cases are cover by the first statement. ByProposition 10.24 above the induced sequence of global sections is exact; thatis, the upper horizontal sequence in the diagram below is exact. The threevertical maps in the diagram are the natural maps from Lemma 10.7 on page197. Since F and H both are quasi-coherent sheaves, the two outer vertical mapsare isomorphisms, and the snake lemma implies that the middle vertical map isan isomorphism as well. Hence G is quasi-coherent.

0 ČF (X) ČG(X) ČH(X) 0

0 F G H 0

o

Thus also for a general scheme X, the category QCohX is a category withvery nice properties: it is an abelian category with tensor products and internalHom’s.


Quasi-coherence of pullbacks

Recall that for a morphism f : Spec B Ñ Spec A of affine schemes, the pullbackof a quasi-coherent sheaf ĂM is again quasi-coherent which follows from theformula

f ˚(ĂM) = ČMbA B

in Theorem 10.14. In fact, the same conclusion holds quite generally:

Proposition 10.46 Let f : X Ñ Y be a morphism of schemes.

i) If G is a quasi-coherent sheaf on Y, then f ˚G is quasi-coherent on X;ii) If moreover X and Y are Noetherian, then f ˚G is coherent if G is.

Proof: The first of these statements follows from the affine case and the formulaabove since being quasi-coherence is a local property. Moreover, since MbA B isa finitely generated B-module if M is a finitely generated A-module, the secondfollows from the affine case as well. o

Quasi-coherence of pushforwards

Likewise, we showed that for a map f : Spec B Ñ Spec A, the pushforward f˚Fis quasi coherent if F is quasi-coherent (since f˚ĂM = ĂMA). The same holds truefor a large class of morphisms:

Theorem 10.47 Let f : X Ñ Y be a morphism of schemes and that F is aquasi-coherent sheaf on X. If X is Noetherian, then the direct image f˚F isquasi-coherent on Y.

Proof: We may assume that Y = Spec B, and X is the quasi-compact and maybe covered it by finitely many open affines Ui. Each intersection Ui XUj is againquasi-compact and we cover it with finitely many open affines Uijk.

For any open V Ď Y, one has the exact sequence

0 Γ( f´1V,F )ś

i Γ(Ui X f´1V,F )ś

i,j,k Γ(Uijk X f´1V,F ).(10.11)

The sequence is compatible with the restriction maps induced from inclusionsUĎV, hence gives rise to the following exact sequence of sheaves on X:

0 f˚Fś

i fi˚F |Ui

ś

i,j,k fijk˚F |Uijk (10.12)

where fi = f |Ui and fijk = f |Uijk . Now, each of the sheaves fi˚F |Ui and fij˚F |Uij

are quasi-coherent by the affine case of the theorem (Proposition 10.13 onpage 199. They are finite in number as the covering Ui is finite. Hence

ś

i fi˚F |Ui

andś

i,j fij˚F |Uij are finite products of quasi-coherent OX-modules and thereforethey are quasi-coherent. Now, the sheaf f˚F equals the kernel of a homomor-


phism between two quasi-coherent sheaves, and so the theorem follows fromProposition 10.26 on page 209. o

The following example shows that some hypotheses on X (e.g. it be Noethe-rian) is necessary for the proposition to hold:Example 10.48 Let X =

š

iPI Spec Z be the disjoint union of countably infinitelymany copies of Spec Z and let f : X Ñ Spec Z be the morphism that equals theidentity on each of the copies of Spec Z which constitute X. Then f˚OX is notquasi-coherent. Indeed, the global sections of f˚OX satisfy

Γ(Spec Z, f˚OX) = Γ(X,OX) =ź

iPI

Z.

On the other hand if p is any prime, one has

Γ(D(p), f˚OX) = Γ( f´1D(p),OX) =ź

iPI

Z[p´1].

It is not true that Γ(D(p), f˚OX) = Γ(Spec Z, f˚OX)bZ Z[p´1]. Indeed, ele-ments in

ś

iPI Z[p´1] are sequences of the form (zi pńi)iPI where zi P Z andni P N. Such an element lies in (

ś

iPI Z)bZ Z[p´1] only if the ni’s form abounded sequence, which is not the case for general elements of shape (zi pńi)iPI

when I is infinite. In particular, f˚OX is not quasi-coherent. K

Coherence of pushforwardsFor morphisms of schemes f : X Ñ Y in general it cannot be expected thatthe pushforward of a coherent sheaf is again coherent, even for very ‘nice’morphisms f . A simple example is the following:Example 10.49 Let X = Spec k[t] and consider the structure morphism f : X ÑSpec k, which is induced by the inclusion k Ď k[t]). The sheaf OX is certainlycoherent, but f˚OX is not. Indeed, the latter sheaf equals Ąk[t], and k[t] is certainlynot finitely generated as a k-module. K

However, for finite morphisms, we have a positive result:

Lemma 10.50 Let f : X Ñ Y be a finite morphism of schemes. If F be a quasi-coherentsheaf on X, then f˚F is quasi-coherent on Y. If X and Y are Noetherian, f˚F is evencoherent if F is.

Proof: Since f is finite, we can cover Y by open affines Spec A such thateach f´1 Spec A = Spec B is also affine, where B is a finite A-module. Wethen have f˚F (Spec A) = F (Spec B). Now, since F is quasi-coherent, we haveF |Spec B = ĂM for some B-module, which we can view as an A-module viaf 7(Y) : A Ñ B. Hence f˚F is quasi-coherent. If X and Y are noetherian, and Fis coherent, the module M is finitely generated as an B-module, and hence asan A-module, since B is a finite A-module. o


The categories of coherent and quasi-coherent sheavesOur work in the previous sections imply the following theorem

Theorem 10.51 The category QCohX is an abelian category and is closed underdirect limits.

By definition, being an abelian category entails that the hom-sets Hom(F ,G)are abelian groups; finite direct sums exist; kernels and cokernels of morphismsexist; every monomorphism F Ñ G is the kernel of its cokernel; every epimor-phism is the cokernel of its kernel; and every morphism can be factored into anepimorphism followed by a monomorphism. The hard part is thus in the lastpart of the statement, that any direct limit of quasi-coherent sheaves is againquasi-coherent.

One reason why we prefer the notion of ‘coherence’ used here (rather thanthe one in Hartshorne ([?])) is that the category of coherent sheaves CohX isalso an abelian category, even in the non-noetherian case. Note that it does notcontain all its direct limits, simply because an arbitrary product of coherentA-modules is typically not coherent (not even finitely generated!)

Coherent sheaves can still be regarded as the building blocks of the categoryQCohX. In fact, a common technique is to prove statements about quasi-coherentsheaves by approximating them with coherent sheaves. This is justified by thefollowing

Theorem 10.52 Any quasi-coherent sheaf on a Noetherian scheme is the directlimit of its coherent subsheaves.

We will not go into details about this statement here, but remark that theproof is not too difficult (see EGA I, Section 6.9).Exercise 10.24 Let M and N be coherent modules over A, and let φ : M Ñ Nbe an A-linear map.

i) Show that the kernel, the image and the cokernel of φ are coherent;

ii) Conclude that the full subcategory of ModA whose objects are thecoherent A-modules is an abelian category;

iii) Show by an example that the category of finitely generated module isnot abelian Hint: e.g. check kernels.

iv) Prove that CohX is an abelian category for any scheme X.

M

Exercise 10.25 Let X be a scheme. Show that every OX-module of finite type iscoherent if and only if the structure sheaf OX is. M

Exercise 10.26 (Extension of subsheaves.) Let F be a quasi-coherent sheaf onˇ

the Noetherian scheme X and let UĎX be an open subscheme. Show that any

closed immersions and closed subschemes 213

coherent subsheaf G ĎF |U can be extended to X; that is, there is a coherentsubsheaf G 1ĎF such that G 1|U = G. Prove Theorem 10.31 above. M

10.5 Closed immersions and closed subschemes

Recall that according to Definition 3.17, a closed subscheme of scheme X is aclosed subset ZĎX equipped with a sheaf of rings OZ that makes (Z,OZ)

into a scheme in a way that i˚OZ » OX/I for some sheaf of ideals I ĎOX.In Chapter 2 we considered the prototypical example when X = Spec A andZ = V(I) for some ideal IĎ A; the closed subscheme Z is then isomorphicto Spec(A/I). However, it was not at all clear which ideal sheaves gave riseto closed subschemes, even in the fundamental case of affine schemes. Inthis section we will show that the right condition is that the ideal sheaf bequasi-coherent.

Proposition 10.53 Let X be a scheme and let I ĎOX be a quasi-coherent sheafof ideals. Then the ringed space Z = (Supp(OX/I),OX/I) is a scheme witha canonical closed immersion ι : Z Ñ X.

Proof: To prove this we may assume that X = Spec A is affine. According tothe basic Theorem 10.18 on page 204 the ideal sheaf I is then the tilde of someideal IĎ A, and the support of ĄA/I consists exactly of the primes p such that(A/I)p ‰ 0, or equivalently, the prime ideals so that p P V(I). Hence Z equalsthe closed subset V(I) which is homeomorphic to Spec(A/I). The sheaf of ringson Spec(A/I) is the same as OX/I on Z, and consequently Z is the schemeSpec(A/I). The topological part of the morphism ι is just the inclusion ZĎX,and the algebraic part ι7 : OX Ñ i˚(OX/I) is just the tilde of the quotient mapA Ñ A/I. o

The converse of the previous proposition holds as well:

Proposition 10.54 Let Z Ă X be a closed subscheme of X, given by an idealsheaf I . Then I is quasi-coherent.

Proof: On the open set XzZ, we have I |XzZ » OXzZ, and so I is quasi-coherentthere. Let x P Z. We first find an affine open neighbourhood U = Spec A of xsuch that U X Z is an open affine in Z (recall that Z is itself assumed to be ascheme). To find U pick any affine open set U1 = Spec R Ă X and let V1 Ď U1XZbe an affine open set containing x. Then pick an element s P OX(U1) = R suchthat s = 0 on U1 X ZzV1, while s(x) ‰ 0 (this is possible because U1 X ZzV1 is aclosed set which does not intersect the closure txu´). Then let U = D(s) Ď U1.Note that U X Z = D(s|V1) Ď V1, and it ensues that U X Z is an affine subsetin Z as well. Write U = Spec A and U X Z = Spec B, and let the inclusion


U X Z Ñ U correspond to the map φ : A Ñ B. Let I = Ker φ. We claim that

I |U » rI,

which will show that I is quasi-coherent: indeed, for any distinguished openset D( f ) in U it holds true that

rI(D( f )) = I f = Ker(A f Ñ B f )

= Ker(OU(D( f ))Ñ OZ(ZXD( f ))

)= I(D( f )).

This completes the proof. o

Notice that the closed subset Z can be recovered from the ideal sheaf Iby Z = Supp(OX/I). In particular, this gives the most economic way ofdefining what a closed subscheme of X is: it is a subscheme of the form(Supp(OX/I),OX/I) for some quasi-coherent sheaf of ideals I .

Now we can finally prove Proposition 3.18 from Chapter 3.

Corollary 10.55 Let ZĎX be a closed subscheme given by an ideal sheaf I . Then forall open affines UĎX, the intersection U X Z is affine in Z. Moreover, if U = Spec A,then ZXU » Spec(A/I) for some ideal IĎ A.

Proof: Since I is quasi-coherent, we have I = rI for some ideal IĎ A. Then foreach open affine U, we have

OZ|U = Coker(I |U Ñ OX|U) = Coker(rI Ñ rA) = ĄA/I.

It follows that (Y,OY) = (V(I), ĄA/I) = Spec(A/I). o

Corollary 10.56 Let X = Spec A be an affine scheme. Associating the closed sub-scheme Spec (A/I) with the ideal I gives a one-to-one correspondence between the setof ideals of A and the set of closed subschemes of X. In particular, any closed subschemeof an affine scheme is also affine.

Induced reduced scheme structureWe have seen that on any open subset U Ď X of a scheme X there is a naturalscheme structure induced from that of X, the structure sheaf simply being therestriction of OX. For W Ď X a closed subset, there will in general be severalquasi-coherent ideal sheaves I corresponding to W. For instance, the affinescheme Spec k[x]/(x) is naturally a proper subscheme of Spec k[x]/(x2), but ofcourse, they have the same underlying topological space. So, in contrast withthe open ‘subschemes’ the underlying topological space does not determine thescheme structure. However there is one which is in some sense the ‘smallest’one:


Proposition 10.57 (Induced reduced scheme structure) Suppose thatX is a scheme and that WĎX a closed subset. There exists a unique closedsubscheme ZĎX such that

i) Z is reduced;ii) The underlying topological space of Z is W.

Proof: Let I ĎOX be the sheaf of ideals defined by

I(U) = t s P OX(U) | s(x) = 0 for all x P U XW u.

where we recall that s(x) denotes the class of s in k(x) = OX,x/mx We contendthat I is quasi-coherent. On an open affine subscheme U = Spec A we haveW XU = V(I) for a unique radical ideal I Ď A, and it holds true that I(U) = I;indeed, assume that f P A maps to zero in Ap/pAp for all p P UXW; that is, forall prime ideals pĎ A containing I. Since the preimage of pAp in Ap equals p, itfollows that f P

Ş

IĎ p p, but this intersection precisely equals?

I = I as I in theoutset was radical. Hence I(U) = I.

Moreover, for D(g)ĎU, we have I(D(g)) = Ig by the same argument, andso rI and I are equal as sheaves on U, and hence I is a quasi-coherent sheaf ofideals.

Now define Z to be the closed subscheme associated with the ideal sheaf I .Then Z is reduced and has the same underlying topological space as W (theseare local statements and we just checked them on the open affines). Finally, if Zand Z1 are two closed subschemes satisfying i) and ii) in the proposition, theirideal sheaves I and I 1 define the same radical ideal I(U) = I 1(U) = I on anyopen affine subscheme U = Spec A, and so they are equal. o

The scheme Z comes with a canonical morphism of schemes r : Z Ñ Xdefined as follows: We define r by the inclusion Z ãÑ X on the level of topologicalspaces. On the level of sheaves, we define f 7 : OX Ñ r˚OZ over an open setU Ă X to be the quotient map OX(U)Ñ (OX/J )(U). As the induced map onstalks is a quotient map as well, it is a local homomorphism, and we obtain amorphism rX = (r, r#) : Z Ñ X of schemes.

In particular, we may apply this construction to X = Z. We denote theresulting scheme by Xred and refer to it as the reduced scheme associated with X.The scheme Xred and the morphism rX : Xred Ñ X satisfy the following universalproperty, which among other things, entail that Xred depends functorially on X(see Exercise 10.31 below).

Proposition 10.58 Let f : Y Ñ X be a morphism of schemes, with Y reduced.Then f factors uniquely through the natural map rX : Xred Ñ X, i.e. there existsa unique morphism g : Y Ñ Xred such that f = r ˝ g.


Proof: The question is easily reduced to case of affine schemes, where it followsfrom the fact that a map of rings A Ñ B where B is without nilpotents, factorsunambiguously through A/

?0. o

Exercises(10.27) Show the following:

i) The skyscraper sheaf of k on A1k at the origin 0 is quasi-coherent;

ii) The skyscraper sheaf of k(T) on A1k at the origin 0 is not quasi-coherent.

(10.28) Let A3k = Spec k[x, y, z] and consider the twisted cubic curve C given by

the idealI = (y´ x2, z´ x3)

Let π : C Ñ A1k = Spec k[z] be the projection from the line L = V(x, y).

i) Show that π is a finite morphism;

ii) Compute π˚OC, π˚OA1k

and π˚J where J is the ideal sheaf of theclosed point 0 P A1

k .

(10.29) Let f : X Ñ Y be a morphism of schemes and let x P X be a point. Wesay that:

o A quasi-coherent sheaf F on X is flat over Y at x if Fx is flat as a OY, f (x)-module (where Fx is considered as a OY, f (x)-module via the natural mapf 7x : OX,x Ñ OY, f (x));

o F is flat if it is flat at every point in X;

o f is flat if OX is flat over Y

i) Show that open embeddings are flat. What about closed immersions?

ii) Show that a morphism of schemes Spec B Ñ Spec A is flat if and onlyif the map of rings A Ñ B is flat. More generally, a quasi-coherentsheaf ĂM on Spec B is flat over Spec A if and only if M is flat as anA-module;

iii) Which of the morphisms in Exercise 7.22 are flat?

iv) Prove that the blow-up morphism π : Bl0A2 Ñ A2 is not flat.

(10.30) Prove that the morphism r : Xred Ñ X is a closed immersion.(10.31) Functoriality of (´)red . If f : X Ñ Y is a morphism, show that there isa unique morphism fred : Xred Ñ Yred so that fred ˝ rX = rY ˝ fred. Show thatassignments X ÞÑ Xred and f ÞÑ fred defines a functor Sch to RedSch which


is adjoint to the inclusion functor RedSch Ñ Sch, where RedSch is the fullsubcategory of Sch whose objects are the reduced schemes.(10.32) Prove Proposition 10.37

(10.33) Morphisms to a closed subscheme. Let Z be a closed subscheme of X givenˇ

by sheaf of ideals I . Suppose f : Y Ñ X is a morphism of schemes. Show that ffactors through a map g : Y Ñ Z if and only if

i) f (Y) Ď Z;

ii) I Ď Ker( f 7 : OX Ñ f˚(OY)).

For a morphism of schemes f : Y Ñ X, we can define the scheme-theoreticimage of f as a subscheme Z Ď X satisfying the universal property that if ffactors through a subscheme Z1 Ď Z, then Z Ď Z1. To define Z it is is temptingto use the ideal sheaf I = Ker(OX Ñ f˚(OY)) — but this may fail to be quasi-coherent for a general morphism f . However, one can show that there is alargest quasi-coherent sheaf of ideals J contained in I , and we then define Z tobe associated to J .(10.34) Noetherian induction. Let X is a scheme. The closed subschemes forma partially ordered set when one lets ZĎZ1 mean that the closed immersionZ ãÑ X factors through the immersion Z1 ãÑ X.

i) Show that ZĎZ1 if and only it I(Z1)Ď I(Z);ii) Assume X to be Noetherian. Show that any non-empty set Σ of closed

subschemes contains a minimal element.

(10.35) Generic freeness of coherent sheaves . Assume that X is a reduced andˇ

irreducible scheme and let F be a coherent sheaf on X. Then F is ‘genericallyfree’, or phrased differently, ‘up to coherent sheaves with proper support it maybe approximated by a free sheaf’. In precise terms, show that there is a coherentsheaf H on X and a map α : F Ñ H with the two properties

i) Both supports Supp Ker α and Supp Coker α are proper subschemesof X;

ii) There is an integer and an inclusion OrX ĎH of a free sheaf such that

the quotient H/OrX has proper support.

(10.36) An ideal sheaf which is not quasi-coherent. Let X = Spec k[T] = A1k and

consider the origin P P X = A1k corresponding to the maximal ideal (T) Ă k[T].

Define the presheaf I of OX by for each open subset UĎX lettingI(U) Ă OX(U)

be given as

I(U) =

#

OX(U) if P R U;

0 if P P U.

a) Show that I is an ideal sheaf, and Supp(OX/I) is not a closed subset of X.

b) Show directly that I is not quasi-coherent by showing that I(X) = 0, butI ‰ 0.

M

Chapter 11

Locally free sheaves

The most important examples of quasi-coherent sheaves are the locally freesheaves. As the name suggests, these are sheaves which are locally isomorphicto a direct sum of copies of the structure sheaf of the scheme. Because of this‘freeness’ property, these sheaves are in many respects the nicest examples ofsheaves on a scheme and the easiest to work with. They are also the algebraiccounterpart to the vector bundles in topology.

An OX-module E is called free if it is isomorphic to a direct sum of copies ofOX. It is Locally free sheaf

(lokalt fritt knippe)locally free if there exists a trivializing cover; that is, an open cover tUiuiPI

such that E |Ui is free for each i. The rank of E at a point x P Ui is the numberrx(E ) of copies of OUi needed (this may be finite or infinite). If X is connected,the rank of E will be the same everywhere, but in general we allow variation. Alocally free sheaf of rank one is called an Invertible sheaf

(invertible knipper)invertible sheaf .

Example 11.1 The sheaf OrX =

Àri=1 OX is a locally free sheaf of rank r. As this

is globally a free sheaf, it is sometimes called ‘trivial’. K

If E is a locally free sheaf, the stalk Ex is a free OX,x-module for every x P X.In fact, under some coherence conditions, the converse holds:

Lemma 11.2 Suppose that X is locally Noetherian scheme. A coherent sheaf E on Xhaving the property that Ex » Or

X,x for every x P X for some fixed r, is locally free.

However, the converse of this statement does not hold in general. A simplecounterexample appears already on the spectrum of a DVR, a continuation ofExample 5 on page 189:Example 11.3 Let A be dvr with fraction field K, and let x and η be respectivelythe closed and the open point of X = Spec A. Let E be the OX module definedby Γ(X, E ) = A and Γ(tηu, E ) = K, and the restriction map being zero. Then E

is an OX-module with exactly the same stalks as the structure sheaf OX, but itis not locally free (in fact, it is not even quasi-coherent). K

Exercise 11.1 Prove Lemma 11.1. Mˇ

locally free sheaves and projective modules 219

11.1 Locally free sheaves and projective modules

On an affine scheme X = Spec A every quasi-coherent OX-module E is isomor-phic to ĂM for some A-module M. Thus a natural question is which A-modulesgive rise to locally free sheaves. The main result of this section is that E is locallyfree of finite rank if and only if M is finitely generated and projective.

We recall a few basic facts about projective modules (for a more extensivetreatment see Chapter ?? in [?]). An A-module M is called projective if thereis another module N so that M‘ N » Ar is free, and M being projective canfurther be characterized by saying that the functor N ÞÑ HomA(M, N) is exact.It is clear that free modules have this property, but there are many examples ofprojective modules which are not free. However, over local rings the two notionsare the same for finitely generated modules:

Lemma 11.4 Let A be a local ring with maximal ideal m and M a finitely generatedprojective A-module. Then M is free.

Proof: This is a standard application of Nakayama’s lemma. Let k = A/mdenote the residue field, and consider the module M bA k = M/mM. SinceM is finitely generated, this is a finite dimensional vector space over k. Letm1, . . . , mr P M denote a collection of elements in M that map to a basis forMbA k. We obtain a map φ : Ar Ñ M sending the standard basis vector ei tomi for each i = 1, . . . , r. Note that φb idk is an isomorphism, so by Nakayama’slemma φ is surjective. We thus get a short exact sequence

0 Ñ K Ñ Ar φÝÑ M Ñ 0,

where K = Ker φ. When M is a projective module, this sequence splits [?]. Henceit stays exact when tensorized by k. Again, since φb idk is an isomorphism,we get that K bA k = 0, and hence K = 0, once more by Nakayama’s lemma(note that K is finitely generated, being a direct summand of a finitely generatedmodule). It follows that M » Ar is free. o

Proposition 11.5 Let X = Spec A where A is Noetherian, and let F = ĂM bea coherent sheaf. The following are equivalent:

i) F is locally free;ii) Fx is a free OX,x-module for all x P X;

iii) M is locally free, i.e., Mp is free for all p P Spec A;iv) M is projective, i.e. there is a module N such that M‘ N » AI is

free.

Proof: This is really a result in commutative algebra, so we only say a fewwords. We have already seen the equivalence i) ðñ ii). The equivalence


ii) ðñ iii) follows by definition of ĂM, and finally, iii) ñ iv) follows becausebeing ‘projective’ is a local property, i.e. M is projective if and only if Mp is forevery p P Spec A. The implication iv) ñ iii) follows from the lemma above. o

From the proposition local properties of coherent locally free sheaves are ob-tained from corresponding properties of coherent projective modules, and byusing sufficiently fine affine covers, one may even (at least, when maps areglobally defined) reduce to the case of free modules. Recall the hom-sheafHomOX (E ,F ) whose sections over an open U is HomOX(U)(E (U),F (U)) andwhich is compatible with the tilde functor. The case that F = OX is of particularimportance and HomOX (E ,OX) is called The dual of a locally

free sheaf (det duale tilet lokalt fritt knippe)

the dual of E and is denoted E _.For each open affine U, there is a map of OX(U)-modules

HomOX(U)(E (U),OX(U))bOX(U) F (U)Ñ HomOX(U)(E (U),F (U))

given by φb s ÞÑ (x ÞÑ φ(x)s). It is compatible with the localization maps, andtherefore induces a map of OX-modules.

E _bOX F Ñ HomOX (E ,F )

The next proposition summarizes some of the basic properties of locallyfree coherent sheaves. The proofs are immediate, just reduce to the affine andfree case by restricting to a sufficiently fine covering, and for free modules thestatements are well-known. (see also xxx CA).

Proposition 11.6 Let X be a scheme and let E and F be two coherent locallyfree OX-modules.

i) The direct sum E ‘F is locally free of rank rx(E ) + rx(F );ii) The tensor product E bOX F is locally free of rank rx(E ) ¨ rx(F );

iii) The dual sheaf E _ is locally free of rank rx(E ), and the canonicalevaluation map (E _)_ Ñ E is an isomorphism;

iv) The canonical map E _bOX F Ñ HomOX (F ,G) is an isomorphism;and rank of HomOX (F ,G) equals rx(E )rx(F ).

A word of warning: the pushforward of a locally free sheaf is not locallyfree in general. For instance, if i : Y Ñ X is a closed immersion, then F = i˚OY

has Fy = OY,y for y P Y, but zero stalks outside of Y, and can not locally freein general, indeed if F is locally free of positive rank, then Supp(F ) = X. Forpullbacks, however, we have the following:

Proposition 11.7 Let f : X Ñ Y be a morphism of schemes. If G is a locallyfree OY-module, then f ˚G is a locally free OX-module.

Proof: Let Ui be trivialization of G on Y, such that F |Ui »À

I OUi . Then, sincef ˚OY = OX, we see that f´1(Ui) is a trivialization of f ˚G. o


Examples(11.8) Let X = Spec A where A = Z/2ˆZ/2 and consider the module M =

Z/2ˆ (0) which has the structure of an A-module. Then M is projective, sinceif N = (0)ˆZ/2, we have Mb N » A (as A-modules!). However, M is clearlynot free, since any free A module must have at least four elements! The sheafE = M is thus locally free, but not free on X. Note that X consists of two copiesof Spec Z/2. E restricts to the structure sheaf on one of these and to the zerosheaf on the other.(11.9) A less trivial example arises in number theory. We consider A = Z[i

?5]

and the ideal a = (2, 1 + i?

5). Then a direct computation shows that ab a »

Ab A, so a is projective (see Example ?? in CA). However, a is an ideal in A, soit is free if and only if it is principal. We therefore conclude that it is not free.(11.10) Let X = P1

A and consider the sheaves OP1A(m) constructed on page 107.

These sheaves were made by gluing together trivial sheaves rank one, so OP1A(m)

is locally free of rank one; that is, it is invertible. Moreover, we showed L » OP1A

for m ‰ 0.(11.11) Consider the ‘squaring-morphism’

f : P1k Ñ P1

k

from Example 10 on page 179. We claim that the pushforward f˚OP1k

is locallyfree of rank two.

Over the local chart U0 = Spec k[u] the map f is induced by k[u] ÞÑ k[t]with u ÞÑ t2, and over the chart U1 = Spec k[u´1] it is given by the mapk[u´1]Ñ k[t´1] such that u´1 ÞÑ t´2.

It ensues that the restriction f˚OP1 |U0 to U0 equals the tilde of k[t] as ak[u]-module which clearly is free with basis 1 and t; indeed, one has k[t] =k[u]‘ k[u]t. In a symmetric way, on the chart U1 = Spec k[u´1] the pushforwardf˚OP1 restricts to the tilde of the module k[u´1]‘ k[u´1]t´1. Hence f˚OP1 islocally free of rank 2.

In fact, one can readily check that there is an isomorphism f˚OP1 » OP1 ‘

OP1k(´1) where OP1

k(´1) is the invertible sheaf constructed in Example 5.3.

Indeed, the factors k[u] and k[u´1] patch up over U0 XU1 to give OP1k, whereas

for the other factor the gluing map is multiplication by u´1 since the equalityq(u´1)t´1 = q(u´1)u´1t holds true.(11.12) (The tangent bundle of the n-sphere) Let X = Spec A where we put A =

R[x0, . . . , xn]/(x20 + ¨ ¨ ¨+ x2

n ´ 1), and consider the A-module homomorphismf : An+1 Ñ A given by f (ei) = xi. Then M = Ker f gives rise to a quasi-coherentsheaf T = ĂM. Any element in the kernel corresponds to a vector of elementsv = (a0, . . . , an) P An+1 so that

a0x0 + ¨ ¨ ¨+ anxn = 0

invertible sheaves and the picard group 222

On U = D(x0) we may divide by x0, and solve for a0, so v is uniquely determinedby the elements (a1, . . . , an). Conversely, given any such an n-tuple of elementsin A, we may define an element v P Mx0 using the above relation. In particular,Mx0 » An. A similar argument works for the other xi, showing that T is locallyfree. It is a hard theorem that T is not free, if n R t0, 1, 3, 7u.

K

Exercise 11.2 Let X = Spec A, where A =ś8

i=0 Z. Show that M = Z isˇ

naturally an A-module which is projective, but not free. M

Exercise 11.3 (The projection formula.) Let f : X Ñ Y be a morphism of schemes,F an OX-module, and E a locally free sheaf of finite rank. Show that there is anatural isomorphism

f˚(F b f ˚E ) » f˚(F )b E .

M

11.2 Invertible sheaves and the Picard group

Recall that an Invertible sheaves(invertible knipper)

invertible sheaf on a scheme X is a coherent locally free sheafof rank one. We usually write L for such sheaves (they correspond to ‘linebundles’, as we will see later). By definition, L is invertible whenever thereexists a covering U = tUiu and isomorphisms φi : OUi Ñ L|Ui . We say thatgi = φi(1) P L(Ui) is a local generator for L. By Lemma 11.1 on page 218 acoherent OX-module L is invertible if and only if the stalk Lx is isomorphic toOX,x for every x P X.

Proposition 11.13 Let X be a scheme and L and M two invertible sheaves onX. Then we have

i) LbOX M is also an invertible sheaf. If g and h are local generators forL and M respectively, then gb h is a local generator for LbOX M;

ii) HomOX (L,OX) is invertible and HomOX (L,OX)bOX L » OX. Ifg is a local generator for L, then ψg defined by ψg(ag) = a is a localgenerator for HomOX (L,OX);

iii) Hom(L, M) » Hom(L,OX)bM.

Proof: Proof of i): We may find a common trivialization of L and M, so that Xmay be covered by open sets tUu where there are isomorphisms φ : OU Ñ L|Uand ψ : OU Ñ M|U . Over such a U, we have an isomorphism OU » OU bOU »

L|U bM|U given by 1 ÞÑ 1b 1 ÞÑ φ(1)b ψ(1). This shows i).For a proof of ii), note that the fact that Hom(L,OX) is invertible, can be seen

by restricting it to an open U where L|U » OU , and observing the isomorphismsHom(L|U ,OU) » Hom(OU ,OU) » OU .

invertible sheaves and the picard group 223

And iii) follows from Proposition 11.4. o

This proposition explains the term ‘invertible’. Indeed, the tensor productacts as a sort of binary operation on the set of invertible sheaves; L b M isinvertible if L and M are, and the tensor product is associative. Tensoring aninvertible sheaf by OX does nothing, so OX serves as the identity. Moreover, foran invertible sheaf L we will define L´1 = HomOX (L,OX); by the proposition,L´1 is again invertible, and serves as a multiplicative inverse of L under b. Wecan make the following definition:

Émile Picard(1856–1941)

Definition 11.14 Let X be a scheme. The Picard group Pic(X) is the group ofisomorphism classes of invertible sheaves on X under the tensor product.

Note that it is the set of isomorphism classes of invertible sheaves that forma group, not the invertible sheaves themselves: LbOX L´1 is isomorphic, butstrictly speaking, not equal to OX. Note also that Pic(X) is an abelian groupbecause LbOX M is canonically isomorphic to MbOX L.Example 11.15 Let X = Spec Z. If E is any coherent sheaf on X, then E = ĂM forsome finitely generated Z-module M, and by the structure theorem for finitelygenerated abelian groups, we may write M = Zr ‘ T, where T is a fine directproduct of groups of the form Z/nZ. If E in addition is required to be locallyfree, it must hold that T = 0 (otherwise, some of the stalks would not be free).Thus E = ĂZr = Or

X, and we conclude that every coherent locally free sheafSpec Z is trivial. In particular, we get that

Pic(Spec Z) = 0.

On the other hand, Pic(Z[?´5] ‰ 0, by Example 4. K

Example 11.16 Locally free sheaves on the affine line. The argument of the previousexample in fact applies over any pid A: every coherent sheaf on X = Spec Amust have the form ĂM for M = Ar ‘ T where T is a finitely generated torsionmodule, and if we require ĂM to be locally free, the torsion part must vanish; i.e.it must hold that T = 0. In particular, this applies to locally free sheaves on theaffine line A1

k = Spec k[x]:

Proposition 11.17 Any coherent locally free sheaf over A1k is trivial. Hence,

in particular, it holds that Pic(A1k) = 0.

In higher dimension the Quillen–Suslin theorem asserts that any locally freesheaf on An

k is trivial. This is a much deeper result than the above. In particularthe Quillen–Suslin implies that Pic(An

k ) = 0; we will see a direct proof of thelatter statement in Chapter 15. K

extending sections of invertible sheaves 224

Invertible sheaves on P1k

On page 107 in Chapter 5 we constructed the family OP1A(m) of sheaves on the

projective line over a ring A. They are all invertible, as we showed in Chapter 5,and in this section we intend to show there are no others when k is a field.

Recall that P1k is obtained by gluing together the two open affine subsets

U0 = Spec k[u] and U1 = Spec k[u´1] along V = Spec k[u, u´1]. Given aninvertible sheaf L on P1, the restriction of it to each of the two opens must betrivial since Pic(A1

k) = 0, so there are isomorphisms φi : L|Ui Ñ OUi . Over theintersection V = U0 XU1 we thus obtain two isomorphisms φi|V : L|V Ñ OV .In particular, the composition φ1|V ˝ φ0|

´1V : OV Ñ OV is an isomorphism. Like

any such map, it is induced by a module homomorphism k[u, u´1]Ñ k[u, u´1]

which is just multiplication by some unit in k[u, u´1]. But all units in k[u, u´1]

are of the form αum for an integer m and non-zero scalar α, the latter can beignored (incorporate it in one of the φi’s), and we recognize L to be the sheafOP1

k(m) from Chapter 5.With the present set-up we also obtain in a natural way an isomorphism

OP1k(m)bOP1

k(m1) » OP1

k(m + m1): the patching map over V for the tensor

product equals the tensor product of the two patching maps (which are multi-plication by sm and sm1 respectively), and when we identify OV bOV with OV ,it becomes the product of the two; that is, it becomes multiplication by sm+m1 . Inparticular, it holds that OP1

k(m)bOP1

k(´m) » OP1

k.

Back in Chapter 5 we verified that the sheaves OP1k(m) are not isomorphic

when m ě 0; e.g. since they have different spaces of global sections, and whatwe just did, extends this to all m. We thus have shown:

Proposition 11.18 Every invertible sheaf on P1k is isomorphic to OP1

k(m) for

some m P Z, and sending OP1k(m) to m yields an isomorphism Pic P1

k » Z.

We will prove a generalization of this in Proposition 15.18.

11.3 Extending sections of invertible sheaves

Let F be an OX-module on a scheme X and let x P X be a point. We will callthe fiber of F at x the k(x)-vector space F (x)

F (x) = Fx/mxFx » Fx bOX,x k(x)

If UĎX is an open subset containing x and s P Γ(U,F ) is a section of F over U,we shall denote by s(x) the image of the germ sx P Fx in the fibre F (x)—this inclose analogy with what we called the ‘value’ of a function defined near x.

extending sections of invertible sheaves 225

Definition 11.19 L be an invertible sheaf on the scheme X, and suppose s PΓ(X, L) is a global section. We define the open set Xs by

Xs = t x P X | f (x) ‰ 0 u.

Equivalently, Xs is the set of points x where s R mxLx.

The set Xs is indeed an open subset of X: the sheaf L is locally free, soevery point has an open affine neighbourhood U such that L|U » OX|U , and wemay safely assume that L = OX with X = Spec A. This brings us back to the‘function case’: the section s is an element in A, and clearly Xs = XzV(s).

In Chapter 10 we established a lemma (Lemma 10.16 on page 202) aboutextending sections of quasi-coherent modules on affine schemes. An analogueversion, which involves an invertible sheaf, is valid over Noetherian schemes andwill be of significant importance later on when we study maps into projectivespaces.

Lemma 11.20 Suppose X is a Noetherian scheme. Let F be a quasi-coherent sheaf onX and L an invertible sheaf. Suppose s P Γ(X, L). Then:

i) If a section t P Γ(X,F ) restricts to zero on Xs, then there is an integer Nsuch that tb sN P Γ(X,F b LN) is zero (on all of X).

ii) Suppose t P Γ(Xs,F ). Then there is an integer N such that tb sN extendsto a global section of F b LN .

Proof: Proof of i): Suppose t restricts to zero on Xs. Since X is quasi-compact,we may cover X by finitely many open affines Ui such that there is a collectionof trivialization isomorphisms φi : L|Ui » OX|Ui , and they induce isomorphismsF bOX L|Ui » F |Ui . Then t maps to zero in Γ(Ui X Xs) = Γ(Ui,F )s|Ui

. Now,Ui is affine, and we may consider s|Ui to be an element of OX(Ui) via theisomorphism φi, so it holds that Ui X Xs = D(s|Ui)). Hence there is a power ofs that annihilates t|Ui . In other words this says that tb sNi |Ui = 0 for some Ni.Taking N = max Ni, we find that tb sN is zero on all of X, as desired.

Proof of ii): For each i we know that some power tb sNi extends to a sectionover all of Ui (as above, because Ui is affine taking sections over distinguishedopen subsets corresponds to localization). Let M = max Ni. Then tb sM extendsto sections ti P Γ(Ui,F b LM). A potential problem is that the ti’s might notnecessarily agree on Ui XUj, hindering them to be glued together. However,it holds that ti = tj on Ui XUj X Xs since they extend something defined onXs, namely the section t. Now, Ui XUj X Xs is also Noetherian, part i) of theproposition applies, and we may conclude that there are natural numbers Mij

such that (ti ´ tj)b sMij = 0 P Γ(Ui XUj,F b LM+Mij). Then N = M + max Mij

does the trick: the ti b smax Mij can be glued together to a section that extendstb sN . o

operations on locally free sheaves 226

Remark 11.21 The same proof works also for an invertible sheaf L on a quasi-compactand separated scheme X.

11.4 Operations on locally free sheaves

Most constructions for vector spaces and free modules have analogies for locallyfree sheaves. For instance, one can define the The tensor algebra

(tensoralgebraen)tensor algebra T(F ) for a locally

free sheaf F . It is the sheaf of graded algebrasÀ

ně0 Tn(F ), where Tn(F )

stands for the n times iterated tensor product Fbn = F bOX . . . bOX F . Fromcommutative algebra we know that the n-fold tensor product ArbA . . .bA Ar isfree and isomorphic to Anr. So restricting Tn(F ) to an open affine Spec A overwhich F is trivial, we see that Tn(F ) is locally free of rank rn if F is of rankr. There are also natural ‘multiplication maps’ Tn(Ar)bA Tn1(Ar)Ñ Tn+n1(Ar)

acting according to the rule

(x1 b . . . b xn)b (y1 b ¨ ¨ ¨ b yn1)Ñ x1 b . . . b xn b y1 b ¨ ¨ ¨ b yn1 ,

and these induce sheaf maps Tn(F )bOX Tn1(F )Ñ Tn+n1(F ), which give T(F )

the structure of a graded OX-algebra.We let Sym(F ) = T(F )/I denote the The symmetric algebra

(den symmetriskealgebraen)

symmetric algebra of F where I is theideal in the tensor algebra T(F ) generated by expressions sb s1 ´ s1 b s where sand s1 are sections of F over some open. It inherits a grading from T(F ), andwe let Sn(F ) denote the part of degree n; this is the n-th symmetric power

(n-te symmetriskepotens)

n-th symmetric power ofF . Over an open set U, the tensor power Tn(F )(U) is generated by all tensorproducts s1 b ¨ ¨ ¨ b sn where si P F (U). Similarly, Sn(F ) is generated by theclasses of elements s1 b ¨ ¨ ¨ b sn modulo I; we write these as s1 ¨ ¨ ¨ sn.

Similarly, we defineŹ

F as the OX-module T(F )/J where J is the idealgenerated by all products sb s where s is a local section of F . This is sometimescalled the

The exterior algebra(ytrealgebraen)

exterior algebra. This is again graded, and a graded pieceŹn F is called

the n-th wedge product(n-th ytreprodukt)

n-th wedge product or the

The n-the alternatingproduct (det n-tealternerende produkt)

n-the alternating product. We write local sections ass1 ^ ¨ ¨ ¨ ^ sn. In these notations, we have s^ s = 0 and s^ t = (´1)klt^ s fors P

Źk F , t PŹl F , so the wedge product in

Ź

F is skew-commutative.

operations on locally free sheaves 227

Proposition 11.22 (Facts about locally free sheaves) Let X be ascheme.

i) The set of locally free sheaves is closed under direct sums, tensorproducts, symmetric products, exterior products, duals, and pullbacks;

If F is locally free of rank r then

ii) Tn(F ) is locally free or rank rn (locally spanned by all elementsm1 b ¨ ¨ ¨ bmn where mi P F );

iii) Sn(F ) is locally free of rank (n+r´1r´1 ) (locally spanned locally by the

elements mn11 ¨ ¨ ¨m

nrr where mi P F (U) and

ř

ni = n);iv)

Źn F is locally free of rank (rn) (spanned locally by elements mi1 ^

¨ ¨ ¨ ^min where i1 ă i2 ă ¨ ¨ ¨ ă in;

If 0 Ñ F 1 Ñ F Ñ F2 Ñ 0 be an exact sequence of locally free sheaves of ranksn1, n, n2 respectively. Then

v)Źn F »

Źn1 F 1 bOX

Źn2 F2.

Note that when F has rank r, one hasŹr+1 F = 0. Moreover,

Źr F isa locally free sheaf of rank (r

r) = 1, i.e. an invertible sheaf. This sheaf isusually called the Determinant of locally

free sheaves(determinanten tillokalt frie knipper)

determinant of F and denoted detF . Assertion v) may bephrased as the determinant being multiplicative over short exact sequences. Thedeterminant will be very important in Chapter ?? when we define the canonicalbundle of a variety.

Chapter 12

Sheaves on projective schemes

Projective schemes are to affine schemes what projective varieties are to affinevarieties. The construction of the projective spectrum Proj R is similar to that ofthe affine spectrum Spec R: the underlying topological space is defined with thehelp of prime ideals and the structure sheaf from localizations of R. However,there are some fundamental differences between the two: in the proj-constructionone only considers graded rings R, and only homogeneous prime ideals that donot contain the irrelevant ideal R+. As we saw, this reflects the construction ofthe projective spectrum Proj R as a quotient space

π : Spec RzV(R+)Ñ Proj R.

Given this, we can pull back a quasi-coherent sheaf to Spec R ´ V(R+) andextend it to a sheaf on Spec R via the inclusion map. Thus, it is natural toexpect that quasi-coherent sheaves on Proj R should be in correspondence with‘equivariant’ modules on Spec R; i.e. the graded1 R-modules. The irrelevantsubscheme V(R+) complicates the picture and makes the classification a littlebit more involved than the one for affines schemes. In particular, we will seethat different graded R-modules may correspond to the same quasi-coherentsheaf on Proj R.

Another important feature of Proj R is that it comes equipped with a canon-ical invertible sheaf which we will denote by OProj R(1). This is the geometricmanifestation of the fact that R is graded. Unlike the case of affine schemes,Proj R can typically not be recovered from the global sections of the struc-ture sheaf. It is the sheaf OProj R(1), or rather, the various tensor powersOProj R(d) = OProj R(1)bd, that will play the role of the affine coordinate ring inthe affine case. So it is rather from the pair (Proj R,OProj R(1)) one may hope torecover R.

1In the model case of the projective spaces, the variety Pn is the quotient of Anzt0u by the group k˚

acting by scalar multiplication, so in this case, the notion ‘equivariant’ is precise and pertinent.

the graded tilde-functor 229

12.1 The graded tilde-functor

Let R be a graded ring and let GrModR denote the category of graded R-modules. Just as in the case of the affine spectrum Spec A, we shall set upa tilde-construction which produces sheaves on Proj R from graded R-modules,and in this way gives a functor GrModR to ModOX . However, in contrast to theaffine case, this will not be an equivalence of categories.

Homogenization & dehomogenizationBack in in Chapter 12 on page 171, we utilized a homogenization-dehomogeniza-tion process to fabricate the structure sheaf on Proj R, and we shall rely on asimilar technique in the tilde-construction.

Recall that if for two homogeneous elements f , g P R+ there is an inclusionD+(g) Ď D+( f ) of the corresponding distinguished sets, then gr = v f forsome homogeneous v P R and some natural number r (indeed, D+(g) Ď D+( f )is equivalent to

a

(g)Ďa

( f )). And as f becomes invertible in Rg, there is acanonical map M f Ñ Mg between the localized modules; it respects the gradingssince both f and g are homogeneous, and its action on the degree zero partsyields a canonical map ρ f ,g : (M f )0 Ñ (Mg)0. That map sends an element x fń

with x homogeneous and deg x = n deg f to the element vnxgńr.Letting B be the basis for the Zariski topology whose elements are the

distinguished open subsets, this permits us to define a B-presheaf ĂM: sectionsover D+( f ) are to be given by

ĂM(D+( f )) = (M f )0,

and the restriction maps ĂM(D+( f )) Ñ ĂM(D+(g)), when D+(g)ĎD+( f ), areto be the maps ρ f ,g above (the two requirements to be a presheaf are easilyverified).

Recall the canonical isomorphism D+( f ) » Spec (R f )0 from Proposition 9.5on page 172, and it will be important to see what sheaf ĂM will yield whenrestricted to D+( f ) and transported to Spec (R f )0. The answer is given in thefollowing proposition.

Proposition 12.1 Under the isomorphism between D+( f ) and Spec (R f )0

one has ĂM|D+( f ) »Č((M f )0).

Recall that a distinguished subset D+(g) of D+( f ) is mapped isomorphicallyonto the distinguished open subset D(u) of Spec (R f )0 where u = gdeg f f´deg g

(which is the simples degree zero element in R f one can fabricate out of f andg). The propostion ensues effortlessly from the following lemma, wich has aslightly technical proof:


Lemma 12.2 With the notation above, the canonical homomorphism ρ f ,g : (M f )0 Ñ

(Mg)0 induces an isomorphism ((M f )0)u » (Mg)0;

Proof: The element u is invertible in (Rg)0, so the map ρ f ,g : (M f )0 Ñ (Mg)0

factors via a mapρ : ((M f )0)u Ñ (Mg)0,

which explicitly is given as

ρ(

x fńu´m) = x f m deg gńg´m deg f

where x P M is homogeneous of degree n deg f . We contend that this is anisomorphism, and begin with showing that ρ surjective. To that end, note thateach element in P (Mg)0 is on the form y ¨ g´l with y P M homogeneous anddeg y = l deg g. With an integer m so large that m deg f ě l, one has the equality

g´ly = u´mumg´ly = u´m(gm deg f´ly) f´m deg g.

The right side is an element in ((R f )0)u since m deg f ´ l ě 0, and the equalitymeans it maps to g´ly.

To see that the map ρ is injective assume that an element x fń P (M f )0 tozero in (Mg)0; this means that there is an integer l ą 0 so that glrvnx = 0 in M.Multiplying up by appropriate powers of v and f , we get a relation in M of theform g(l+n)rx = 0, and consequently it holds that u(l+n)rx = 0 P (M f )0. But thenx fń = 0 in ((M f )0)u, and we are through. o

As an immediate consequence of Proposition 12.1 we obtain the desired

Proposition 12.3 The B-presheaf ĂM is a B-sheaf, and extends to a quasi-coherent sheaf on Proj R; which we continue to denote ĂM.

Properties of the tilde-functorAs is the case for the tilde-construction for affine spectra, the assignment M ÞÑ ĂMis functorial and gives a functor GrModR Ñ QCohProj R. This is close to obviousas a map M Ñ N homogeneous of degree zero keeps being homogeneous ofdegree zero when localized and so induces maps (M f )0 Ñ (N f )0.

In some apects the projective tilde-funstor behaves as the affine one, butin other apsects, the behaviour deviates seriously; the most striking differencebeing that different modules may yield isomorphic sheaves, and this is inherent,not accidental.

The following proposition summarizes the basic properties of the tilde-functor.


Proposition 12.4 Let R be a graded ring. The functor GrModR Ñ QCohProj R

that sends M to ĂM has the following properties:

i) The tide-functor is additive and exact and commutes with direct limits.ii) Sections over distinguished open sets: for f P R homogeneous the

equaity Γ(D+( f ), ĂM) = (M f )0 holds tue;

iii) Stalks: for each p P Proj R it holds that ĂMp = (Mp)0;iv) When the ring R is Noetherian and M is finitely generated, then ĂM

is coherent.

Proving these properties is straightforward, since most of them can bechecked locally. Using the isomorphisms between D+( f ) and Spec(R f )0 wereduce immediately to the affine case.

It is important to note that, unlike the affine case, the tilde-functor is notfaithful, as several modules can correspond to the same sheaf. This is not sosurprising and is rooted in the fact that primes in V(R+) are thrown away inthe Proj-construction, which has the effect that modules supported in V(R+)

necessarily give the zero sheaf when exposed to the tilde-functor. For any integerd we let Mąd be the R-module Mąd =

À

iąd Mi (it is an R-module because ofthe standing hypothesis that R be positively graded).

Lemma 12.5 Assume that R is an graded ring and let M and N be two graded R-modules,

i) If Supp MĎV(R+), then ĂM = 0;ii) Assume that Mąd » Nąd for some d. Then ĂM » rN.

Proof: To prove i), suppose that Supp MĎV(R+). Statement iii) of Proposi-tion 12.4 above then entails that ĂM = 0 since Mp = 0 for all p P Proj R.

To prove ii), note that the quotient M/Mąd is killed by the power (R+)d andconsequently has support in V(R+). By i) its tilded sheaf vanishes, and henceĆMąd = ĂM. As this holds for both M and N we are through. o

Example 12.6 On X = Proj k[x0, x1], the module M = k[x0, x1]/(x20, x2

1) hasĂM = 0, but it is non-zero. K

The next lemma is sometimes be useful for working with the localization of Mwhen R is generated in degree one. It says essentially that we are allowed to‘substitute 1 for f ’ when restricting a module to an affine chart D+( f ) Ă Proj R.

Lemma 12.7 Suppose that M is a graded R-module and that f P R homogeneous ofdegree one. Then there are natural isomorphisms of (R f )0-modules

(M f )0 » M/( f ´ 1)M » MbR R/( f ´ 1)R.

In particular, (R0) f » R/( f ´ 1)R.


Proof: The element f acts as the identity on the R-module M/( f ´ 1)M, soM/( f ´ 1)M is a module over R f . Plainly sending x f´r to x yields an R f -linear homomorphism M f Ñ M/( f ´ 1)M, as one easily verifies, and restrictingit to the degree zero piece one obtains an (R f )0-homomorphism (M f )0 Ñ

M/( f ´ 1)M. It is surjective: the class a homogeneous element x is the image ofx f´deg x, and every element in M/( f ´ 1)M is the sum of classes of homogenouselements. To check it is injective, assume that x f´deg m maps to zero; i.e. thatx = ( f ´ 1)y for some y P M. Expanding y in homogeneous components wemay write y =

ř

sďiďt yi with neither ys nor yt equal to zero. Then

x = ( f ´ 1)y = ýs +t´1ÿ

s( f yi ´ yi+1) + f yt.

Because x is homogeneous and ys ‰ 0, we may infer that ys = ´x, but alsothat f yt = 0 and yi+1 = f yi. A straightforward induction then yields equalitiesyt = f t´sys = ´ f t´sx; consequently x is killed by a power of f and vanishes inM f . o

Example 12.8 That f is of degree one is essential. To give an example where theabove lemma fails, let M = R = k[x] and f = x2. We find k[x]x2 = k[x, x´2] =

k[x, x´1] so that (k[x]x2)0 = k. But k[x]/(x2 ´ 1) » k‘ k. K

Tensor product & Hom’sLet M and N be two graded modules over the graded ring R. There is a naturalway of giving the tensor product a graded structure; a decomposable tensorxb y is precisely homogenous when x and y are, and, of course, it is of degreedeg x + deg y. Homogenous tensors will be the sums of decomposables of thesame degree; i.e. they form the image

À

i+j=n MibR0 Mj, and this will be thegraded piece of MbR N of degree n. One may check that MbR N as an R0-module is the direct sum of these graded parts (that they generate is obvious;that the pairwise intersections are zero is slightly more subtle).

The tilde-functor is in the case of affine spectra well-behaved when it comesto tensor products in that ĂMbOSpec A

rN = ČMbA N. In the projective howevercase‚ this is not always so. Unless R is generated in degree one, curiousphenomena take place. The following simple example may be instructive, whichalso illustrates the subtlety of the proj-construction for rings not generated indegree one.Example 12.9 Again we consider the ring R = k[x2] where k is a field andx2 is of degree two. Then Proj R is reduced to one point with structure sheafOProj R = k, and, moreover, it is covered by the sole distinguished set D+(x2).

Consider the R-module M =À

iě0 k ¨ x2i+1, which is nothing but the sub-module of k[x] consisting of polynomials that having non-vanishing terms onlyof odd degrees. We contend that (Mx2)0 = 0. Indeed, elements of Mx2 are

serre’s twisting sheaf o(1) 233

linear combinations of terms shaped like x2i+1x´2s, and none of these can be ofdegree zero. So ĂM = 0, and thus also ĂMbOProj R

ĂM = 0. On the other hand, themodule MbR M possesses elements of degree zero when localized at x2; in fact,all elements are of even degree since they are sums of terms like p(x)b q(x).Hence (MbR M)x2)0 ‰ 0; or in other words, ČMbR M ‰ 0.

It may seem paradoxical that redefining the grading on k[x2] by givingx2 degree one, the tilde-construction and tensor product will commute; theexplanation is that the ‘counter-example’ M above is no more a graded module!Well, the only sensible degree one could give x and still make the example work,would be 1/2, which is not allowed.

Note that the example also illustrates that the converse of Lemma 12.5 doesnot hold unconditionally (but, again as we shall see, it holds true when R isgenerated in degree one). K

Let us proceed to compare ČMbR N with ĂMbOXrN. For each homogeneous el-

ement f P R there is map (M f )0ˆ (N f )0 Ñ ((MbR N) f )0 sending (x/ f n, y/ f m)

to (xb y)/ f m+n+b. It is obviously (R f )0-bilinear, and consquently there is aninduced map

(M f )0b(R f )0(N f )0 Ñ ((MbR N) f )0.

Since maps between B-sheaves induce maps between sheaves, we get a naturalmap

ĂMbOProj RrN Ñ ČMbR N, (12.1)

which, however, as the Example 3 above shows, it is not always an isomorphism;but one has the folowing:

Proposition 12.10 Suppose R is generated in degree one. Then the naturalmap (12.1) is an isomorphism.

Proof: By assumption, Proj R is covered by open affines of the form D+( f )where f has degree one. For such an f , the functor M Ñ (M f )0 coincides withthe tensor-functor (´)bR R/( f ´ 1)R by the previous lemma. Furthermore, oneof the standard properties of the tensor product gives that

(MbR (R f )0)b(R f )0(N bR (R f )0) » (MbR N)bR (R f )0.

This isomorphism provides the inverse to the natural map (M f )0b(R f )0(N f )0 Ñ

((MbR N) f )0 defined above. Then, since the map from (12.1) restricts to anisomorphism on each D+( f ) for f P R of degree one, it is an isomorphism. o

12.2 Serre’s twisting sheaf O(1)

Arguably the most interesting sheaf on X = Proj R is the so-called twisting sheaf,denoted by OX(1). This is a generalization of the tautological sheaf on Pn

k , andconstitutes a geometric manifestation of the fact that R is a graded ring. It was


introduced in the groundbreaking paper [?] by Jean Pierre Serre. Elements in Rdo not define ‘regular functions’ on Proj R, and we shall see that in good casesRd will be the space of sections of the tensor power OProj R(d) when d ě 0, andthis is a means of recovering the ring R.

Jean-Pierre Serre(1928 – )

Let M be a graded module over the graded ring R. For each integer n, wewill define an R-module M(n) as follows: As an underlying R-module M(n) isjust M, but the grading is shifted by n:

M(n)d = Md+n.

Thus N = M(n) is a graded R-module with N0 = Mn, N1 = Mn+1 and so on.The construction is functorial and is called the shift-functor or the twist-functor.Note that elements from Md considered as element in M(n) will be of degree dń. The particular case when M = R, this gives the naturally a graded and free R-module R(n) generated by the element 1 P Rń. Note that M(n) = MbR R(n):both have M as underlying module, and

À

i+j=d MibR0 R(n)j = Md+n.Applying the tilde-functor to R(n) gives us a quasi-coherent OProj R-module

on Proj R:

Definition 12.11 For an integer n, we define

OX(n) = ĆR(n).

For a sheaf of OX modules F on X, we define the twist by n by F (n) =

F bOX OX(n).

To each element r P Rd there is a corresponding section in Γ(X,OX(d)). Thisis so because, according to Proposition 12.4, we can think of an element ofΓ(X,OX(d)) as a collection of pairs (r f , D+( f )) with r f P ((R f )d)0 matching onthe overlaps D+( f )XD+(g), where f and g run through a set of homogeneousgenerators for R+. Hence we can define an R0-module homomorphism

Rd Ñ Γ(X,OX(d))

by r ÞÑ t(r/1, D+( f ))u. The element r/1 is just the image of r under thecanonical localization map R Ñ R f , and it is of degree zero as degrees areshifted by ´d. This also makes it clear that on the overlaps D+( f g) the twoelements (r/1, D+( f )) and (r/1, D+(g)) become equal, and so we obtain anactual global section of O(d). Abusing notation, we will also denote this sectionby r.

In the special case that the element f is of degree one; that is, when f P R1,we have the equality (R(n) f )0 = f n(R f )0; indeed, for each N the equalitiesr fŃ = r f n ¨ fŃ+n with r P RN+n hold true. Thus, on the distinguished affineopen set D+( f ) it holds true that OX(n)|D+( f ) = f nOX|D+( f ). In particular,


OX(n)|D+( f ) » OD+( f ). In other words, if R is generated in degree one, the sheafOX(n) is locally free of rank one, that is, it is an invertible sheaf.

Proposition 12.12 When R is generated in degree one, the sheaf OX(n) isinvertible for every n. Moreover, there are canonical isomorphisms

OX(m + n) » OX(m)bOX OX(n).

Proof: Indeed, if R is generated in degree one, Proposition 12.7 shows thatOX(m)bOX(n) is the sheaf associated to R(m)bR R(n) » R(n + m); that is,associated to OX(n + m). o

So this is a big difference between affine schemes and projective schemes:Proj R comes equipped with lots of invertible sheaves.Example 12.13 P1

A once more. Let X = Proj R where R = A[x0, x1] be theprojective line P1

A over A. Let us compute the global sections of the sheafOX(d). Our scheme X is covered by the two distinguished opens D+(x0) =

Spec A[x1x0´1] and D+(x1) = Spec A[x0x1

´1], and the following hold true

Γ(D+(x0),OX(d)) = (Rx0)d = A[x1x0´1]xd

0

andΓ(D+(x1),OX(d)) = (Rx1)d = A[x0x1

´1]xd1.

On the overlap D+(x0)XD+(x1) = D+(x0x1), we have

Γ(D+(x0x1),OX(d)) = A[x0x´11 , x1x´1

0 ]xd0 = A[x0x´1

1 , x1x´10 ]xd

1

and we find that two sections

s0 = p0

(x1

x0

)xd

0 and s1 = p1

(x0

x1

)xd

1

agree on the overlap if and only if

xd1 p1

(x0

x1

)= xd

0 p0

(x1

x0

). (12.2)

Here p0 and p1 are polynomials with coefficients in A. Thus for such anequality to hold in A[x0, x´1

0 , x1, x´11 ], we immediately see that d must be non-

negative; that p0 and p1 must have degree d and that each side of (12.2) mustbe a homogeneous polynomial of degree d. Conversely, any homogeneouspolynomial of degree d

P(x0, x1) = a0xd0 + a1xd´1

0 x1 + ¨ ¨ ¨+ adxd1

the associated graded module 236

gives rise to a global section through the assignments p0 = x´d0 P and p1 =

x´d1 P. Thus Γ(X,OX(d)) can be naturally identified with the A-module of

homogeneous polynomials in x0, x1 of degree d. K

As alluded to above, the main point of the sheaves OX(d) is that they helpus recover the ring R; for instance, while xd

0 does not correspond to a regularfunction on X = Proj k[x0, x1], it gives a section of the sheaf OX(d).

12.3 The associated graded module

We have associated to a graded R-module M a sheaf ĂM on X = Proj R. Toclassify quasi-coherent sheaves on X we would, as in the case of affine schemes,like to give some sort of inverse to this assignment. However, as opposed tothe case for X = Spec A, simply using the global sections functor will not work.Indeed, even for F = OP1

kon P1

k , it holds that Γ(P1k ,F ) = k, from which we

certainly cannot recover F . The remedy is to look at the various Serre twistsF (d) of F ; in fact all of them at once:

Definition 12.14 Let R be a graded ring and let F be an OX-module onX = Proj R. We define the graded R-module associated to F , denoted Γ˚(F )

asΓ˚(F ) =

à

dPZ

Γ(X,F (d)).

In particular, from X alone we get the associated graded ring

Γ˚(OX) =à

dPZ

Γ˚(X,OX(d)).

The associated graded module has the structure of an R-module: If r P Rd,we have a corresponding section r P Γ(X,OX(d)) (abusing notation, as before).So if σ P Γ(X,F (n)), then we may define r ¨ σ P Γ(X,F (n + d)) as rb σ via theisomorphism F (n)bO(d) » F (n + d).

If R is generated in degree one, and M is a graded R-module, we can definea homomorphism of graded R-modules

α : M Ñ Γ˚(ĂM)

To define it, it is useful to think of elements in Γ(X, ĂM(n)) as a collection ofelements (m f , D+( f )) for m P (M f )n and f P R1, matching on the variousoverlaps.

With this in mind, we can send an element m P Md to the collection given by(m/1, D+( f )), where f ranges over the degree one piece R1. On the overlapsD+( f )X D+(g) = D+( f g) the two elements (m/1, D+( f )) and (m/1, D+(g))become equal so this defines an actual global section of ĂM(n). It is clear thatthis is a graded homomorphism. Moreover, it is functorial in M.

the associated graded module 237

Proposition 12.15 Let R be a graded integral domain, finitely generated overR0 in degree 1 by elements x0, . . . , xn, and let X = Proj R. Then

i) Γ˚(OX) =Şn

i=0 Rxi Ă K(R);ii) If each xi is a prime element, then R = Γ˚(OX).

Proof: Cover X by the opens Ui = D+(xi). We have, since Γ(D+(xi),O(m)) »

(Rxi)m, that the sheaf axiom sequence takes the following form

0 Ñ Γ(X,O(m))Ñnà

i=0(Rxi)m Ñ

à

i,j(Rxixj)m

Taking directs sums over all m, we get

0 Ñ Γ˚(OX)Ñnà

i=0Rxi Ñ

à

i,jRxixj

So a section of Γ˚(OX) corresponds to an (n + 1)-tuple (t0, . . . , tn) PÀn

i=0(Rxi)

such that ti and tj coincide in Rxixj for each i ‰ j. Now, the xi are not zero-divisors in R, so the localization maps R Ñ Rxi are injective. It follows thatwe can view all the localizations Rxi as subrings of Rx0...xn , and then Γ˚(OX)

coincides with the intersectionnč

i=0

Rxi Ă R0[x0, x´10 , ¨ ¨ ¨ , xn, x´1

n ].

In the case each xi is prime, this intersection is just R. o

Corollary 12.16 Let X = PnA = Proj A[x0, . . . , xn] for a ring A. Then

Γ˚(OX) » A[x0, . . . , xn]

In particular we can identify Γ(PnA,O(d)) with the A-module generated by homoge-

neous degree d polynomials.

When R is not a polynomial ring, it can easily happen that Γ˚(OX) is differentthan R. Here is a concrete example:Example 12.17 A quartic rational curve. Let k be a field and let R be the k-algebraR = k[s4, s3t, st3, t4] Ă k[s, t]. Note that the monomial s2t2 is missing from the setof generators of R. Define the grading such that R1 is the vector space generatedby s4, s3t, st3 and t4.

We can also think of R as the graded ring

R = k[x0, x1, x2, x3]/(x20x2 ´ x3

1, x1x23 ´ x3

2, x0x3 ´ x1x2).

The radical of the ideal (s4, t4) equals R+ so Proj R is covered by U0 = D+(s4) =

D+(x0) and U1 = D+(t4) = D+(x3); that is, Proj R = U0 YU1. Furthemore wehave

U0 = Spec(Rx0)0 and U1 = Spec(Rx3)0.

quasi-coherent sheaves on proj r 238

Here (Rx0)0 = k[ts´1, t3s´3, t4s´4] = k[ts´1] and similarly (Rx3)0 = k[st´1]. SoProj R is in fact isomorphic to P1

k . We have thus shown that X = Proj R embedsas a rational (degree 4) curve in P3

k .What is Γ(X,OX(1))? On the two opens we find OX(1)(U0) = k[ts´1] ¨ s4

and OX(1)(U1) = k[st´1] ¨ t4. So using the sheaf sequence, we get

0 Ñ Γ(X,OX(1))Ñ k[ts´1]s4 ‘ k[st´1]t4 Ñ k[st´1, ts´1]s4.

Note that the monomial s2t2 belongs to both the rings k[st´1]t4 and k[ts´1]s4,and so defines an element in Γ(X,OX(1)). In fact,

Γ(X,OX(1)) = kts4, s3t, s2t2, st3, t4u

even though R1 = kts4, s3t, st3, t4u.In this example, the graded ring Γ˚(OX) = k[s4, s3t, st3, t4] is the integral

closure of R. Exercise 12.2 below shows that this is not a coincidence. K

Example 12.18 Let X = P1k . The sheaves OX(n) give another example why one

has to sheafify in the definition of the tensor product. If T denotes the naivepresheaf

T(U) = OX(´1)bOX OX(1)

Then clearly Γ(X, T) = 0bk k2 = 0. However, the sheafification T+ is isomorphicto OX, so Γ(X, T+) = k, and thus T ‰ T+. K

Exercise 12.1 Let k be a field and let R = k[x0, . . . , xn]. Let π : An+1 ´ 0 ÑPn

k = Proj R denote the ‘quotient morphism’ from Exercise 9.9. Show that for agraded R-module M, we have

π˚(ĂM|An+1k ´0) =

à

nPZ

ĂM(d)

M

Exercise 12.2 Let R be a graded Noetherian integral domain generated inˇ

degree one. Show that R1 = Γ˚(OX) is an integral extension of R. (Hint: Usethe Cayley–Hamilton theorem.) M

12.4 Quasi-coherent sheaves on Proj R

As before, we assume that R is a graded Noetherian ring generated in degree1. The main theorem of this section says that any quasi-coherent sheaf F onX = Proj R is the graded „ of some graded R-module M. Not surprisingly, thisR-module is exactly the associated graded module M = Γ˚(F ). The followingis the main theorem of this section:

Proposition 12.19 Let R be a graded ring, finitely generated in degree 1 overR0. Suppose F is a quasi-coherent sheaf on Proj R and let M = Γ˚(F ). Thenthere is a canonical isomorphism

β : ĂM Ñ F (12.3)

quasi-coherent sheaves on proj r 239

Consider the R-module M = Γ˚(F ). The sheaf ĂM is naturally a quasi-coherent sheaf of OX-modules. Let us first define the map 12.3. As usual, itsuffices to define this map over the distinguished opens D+( f ), and since weassume that R is generated in degree 1, we may assume that f P R1. On D+( f ), asection of ČΓ˚(F ) is represented by a fraction m/ f d where m P Md = Γ(X,F (d)).If we regard f´d as a section in OX(´d)(D+( f )), then the tensor product mb f´d

can be regarded as a section of F via the isomorphism F (d)bO(´d) » F . Thisallows us to define (12.3) by associating m/ f d to mb f´d. Note that this map isa map of OX-modules.

Proof (of Theorem 12.13): Since R is generated by R1 over R0, the open setsD+( f ) with f P R1 cover X. To show that (12.3) is an isomorphism of sheaves, itis sufficient to prove it on such an open.

Let f P R1, and consider it as a section of Γ(X,O(1)). Then taking L = O(1)in Lemma 11.11, point (i) there says that if an element s of Γ(D+( f ),F ) isgiven, we can find some element t of Γ˚(F )N (for N sufficiently large) such thattb fŃ P Γ(D+( f ),F ) equals s. This implies that the map β is surjective.

For injectivity, suppose s P Γ(X,F (n)) is such that sb fń = 0 on D+( f ),i.e. s/ f n P (M f )0 is in the kernel of (12.3) on the D+( f )-sections. Then thelemma implies that there is a power f N with sb f N P Γ(X,F (n + N)) = 0. Thismeans that s/ f n = 0 in (M f )0 by the definition of localization, and so the mapis injective. o

We have now defined two functors

„: GrModR Ñ QCohX

andΓ˚ : QCohX Ñ GrModR

Since β : ČΓ˚(F )Ñ F is an isomorphism, it follows that „ is essentially surjective.However, unlike the affine case, the functors do not give mutual inverses. Thisis because, as we have seen, that „ is not faithful; the „ of any module M whichis finite over R0 is the zero sheaf.

We can define an equivalence relation on graded modules by setting M „ NifÀ

iěi0 Mi »À

iěi0 Ni for some i0 P Z. For two finitely generated gradedR-modules M, N we have M „ N if and only if ĂM » rN, so we have identifiedprecisely the ‘kernel’ of the functor „.

Putting everything together, we find

closed subschemes of projective space 240

Theorem 12.20 Let R be a graded ring, finitely generated in degree 1 over R0

and let X = Proj R. Then the functors

„: GrModR Ñ QCohX

andΓ˚ : QCohX Ñ GrModR

satisfy ČΓ˚(F ) = F for all F P QCohX, and give an equivalence between thecategories of quasi cohoherent sheaves on X and graded R-modules modulo theequivalence relation M „ N.

12.5 Closed subschemes of projective space

Having discussed what quasi-coherent sheaves are on projective spectra, we willnow use this to study closed subschemes. We saw earlier that given a gradedideal I Ă R we could associate a closed subscheme V(I) Ă Proj R and a closedimmersion Proj(R/I) Ñ Proj R. On the other hand, we also saw above thatmany graded modules M could give rise to the same quasi-coherent sheaf ĂM.This is also the case for graded ideals , as we shall see, but luckily we are againable to completely identify which ideals give rise to the same closed subscheme.

In the discussion it will be convenient to introduce the saturation of an ideal.The upshot will be that this will serve as the ‘largest’ ideal corresponding toa given subscheme. We fix an ideal B Ă R (the case to have in mind is theirrelevant ideal B = R+). Then for a graded ideal I Ă R, we define the saturationof I with respect to an ideal B is defined as the ideal

I : B8 :=ď

iě0

I : Bi = tr P R|Bnr P I for some n ě 0u.

We say that I is B-saturated if I = I : B8 and more concisely, saturated if it isR+-saturated. We will here denote I : (R+)8 by I. It is not hard to check that Iis homogeneous if I is.Example 12.21 In R = k[x0, x1], the (x0, x1)-saturation of (x2

0, x0x1) is the ideal(x0). Note that both (x0) and (x2

0, x0x1) define the same subscheme of P1k , but in

some sense the latter ideal is inferior, since it has a component in the irrelevantideal (x0, x1). This example is typical; the saturation is a process which throwsaway components of I supported in the irrelevant ideal. K

closed subschemes of projective space 241

Proposition 12.22 Let A be a ring and let R = A[x0, . . . , xn].

i) If Y is a closed subscheme of PnA = Proj R defined by an ideal sheaf I ,

then the idealI = Γ˚(X, I) Ă R

is a homogeneous saturated ideal. In this setting, Y corresponds to thesubscheme Proj(R/I)Ñ Proj R.

ii) Two ideals I, J defined the same subscheme if and only if they have thesame saturation.

In particular, there is a 1-1 correspondence between closed subschemes i : Y ÑPn

A and saturated homogeneous ideals I Ă R.

Proof: (i) Let i : Y Ñ PnA be a closed subscheme of Pn

A = Proj R and letI Ă OPn

Adenote the ideal sheaf of Y. Using the fact that global sections is

left-exact, we have Γ˚(I) Ă Γ˚(OPnA) = R. I = Γ˚(I) is naturally a graded

R-module, so in fact I is a homogeneous ideal of R.Any such ideal I gives rise to a closed subscheme i1 : Proj(R/I)Ñ Pn

A andhence an ideal sheaf J satisfying rI = J . By Proposition 12.13, we also haverI = I , so the two quasi-coherent ideal sheaves coincide and i is indeed the sameas i1. By construction I = Γ˚(rI).

Let us show that I is saturated. Let f P I be homogeneous of degree q. Thenthere is an m such that f ¨ xm

i P Iq+m for all i. Since f1 =

f ¨xmi

xmiP (Ixi)q for all i, the

f ¨xmi

xmiP Γ(Ui, I) glue to an element s in Γ(X, I(q)) = Iq. Since f and s restrict to

the same element in each Γ(Ui, I), we see that f P Iq. Since I is homogeneous,we are done.

(ii) If I, J define the same subscheme, they have the same ideal sheaf I andso I = Γ˚(X, I) = J. o

Example 12.23 Let k be a field and let R = k[u, v]. Moreover introduce thegraded ring S = R(n) = k[un, un´1v, . . . , vn]. We have a graded surjection

φ : k[x0, . . . , xn]Ñ S

given by xi ÞÑ uivní for i = 0, . . . , n. The ideal I = Ker φ is generated by the2ˆ 2-minors of the matrix (

x0 x1 . . . xn´1

x1 x2 . . . xn

)

Thus we have an embedding of P1k = Proj S into Pn with image V(I). The

image is called a rational normal curve of degree n. Note that for n = 2, the imageof P1

k Ñ P2k is the conic given by x2

1 = x0x2. K

Exercise 12.3 Check that the saturation I is homogeneous if I is. M

the segre embedding 242

12.6 The Segre embedding

Recall that for affine schemes X = Spec B, Y = Spec C over S = Spec A, the fiberproduct XˆS Y was defined as Spec(BbA C). There is a similar statement forProj:

Theorem 12.24 Let R, R1 be graded rings with R0 = R10 = A. Let S =À

ně0(Rn b R1n). Then

Proj S » Proj RÂ Proj R1

Exercise 12.4 Prove Theorem 12.16. Hint: Prove that S fbg » (R f )0 bA (R1g)0ˇ

for f P R and g P R1. Then compare both sides by gluing together fibre productsover distinguished open sets. M

Corollary 12.25 Let A be a ring and let m, n ě 1 be integers. Then there is a closedimmersion

σm,n : PmA Â Pn

A Ñ Pmn+m+n

Proof: Consider the A-algebra S =À

ně0(Rn b R1n) above, where R =

A[x0, . . . , xm] and R1 = A[y0, . . . , yn] are the polynomial rings. Consider thefollowing morphism of graded A-algebras.

A[zij]0ďiďm,0ďjďn Ñ A[x0, . . . , xm]b A[y0, . . . , yn]

zij ÞÑ xi b yj.

It is clear that S is generated as an R0 b R10-algebra by the products xi b yj, sothe map is surjective and thus we get the desired closed immersion. o

Example 12.26 Let R = k[x0, x1], R1 = k[y0, y1]. Then uij = xiyi defines anisomorphism

S =à

ně0(Rn b R1n)Ñ k[u00, u01, u10, u11]/(u00u11 ´ u01u10)

This recovers the usual embedding of P1k ˆk P1

k as a quadric surface in P3k .

A smooth quadric surface

K

locally free sheaves on1

243

12.7 Locally free sheaves on P1

In 1955, Grothendieck wrote his paper "Sur la classification des fibres holomor-phes sur la sphere de Riemann", showing that any locally free sheaf on theprojective over a field splits as a sum of invertible sheaves:

Theorem 12.27 Let X = P1k and let E be a locally free sheaf of rank r. Then

there are integers a1, . . . , ar such that

E » OP1k(a1)‘ ¨ ¨ ¨ ‘OP1

k(ar) (12.4)

Grothendieck’s proof was sheaf-theoretic, but in fact this is a rather elemen-tary result which has been rediscovered and reproved several times. For instance,Grothendieck was not aware of the following result, due to Dedkind–Weberfrom 1882. Dedekind, Weber.

Theorie deralgebraischenFunktionen einerVeränderlichen’,Crelle’s Journal, 1882

Lemma 12.28 (Dedkind–Weber) Let k be a field and let A P GLr(k[x, x´1]). Thenthere exist matrices B P GLr(k[x]) and C P GLr(k[x´1]) such that

BAC =

xa1 0. . .

0 xar

(12.5)

This lemma is completely elementary, and can be proved by induction on r withonly basic row-operations on matrices (see for instance Exercise ?? in [?])

In any case, Theorem 12.18 follows immediately from the description ofquasi-coherent sheaves on P1

k from Example 14. In the notation of that example,we have M0 = k[x]r, M1 = k[x´1]r and τ : k[x˘1]r Ñ k[x˘1]r. The lemma aboveimplies that after changing bases, the map τ is given by a diagonal matrix 12.5.Hence E splits as (12.4).Exercise 12.5 Prove Lemma 12.19 for r = 2. M

12.8 Two important exact sequences

HypersurfacesLet R = k[x0, . . . , xn] and Pn

k = Proj R. Let F P R denote an homogeneouspolynomial of degree d ą 0. F determines a projective hypersurface X = V(F),which has dimension n´ 1. We then have an isomorphism

R(´d)Ñ I(X)

given by multiplication with F. Note the shift here: The constant ‘1’ gets sent toF should have degree d on both sides! This gives the sequence of R-modules

0 Ñ R(´d)Ñ R Ñ R/(F)Ñ 0

two examples of locally free sheaves 244

We have ČR(´d) = OPnk(´d) and Č(R/F) = i˚OX, where i : X Ñ Pn

k is theinclusion, so we get the exact sequence of sheaves

0 Ñ OPnk(´d)Ñ OPn

kÑ i˚OX Ñ 0

Complete intersectionsLet F, G be two homogeneous polynomials without common factors of degreesd, e respectively. Let I = (F, G) and X = V(I) Ă Pn

k . X is called a ‘completeintersection’ – it is the intersection of the two hypersurfaces V(F) and V(G). Tostudy X we have exact sequences

0 Ñ R(´d´ e) αÝÑ R(´d)‘ R(é)

βÝÑ I Ñ 0

The maps here are defined by α(h) = (´hG, hF) and β(h1, h2) = h1F + h2G.These maps preserve the grading.

To prove exactness, we start by noting that α is injective (since R is an integraldomain) and β is surjective (by the defintion of I). Then if (h1, h2) P Ker β, wehave h1F = ´h2G, which by the coprimality of F, G means that there is anelement h so that h1 = ´hG, h2 = hF.

Applying „, we obtain the following exact sequence

0 Ñ OPnk(´d´ e)Ñ OPn

k(´d)‘OPn

k(é)Ñ IX Ñ 0

These sequences are fundamental in computing the geometric invariantsfrom X. We will see several examples of this later.

12.9 Two examples of locally free sheaves

Projective spaceLet k be a field and write Pn = Proj R where R = k[x0, . . . , xn]. Consider themap of graded modules φ : R(´1) Ñ Rn+1 sending 1 P R to the element(x0, . . . , xn) P Rn+1. This map is clearly injective, so we get an exact sequence

0 Ñ R(´1)φÝÑ Rn+1 Ñ M Ñ 0

where M = Coker φ. Applying „, we get an exact sequence of sheaves

0 Ñ OPnk(´1)Ñ On+1

PnkÑ E Ñ 0 (12.6)


where E = ĂM. We claim that E is locally free of rank n. Indeed, on thedistinguished open set D+(x0) = Spec(Rx0)0, we have

E (D+(x0)) =

(nà

i=0Rx0 /(x0e0 + ¨ ¨ ¨+ xnen)

)0

=

(nà

i=0k[

x1

x0, . . . ,

xn

x0]ei

)/(e0 +

x1

x0e0 ¨ ¨ ¨+

xn

x0en)

»

nà

i=1k[

x1

x0, . . . ,

xn

x0]ei

Hence E |D+(x0) » OnU0

. By a symmetric argument, E is free also on the otherD+(xi), so it is locally free of rank n. We will show in Section 14.6 that E is notfree, and in fact not even isomorphic to a direct sum of invertible sheaves.

The four-dimensional quadric hypersurfaceLet k be a field and let R = k[p01, p02, p03, p12, p13, p23]. Consider the matrix

M =

p12 p13 p23 0´p02 ´p03 0 p23

p01 0 ´p03 ´p13

0 p01 p02 p12

Let us consider the closed subschemes in P5 = Proj R defined by the conditionsthat this matrix has a given rank. Note that M has rank ď 3 precisely when thedeterminant vanishes. In fact, this matrix M has the special property that thedeterminant is a square: det M = q2 where

q = p01 p23 ´ p02 p13 + p03 p12

We define the Grassmannian variety(Grassmann-varietet)

Grassmannian Gr(2, 4) as the hypersurface defined by q in P5.Note that Gr(2, 4) is a projective variety of dimension 4.

The locus of points where M has rank 2 is given by the ideal generated bythe 2ˆ 2-minors, which by direct calculation has radical equal to the irrelevantideal R+. Consider the exact sequence

0 Ñ R(´1)4 MÝÑ R4 Ñ Coker M Ñ 0

Applying „ we obtain an exact sequence of sheaves

0 Ñ OP5(´1)4 Ñ O4P5 Ñ F Ñ 0 (12.7)

where F = ČCoker M.Consider the quadric hypersurface X = V(q) and let i : X Ñ P5 denote the

inclusion. Applying, i˚ we arrive at an exact sequence of sheaves on X


OX(´1)4 Ñ O4X Ñ E Ñ 0

where E = i˚F (recall that i˚ is only right-exact). Now the discussion aboveshows that E is locally free of rank 2 (as it has rank 2 at all closed points). Thesheaf E is known as the universal quotient bundle on the Grassmannian Gr(2, 4).

The Hilbert syzygy theoremLet k be a field and let R = k[x0, . . . , xn]. Then if M is a finitely generated gradedR-module, then Hilbert Syzygy theorem says that there is a finite free resolution(that is, an exact sequence)

0 Ñ Fn Ñ ¨ ¨ ¨ Ñ F1 Ñ F0 Ñ M Ñ 0

where Fbk =Àbk

i=1 R(´di) is a free graded R-module. Fi is called the i-th syzygymodule of the resolution. The minimal integer n that appears in such a resolutionis called the projective dimension of M.

David Hilbert(1862 – 1943)

If we apply the „-functor here we obtain an exact sequence of sheaves on Pnk

0 Ñ En Ñ ¨ ¨ ¨ Ñ E1 Ñ E0 Ñ ĂM Ñ 0

where Ei =Àbk

i=1 OPnk(´di) is a direct sum of sheaves of the form O(d).

Thus any coherent sheaf can be resolved by locally free sheaves – in factdirect sums of invertible sheaves. This shows why the invertible sheaves O(d)are so important: They are the building blocks of all coherent sheaves on Pn.We already saw some examples such a presentation was convenient. Let us giveone more:Example 12.29 The twisted cubic curve. Let k be a field and consider P3 = Proj Rwhere R = k[x0, x1, x2, x3]. We will consider the twisted cubic curve C = V(I)where I Ă R is the ideal generated by the 2ˆ 2-minors of the matrix

M =

(x0 x1 x2

x1 x2 x3

)

i.e., I = (q0, q1, q2) = (x21 ´ x0x2, x0x3 ´ x1x2,´x2

2 + x1x3).Consider the map of R-modules R3 Ñ I sending ei ÞÑ qi. This is clearly

surjective, since the qi generate I. Let us consider the kernel of this map, thatis, the module of relations of the form a0q0 + a1q1 + a2q2 = 0 for ai P R. Thereare two obvious relations of this form, i.e., the ones we get from expanding thedeterminants of the two matricesx0 x1 x2

x0 x1 x2

x1 x2 x3

x0 x1 x2

x1 x2 x3

x1 x2 x3

appendix: graded modules 247

(So first matrix gives x0q2 ´ x1q1 + x2q2 = 0 for instance). These give a mapR2 M¨ÝÑ R3, where M is the matrix above. This map is injective, and it turns out

that there is an exact sequence of R-modules

0 Ñ R2 MÝÑ R3 Ñ I Ñ 0

Again, if we want to be completely precise, we should consider these as gradedmodules, so we must shift the degrees according to the degrees of the mapsabove

0 Ñ R(´3)2 MÝÑ R(´2)3 Ñ I Ñ 0

This gives the resolution of the ideal I of C. Then applying „, and using the factthat I = rI, we get a resolution of the ideal sheaf of C:

0 Ñ OP3k(´3)2 M

ÝÑ OP3k(´2)3 Ñ I Ñ 0

We will see later in Chapter 14 how to use sequences like this to extract geometricinformation about C. K

12.10 Appendix: Graded modules

In this appendix R will be a graded ring and M and N two graded R-modules.The aim is to explain the not totally simple facts that the tensor product and themodule of homomorphisms HomR(M, N) both are graded modules. The ring Rdecomposes in its graded pieces as the direct sum R =

À

i Ri of R0-modules; asdoes both M and N; that is M =

À

i Mi and N =À

i Ni, both sums being in thecategory ModR0 .

The tensor productWe aim at showing that MbR N is graded module in way that decomposabletensors xb y with x P M and y P N homogeneous are homogeneous andthat deg xb y = deg x + deg y. One may consider the sub R-module thesedecomposable tensors generate, with deg x + deg y fixed, say equal to n, andcall it (MbR N)n. Clearly the sum of these submodules equals the whole tensorproduct MbR N, and as a(xb y) = axb y the requirement that products ofhomomogenous ring element with homogeneous elements and with degreesadding up, is cleary fulfilled. The only condition left to be verified is that thesum is direct; i.e. that (MbR N)n X (MbR N)m = 0.

To that end, note that there is a surjection

MbR0 N Ñ MbR N

simply sending xb y to xb y, whose kernel is generated by elements of theform axb y´ xb ay with x and y running through respectively M and N and a

appendix: graded modules 248

through R. The tensor product over R0 is th easier to handle, it decomposes as adiect sum

MbR0 N =à

i,jMibR0 Nj =

à

n(à

i+j=nMibR0 Nj);

and thus is a graded R0-module. It inherits an R-module structure from themodule structure of M, i.e. the action of R being a ¨ (xb y) = axb y, andthis structure is compatible with the grading given by the decomposition in12.10. Thus to finish the proof, we merely need to check that the kernel is ahomogeneous submodule. So write a =

ř

ai, x =ř

xi and y =ř

yi; then

axb y´ xb ay =ÿ

i,j,k

aixjb yk ´ xjb aiyk

and as each term aixjb yk ´ xjb aiyk is homogeneous and belongs to the kernel,the homogeneous parts of axb y´ xb ay belongs there too, being a sum ofterms aixjb yk ´ xjb aiyk.

Chapter 13

First steps in sheaf cohomology

One of the main challenges when working with sheaves is that surjective mapsof sheaves do not always induce surjections on global sections. We have seenseveral examples of this failure of the global sections functor Γ to be exact whenapplied to a short exact sequence of sheaves

0 F 1 F F2 0.

One has a sequence

0 Γ(X,F 1) Γ(X,F ) Γ(X,F2) (13.1)

which is exact at each stage except on the right. In many situations in algebraicgeometry, knowing that Γ(X,F ) Ñ Γ(X,F2) is surjective is of fundamentalimportance. For instance, if Z Ă X is a subscheme, we would sometimes liketo lift sections of F |Z to sections of F on X (this is often useful in inductionproofs).

The cohomology groups are defined as a partial response to this behavior ofΓ; and in some good situations, these groups allow us to say something aboutthe missing cokernel. This is done by continuing the sequence sequence (13.1),giving rise to a long exact sequence of cohomology:

0 Γ(X,F 1) Γ(X,F ) Γ(X,F2)

H1(X,F 1) H1(X,F ) H1(X,F2)

H2(X,F 1) H2(X,F ) H2(X,F2) ÝÑ ¨ ¨ ¨

In other words, the failure of surjectivity of the above is controlled by the groupH1(X,F 1) and the other groups in the sequence.

Cohomology groups can be defined in a completely general setting, forany topological space X and a (pre)sheaf F on it. With this as input we will

some homological algebra 250

define the cohomology groups Hk(X,F ), which will capture the main geometricinvariants of F . These should also be functorial in F , meaning that we want toconstruct additive functors

Hq(X,´) : AbShX Ñ Ab

F ÞÑ Hq(X,F )

satisfying the following properties:

i) H0(X,F ) = Γ(X,F ) = F (X);

ii) A morphism of sheaves φ : F Ñ G induces for all i group homomor-phisms Hi(X,F ) Ñ Hi(X,G) which are functorial; in other words,each Hi(X,´) is a functor;

iii) For each short exact sequence 0 Ñ F 1 Ñ F Ñ F2 Ñ 0, there are mapsδ : Hi(X,F2)Ñ Hi+1(X, F1) giving a long exact sequence as above.

The maps δ are called connecting homomorphisms. They depend on both i andthe given short exact sequence.

There are several ways to define these groups. The modern approach, andthe one summarized in Hartshorne Chapter III ([?]), takes the approach ofusing derived functors. This is in most respects the ‘right way’ to define thegroups in general, but going through the whole machinery of derived functorsand homological algebra would take us too far astray. We therefore beginwith taking a more down-to-earth approach using Cech cohomology and later onwill reintroduce the cohomology via the Godement resolution. The Godementresolution has the advantage that it is completely canonical, and we can provethe main theorems we need by hand. On the other hand, the Cech resolution,which depends on the choice of a covering of X, is more intuitive and bettersuited for computations. Of course, the two notions of cohomology turn out tobe the same, as we shall prove in Appendix A.

Cohomology of coherent sheaves was first used in complex geometry, andwas subsequently introduced to algebraic geometry by Jean Pierre Serre inthe illustrious paper FAC, where he used Cech cohomology. There are goodreasons for the double introduction, the definition by Godement resulotions iscanonical and directly enjoy all functorial properties, and in fact, is the easiestway to Theorem xxxx below, needed to effectively use Cech cohomology incomputations.

13.1 Some homological algebra

In this section we recall the most rudimentary elements from homologicalalgebra. For a slightly more extensive presentation see Section ?? in CA (or anyof the uncountably many texts on homological algebra).

some homological algebra 251

Complexes of abelian groupsRecall that a complex of abelian groups A‚ is a sequence of groups Ai togetherwith maps between them

¨ ¨ ¨ Ai´1 Ai Ai+1 ¨ ¨ ¨di´2 di´1 di di+1

such that di+1 ˝ di = 0 for each i. A morphism of complexes A‚fÝÑ B‚ is a collection

of maps fp : Ap Ñ Bp making the following diagram commutative:

¨ ¨ ¨ Ai´1 Ai Ai+1 ¨ ¨ ¨

¨ ¨ ¨ Bi´1 Bi Bi+1 ¨ ¨ ¨

di´1

fi´1

di

fi fi+1

In this way, we can talk about kernels, images, cokernels, exact sequences ofcomplexes, etc.

We say that an element σ P Ap is a cocycle if it lies in the kernel of the mapdp i.e., dpσ = 0. A coboundary is an element in the image of dp´1, i.e., σ = dp´1τ

for some τ P Ap´1. These form subgroups of Ap, denoted by Zp A‚ and Bp A‚

respectively. Since dp(dp´1a) = 0 for all a, all coboundaries are cocycles, so thatZp A‚ Ě Bp A‚. The cohomology groups of the complex A‚ are set up to measurethe difference between these two notions. We define the p-th cohomology group asthe quotient group

Hp A‚ = Zp(A‚)/Bp(A‚) = Ker dp/Im dp´1.

One thinks of Hp A‚ as a group that measures the failure of the complex A‚ ofbeing exact at stage p: A‚ is exact if and only if Hp A‚ = 0 for every p.

The following fact is very important:

Proposition 13.1 Suppose that 0 Ñ F‚fÝÑ G‚

gÝÑ H‚ Ñ 0 is an exact

sequence of complexes. Then there is a long exact sequence of cohomologygroups

¨ ¨ ¨ HpF‚ HpG‚ HpH‚

Hp+1F‚ Hp+1G‚ Hp+1H‚ ÝÑ ¨ ¨ ¨

Proof: For each p P Z, consider the commutative diagram

0 Fp Gp Hp 0

0 Fp+1 Gp+1 Hp+1 0

fp

dp

gp

dp dp

fp+1 gp+1

cech cohomology of a covering 252

where the rows are exact by assumption. By the Snake lemma, we obtain asequence

0 Zp(F‚) Zp(G‚) Zp(H‚)

Fp+1/Bp(F‚) Gp+1/Bp(G‚) Hp+1/Bp(H‚) 0

fp gp

δ

fp+1 gp+1

Consider now the diagram

0 Fp/Bp(F‚) Gp/Bp(G‚) Hp/Bp(H‚) 0

0 Zp+1(F‚) Zp+1(G‚) Zp+1(H‚)

fp

dp

gp

dp dp

fp+1 gp+1

where the rows are exact by the above. For the maps in this diagram, HpF‚ =Ker dp and Hp+1F‚ = Coker dp. Hence applying the Snake lemma one moretime, we get the desired exact sequence. o

Complexes of sheavesThe definitions and arguments of the previous subsection apply much moregenerally (to any abelian category). In particular, we make the following sheafanalogue. A complex of sheaves F‚ is a sequence of sheaves with maps betweenthem

¸ ¨ ¨ ¨ Fi´1 Fi Fi+1 ¨ ¨ ¨di´2 di´1 di di+1

such that di+1 ˝ di = 0 for each i. Given such a complex, we define the cohomologysheaves HpF‚ as Ker di/Im di´1. As above, a short exact sequence of complexesof sheaves gives rise to a long exact sequence of cohomology sheaves.

13.2 Cech cohomology of a covering

Let X be a topological space, and let F be a presheaf on it. Let U = tUiu be anopen cover of X, indexed by a linearly ordered set I. We shall assume that I isfinite (and hence we may identify I with a finite segment in N0). As we sawpreviously, if F is a sheaf, the sequence (1.1)

0 Ñ F (X)Ñź

i

F (Ui)Ñź

i,j

F (Ui XUj).

is exact (whether I is finite or not). The Cech complex is essentially the con-tinuation of this sequence; it is a complex obtained by adjoining all the groupsF (Ui1 X ¨ ¨ ¨ XUir) over all intersections Ui1 X ¨ ¨ ¨ XUir where the indices will be


strictly increasing sequences i0 ă ¨ ¨ ¨ ă ip of elements from I (in contrast to theconvention in (1.1) where i, j is just a pair).

Definition 13.2 For a presheaf F on X we define the Cech complex (Cech-komplekset)

Cech complex C‚(U ,F )

of F (with respect to the open covering U ) as

C0(U ,F ) C1(U ,F ) C2(U ,F ) . . .d0 d1 d2

whereCp(U ,F ) =

ź

i0ăi1ă¨¨¨ăip

F (Ui0 X ¨ ¨ ¨ XUip),

and the coboundary maps dp : Cp(U ,F )Ñ Cp+1(U ,F ) by

(dpσ)i0,...,ip+1 =

p+1ÿ

j=0

(´1)jσi0,...ij,...,ip+1|Ui0X¨¨¨XUip

where i0, . . . ij, . . . , ip+1 means i0, . . . , ip+1 with the index ij omitted.

Eduard Cech(1893–1960)

For small p we have

C0(U ,F ) =ź

i0

F (Ui0) and C1(U ,F ) =ź

i0ăi1

F (Ui0 XUi1).

Also note that since the covering is assumed to be finite, say having r elements,Cp(U ,F ) = 0 if p ě r, simply because empty products are zero. So the Cechcomplex is a finite complex.Example 13.3 Let us look at the first few maps in the complex:

i) The coboundary map in degree zero d0 : C0(U ,F )Ñ C1(U ,F ) is givenas follows: if σ = (σi), then

(d0σ)ij = σj ´ σi

where i ă j;ii) The one in degree one d1 : C1(U ,F )Ñ C2(U ,F ) satisfies: if σ = (σij),

then(d1σ)ijk = σjk ´ σik + σij

where i ă j ă k.

K

By direct substitution we see that d1 ˝ d0 = 0 (all the σij cancel). This happensalso in higher degrees as a basic computation using the definition of di shows;that is one has:

Lemma 13.4 dp+1 ˝ dp = 0.


In particular, the C‚(U ,F ) forms a complex of abelian groups. As before, wesay that an element σ P Cp(U ,F ) is a cocycle if dpσ = 0, and a coboundary ifσ = dp´1τ, and denote the sets of these by Zp(U ,F ) and Bp(U ,F ) respectively.The Cech cohomology groups of F with respect to U are set up to measure thedifference between these two notions:

Definition 13.5 The p-th Cech cohomology(Cech -kohomologi)

Cech cohomology of F with respect to U is definedas

Hp(U ,F ) = Zp(U ,F )/Bp(U ,F ) = (Ker dp)/(Im dp´1).

It is not hard to check that a sheaf homomorphism F Ñ G induces amapping of Cech cohomology groups, so we obtain functors F Ñ Hp(U ,F )

from abelian sheaves to abelian groups. In fact, it is clear that it induces mapsCi(U ,F ) Ñ Ci(U ,G) (it does so factor-wise), and an easy computation showsthat the induced maps commutes with the coboundary maps; hence pass to thecohomology.

Examples(13.6) Again it is instructive to consider the group H0(U ,F ). It is governed bythe map d0 : C0(U ,F )Ñ C1(U ,F ), which is simply the usual map

ź

i

F(Ui)Ñź

iăj

F (Ui XUj),

whose kernel is F (X) by the sheaf axioms. It follows that H0(U ,F ) = F (X).(13.7) The most interesting cohomology group is arguably H1(U , F). It is thegroup of cochains σij such that σik = σij + σjk modulo the cochains of the formσij = τj ´ τi.(13.8) The unit circle. Consider the unit circle X = S1 (with the Euclideantopology), and equip it with a standard covering U = tU, Vu consisting oftwo intervals (intersecting in two intervals S and T) and let F = ZX be theconstant sheaf.

Here we have

C0(U ,F ) = ZX(U)ˆZX(V) » ZˆZ C1(U , Z) = ZX(U XV) » ZˆZ.

The map d0 : C0(U , ZX) Ñ C1(U , ZX) is the map Z2 Ñ Z2 given by d0(a, b) =(b´ a, b´ a). Hence

H0(U , ZX) = Ker d0 = Z(1, 1) » Z,


andH1(U , ZX) = Coker d0 = (ZˆZ)/Z(1, 1) » Z.

Students familiar with algebraic topology, may recognize that this gives thesame answer as singular cohomology.(13.9) The projective line. Consider the projective line P1 = P1

k over a field k.It is covered by the two standard affines U0 = Spec k[t] and U1 = Spec k[t´1]

with intersection U0 XU1 = Spec k[t, t´1]. For the structure sheaf OP1 , the Cech-complex takes the form

0 OP1(U0)ÔP1(U1) OP1(U0 XU1) 0

k[t]ˆ k[t´1] k[t, t´1],

d0

»

d

»

where d sends a pair (p(t), q(t´1)) to q(t´1)´ p(t). As we saw in Chapter 5 wehave Ker d = k, and, on the other hand, it is clear that any element of k[t, t´1]

can be written as a sum of a polynomial in t and one in t´1. Hence d is surjective,and we have

H1(U ,OP1) = Coker d = 0.

(13.10) The sheaves OP1(m). Continuing the above example, let us compute theCech -cohomology for F = OP1(m). We use the same affine covering, so theCech -complex still appears as

0 k[t]ˆ k[t´1] k[t, t´1] 0,d

but the coboundary map d0 is different; there is a multiplication by tn in one ofthe restrictions, so the coboundary map is now give by

d(p(t), q(t´1)) = tnq(t´1)´ p(t).

(see Section ??). As we computed in Proposition 5.2, it holds true that Ker d »kn+1 if n ě 0, and Ker d = 0 otherwise. The computation of H1(U ,OP1) isslightly more subtle.

Consider first the case when n ě 0. As before, it is easy to see that anypolynomial in k[t, t´1] can be written in the form tnq(t´1)´ p(t). In fact, thisalso works for n = ´1; indeed, one has t´k = t´1 ¨ t´k+1 ´ 0 and tk = t´1 ¨

0´ tk. Hence H1(U ,OP1(m)) = 0 for n ě ´1. For n ď ´2 however, no linearcombination of the monomials

t´1, t´2, . . . , tn+1

lies in the image, but combinations of all the others do, and it ensues thatH1(U ,OP1(m)) is a k-vector space of dimension ń + 1.

K


Exercises(13.1) Let X = S1 and let U be the covering of X with three pairwise intersectingopen intervals with empty intersection. Show that the Cech -complex is of theform

Z3 d0ÝÑ Z3 Ñ 0.

Compute the map d0 and use it to verify again that Hi(U , ZX) = Z for i = 0, 1as above.(13.2) Let X = S2 and let U be the covering of X with four pairwise intersectingopen sets with empty quadruple intersection; Show that the Cech -complextakes the form

Z4 d0ÝÑ Z6 d1

ÝÑ Z4 Ñ 0

Compute the matrices d0, d1 and show that Hi(U , ZX) = Z for i = 0, 2 andHi(U , ZX) = 0 for i ‰ 0, 2.(13.3) Prove Lemma 13.3.

M

Example: Constant sheavesIn the examples above, we considered S1 and S2 with the standard topologyand the constant sheaves ZX on them, and, in fact, the cohomology of constantsheaves will give singular cohomology for most topological spaces. However,the following proposition shows that constant sheaves are not so interesting inalgebraic geometry, as we would like to study spaces which are irreducible astopological spaces. Then any open set U Ă X is connected and the constantsheaves are effectively constant (in general a constant sheaf AX takes the valueA merely on connected sets).

Proposition 13.11 Let X be an irreducible topological space. Then for anyfinite covering U of X we have for a constant sheaf AX

Hp(U , AX) = 0

for p ą 0.

Proof: In this case the Cech complex takes the formź

i

A Ñź

iăj

A Ñź

iăjăk

A Ñ ¨ ¨ ¨

Note that this complex of groups does not depend on X or the covering U , onlythe index set I plays a role. In particular, the complex is the same as the Cechcomplex of A on a one-point space (which makes it plausible that all the highercohomology should vanish). In this case it is easy to show by hand that anyp-cocycle is the coboundary of some (p´ 1)-cochain for p ą 0.

cech cohomology of a sheaf 257

For instance, given a 1-cocycle g = (gij) Pś

iăj A, let n P I be the largestelement and define the element h = (hi) P C0(U , A) =

ś

iPI A by the assignment

hi =

#

gin when i ă n,

0 when i = n.

The cocycle condition

0 = (d1g)ijn = gij ´ gjn + gin,

where i ă j ă n, translates into 0 = gij ´ hj + hi or in other words gij = hj ´ hi,and by definition it holds that gjn = hj = hj ´ hn. This proves that the cocycleg = (gi,j) is the coboundary of the element h = (hi), and thus that the class ofthat cocycle is zero in H1(U , AX).

The same trick works for higher p ą 0 as well. Let again n P I be the largestelement, and suppose that we are given a cocycle g = (gi0,...,ip) P

ś

i0ă¨¨¨ăipA.

Setting ip+1 = n in the differential, we see that

0 = (dpg)i0¨¨ïpn =

pÿ

j=0

(´1)jgi0¨¨ïj¨¨ïpn + (´1)p+1gi0¨¨ïp .

Now, mimicking what we for p = 2, we define h Pś

i0ă¨¨¨ăip´1A by the assign-

ment

hi0,...,ip´1 =

#

(´1)pgi0,...,ip´1n when ip´1 ă n,

0 when ip´1 = n.

Solving the previous equation for gi0¨¨ïp we see that when ip ă n

(dp´1h)i0¨¨ïp =

pÿ

j=0

(´1)jhi0 ...ij ...ip=

= (´1)ppÿ

j=0

(´1)jgi0¨¨ïj¨¨ïp´1n = gi0¨¨ïp ,

and for ip = n one finds

(dp´1h)i0...ip´1n =

p´1ÿ

j=0

(´1)jhi0 ...ij ...ip´1n + (´1)phi0 ...ip´1 = gi0...ip´1n.

It follows that g = dp´1h, so that the class of g is zero in Hp(U , AX). o

13.3 Cech cohomology of a sheaf

As seen in the examples above, the groups Hp(U ,F ) are easily computable ifone is given a nice open cover of X. Indeed, the maps in the Cech complex are


completely explicit, and computing their kernels and images involves only basicrow operations from linear algebra.

On the other hand, the definition of the cohomology groups is unsatisfactoryfor various reasons. First of all, the groups Hp(U ,F ) depend on the opencover U , whereas we want something canonical that only depends on F . Moreimportantly, it is not clear that the definition above really captures the desiredinformation about F . For instance, U could consist of the single open set X, andso Hi(U ,F ) = 0 for all i ě 1! Finally, it is not at all clear if these groups satisfythe functorial requirements mentioned in the introduction.

There is a fix for all of these problems which involves passing to finer andfiner ‘refinements’ of the covering. However, this introduces a new complicationmaking explicit computation almost impossible, so the Cech cohomology ismost useful when this limit process is not needed; that is if one particular opencovering yields the limit.

We say that an open covering V = tVjujPJ is a Refinement of a cover(forfining av enoverdekning)

refinement of U = tUiuiPI if forevery Vj P V , there is a i P I so that VjĎUi. This defines a preordering on theset of coverings which we denote by V ď U , and the order is directed since iftUiu and tVju are two open covers, the cover tUi XVju is finer than both. If wechose a map σ : J Ñ I so that Vj Ă Uσ(j) for every j (there are several such, butwe single out one), we can define a refinement homomorphism

refUV : Cp(U ,F )Ñ Cp(V ,F )

by setting(refUV (σ))j0 ...jp =

(σε(j0)...ε(jp)

)|Vj0X¨¨¨XVjp

.

A straightforward computation yields that d ˝ refUV = refUV ˝d, so that refinduces a map on cohomology groups:

refUV : Hp(U ,F )Ñ Hp(V ,F ).

Moreover, while the refinement maps between the complexes depend on thechoice of the function σ : J Ñ I, the map induced on between the cohomologygroups does not.

One may then define a group Hp(X,F ) to be the direct limit of all Hp(U ,F )

as U runs through all possible open covers U ordered by refinement. Theresulting groups are indeed canonical, and they turn out to give a good answerfor cohomology

Definition 13.12 The groups Hp(X,F ) are called the Cech cohomologygroups of F . In symbols,

Hp(X,F ) = limÝÑU

Hp(U ,F ).


The main properties of Cech cohomology are summarized in the followingtheorem:

Theorem 13.13 Let X be a topological space and let F be a sheaf on X.

i) The Cech cohomology groups are functors Hi(X,´) : AbShX Ñ Ab;ii) H0(X,F ) = F (X);

iii) Short exact sequences of sheaves induce long exact sequences of coho-mology;

iv) (Leray’s theorem). If F is a sheaf and U is a covering such thatHi(Ui1 X ¨ ¨ Üip ,F ) = 0 for all i ą 0 and multi-indices i1 ă ¨ ¨ ¨ ăip, then

Hi(X,F ) = Hi(U ,F ).

We have already proved the first two of these properties. The next two requiremore work, but the arguments are mostly formal, and we postpone the proofuntil Appendix B. We will nevertheless shortly derive the long exact sequencefor sequences involving quasi-coherent sheaves F on separated schemes, whichwill be our main concern.

The last statement (Leray’s theorem) is very important. It says that eventhough Hi(X,F ) is defined as an infinite directed limit over coverings U , itsuffices to compute it at a covering which is ‘sufficiently fine’ in the sense thatthe higher groups Hi(Ui1 X ¨ ¨ ¨ XUip ,F ) = 0 vanish for i ą 0. In practice, thelatter condition is sometimes rather easy to check: it holds for instance if Fis quasi-coherent and all of the intersections are affine schemes (see Corollary14.2).

The long exact sequence for quasi-coherent sheavesLet X be a scheme and consider a short exact sequence of quasi-coherent sheaves

0 F G H 0.

In Proposition 10.24 we proved that whenever the U = Spec A is an open affinein X, the sequence

0 F (U) G(U) H(U) 0. (13.2)

is exact. This means that if an affine cover U = tUiuiPI has the property thateach intersection

Ui0 X ¨ ¨ ¨ XUip

is affine, as taking products do not disturb exactness, there is an exact sequence

0 Cp(U ,F ) Cp(U ,G) Cp(U ,H) 0,


and consequently the sequence of Cech complexes

0 C‚(U ,F ) C‚(U ,G) C‚(U ,H) 0

is also exact. Thus we are in position to apply Lemma 13.1 to obtain a long exactsequence of Cech cohomology groups

¨ ¨ ¨ Hi(U ,F ) Hi(U ,G) Hi(U ,H) ¨ ¨ ¨ .

If X is a separated scheme, such coverings are cofinal in the directed system ofcoverings (i.e. every open covering has a refinement of the kind), so in fact wehave a proof of iii) — that is, of the existence of the long exact sequence (13.1)for quasi-coherent sheaves on separated schemes.

In general, it may certainly happen that the restriction map (13.2) is notsurjective — one can for instance take the open covering of X with just one openset X. This explains why the Cech cohomology groups Hi(U ,F ) do not givelong exact sequences in general. However, by passing to smaller refinementsV ď U , we may arrange that any section lifts, and we can use the above approachto construct the connecting homomorphisms δ.

Chapter 14

Computations with cohomology

In this chapter, we begin with more explicit computations with sheaf coho-mology. We will prove two main theorems. The first result is that all highercohomology groups of quasi-coherent sheaves on affine schemes vanish. Thisin turn has important foundational consequences for sheaf cohomology. Forinstance, together with Leray’s theorem, it implies that cohomology (which isdefined by a direct limit over the set of all open coverings) can be computedas Cech -cohomology with respect to any affine covering (assuming that thescheme is separated).

The second result is a complete computation of the cohomology groups ofthe sheaves O(d) on projective space Pn

A. Using Hilbert’s syzygy theorem, thisin turn will allow us, at least in principle, to compute sheaf cohomology of anycoherent sheaf on Pn

A.Towards the end of the chapter we will study many explicit examples.

14.1 Cohomology of sheaves on affine schemes

The following result is fundamental in the study of sheaf cohomology groups.It is the first example of a ‘vanishing theorem’ for cohomology. Recalling thatcohomology groups were defined to measure the ‘failure’ of certain desirablestatements (e.g. restr iction maps being surjective), we are in general happy ifcohomology groups are zero.

Theorem 14.1 Let X = Spec A and let F be a quasi-coherent sheaf on X. Thenfor the Cech cohomology one has

Hp(X,F ) = 0 for all p ą 0.

Proof: Recall that we defined the groups Hi(X,F ) by taking the direct limit ofHi(U ,F ) over finer and finer coverings U of X. Since the distinguished open

cohomology of sheaves on affine schemes 262

subsets form a basis for the topology on X, it suffices to prove that

Hp(U ,F ) = 0 for all p ą 0

for a covering U = tD(gi)u where the gi are finitely many elements of Agenerating the unit ideal. (We may consider only finitely many gi since X isquasi-compact.)

As F is quasi-coherent, we may write F = ĂM for some A-module M, andM = Γ(X,F ). Note that F (Dgi) = Mgi , that F (Dgi X Dgj) = Mgi gj and so on,so the sheaves appearing in the Cech complex are products of localizations ofthe module M. Explicitly, the fact that the higher Cech -cohomology groupsvanish is equivalent to the statement that the following sequence is exact:

0 Ñ M Ñź

iPI

Mgi Ñź

iăj

Mgi gj Ñź

iăjăk

Mgi gjgk Ñ . . .

Here the boundary maps are given as alternating sums of localization maps. Forexample,

d1 :ź

iăj

Mgi gj Ñź

iăj,ăk

Mgi gjgk

maps the cochain (σij)ij P Mgi gj to the cochain (σjk ´ σik + σij)i,j,k, viewed as anelement of

ś

Mgi gjgk .Notice that the beginning of the exact sequence

0 Ñ M Ñź

i

Mgi Ñź

iăj

Mgi gj

appeared already in Proposition iii) on page 196 when we computed sections ofthe quasi-coherent module ĂM. The proof for the exactness of this sequence issimilar to the general case.

To prove that the cohomology groups vanish, we must to each cocycle σ

(that is, a cochain σ such that dσ = 0) find a cochain τ which makes σ = dτ acoboundary. The proof is a direct calculation; one constructs the element τ byhand.

To see how this can be done, let us for simplicity consider the case p = 1first. Let σ P

ś

i,j Mgi gj be in the kernel of d. We may write

σij =mij

(gigj)r where mij P M

for some r (since the index set I is finite, we may choose r independent of i andj). The relation dσ = 0 gives the relation

mjk

(gjgk)r ´mik

(gigk)r +mij

(gigj)r = 0

in Mgi gjgk . This implies that we have the following relation in Mgjgk

gr+li mjk

(gjgk)r =gl

i grj mik

(gjgk)r ´gl

i grkmij

(gjgk)r (14.1)


for some l ě 0. Now, as the open sets D(gi) = D(gr+li ) cover X, we have a

relation1 =

ÿ

iPI

higr+li

where hi P A. Let us define the cohain τ = (τj) Pś

Mgj by

τj =ÿ

iPI

higl

imij

grj

.

Inś

Mgjgk we may write this as

τj =ÿ

iPI

higl

i grkmij

(gjgk)r .

We want to show that dτ = σ. This is a basic computation using the relation(14.1) above. We find

(dτ)jk = τk ´ τj

=ÿ

iPI

higl

i grj mik

(gjgk)r ´ÿ

iPI

higl

i grkmij

(gjgk)r

=ÿ

iPI

hi

(gl

i grj mik

(gjgk)r ´gl

i grkmij

(gjgk)r

)

=ÿ

iPI

higr+l

i mjk

(gjgk)r =mjk

(gjgk)r

ÿ

iPI

higr+li

=mjk

(gjgk)r = σjk

as desired. Hence H1(U ,F ) = 0.The proof in the general case is quit similar, but there are more indices to

keep track of. For every cohain σ Pś

i0,...,ipMgi0 ¨¨¨gip

we may then write

σi0,...,ip =mi0,...,ip

(gi0 ¨ ¨ ¨ gip)r where mi0,...,ip P M

for some r (again, since the index set I is finite, we may choose r independent ofi and j), and if σ is a cocycle, the relation dσ = 0 gives the following relation inMgi0 ¨¨¨gip

:

gr+li mi0,...,ip

(gi0 ¨ ¨ ¨ gip)r =

pÿ

k=0

(´1)kgl

i grik

mii0,...,ik ...ip

(gi0 ¨ ¨ ¨ gip)r (14.2)

As the sets D(gi) = D(gr+li ) cover X, we have a relation

1 =ÿ

iPI

higr+li


where each hi P A. Now, define the cochain τ Pś

Mgi0 ¨¨¨gip´1by

τi0,...,ip´1 =ÿ

iPI

higli

mii0,...ip

(gi0 ¨ ¨ ¨ gip´1)r .

Localizing toś

Mgi0 ,...,gip, we may write this as

τi0,...,ip =ÿ

iPI

higli g

rik

mii0,...ip

(gi0 ¨ ¨ ¨ gip)r .

We want to show that dτ = σ. As before, we can check this using the relation(14.2) above:

(dτ)i0¨¨ïp =

pÿ

k=0

(´1)kτi0 ...ik ...ip

=ÿ

iPI

higr+li σi0...ip = σi0 ...ip .

This completes the proof. o

Cech cohomology and affine coveringsAs a corollary of the previous theorem, we see that affine coverings of schemessatisfy the conditions of Leray’s theorem (see Theorem 13.7). This implies

Corollary 14.2 Let X be a noetherian scheme and let U = tUiu be an affine coveringsuch that all intersections Ui0 X ¨ ¨ ¨ XUis are affine. Then

Hi(X,F ) = Hi(U ,F ).

In particular, the theorem applies to any open affine covering on a Noetherianseparated scheme.Example 14.3 The affine line with two origins. Consider the ‘affine line with twoorigins’ X from Example 5.4 on page 109. It is covered by two affine subsetsX1 = Spec k[u] and X2 = Spec k[u] and these are glued together along theircommon open set X12 = D(u) = Spec k[u, u´1] with the identity as gluing map.The Cech complex for this covering looks like

0 k[u]ˆ k[u] k[u, u´1] 0d1 d2

where d1(p, q) = q´ p, and is nothing but the standard sequence that appearedin the example, and as we checked in there, it holds that OX(X) = Ker d1 = k[u].

More strikingly, the cokernel H1(X,OX) = Coker d1 of the map k[u]‘ k[u]Ñk[u, u´1] is rather big. It equals k[u, u´1]/k[u] =

À

ią0 kuí, so that H1(X,OX)

is not finitely generated as a vector space over k. This gives another proof thatX is not isomorphic to an affine scheme. K

cohomology and dimension 265

14.2 Cohomology and dimension

The next result is another ‘vanishing theorem’. It a general result, due toGrothendieck, that the cohomology groups vanish above the dimension ofbase space X, at least for spaces X that are Noetherian and the dimension isinterpreted as the Krull-dimension.

Theorem 14.4 Let X be a Noetherian topological space of dimension n, and letF be an abelian sheaf on X. Then

Hp(X,F ) = 0

for all p ą n.

A proof valid in the general case may be found in Godement (Theorem4.5.12), but we contend ourself with proving it in the special case when X is aquasi-projective variety. We begin with an easy lemma:

Lemma 14.5 Let X be a topological space and let Z Ă X be a closed subset. Then forany abelian sheaf F on Z, it holds true that Hp(Z,F ) = Hp(X, i˚F ).

Proof: Observe that each open cover tUiu of X induces an open cover tUi X Zuof Z, and all open covers of Z arise like this. The lemma then follows from thebasic fact that for each open subset U Ă X it holds that Γ(U, i˚F ) = Γ(ZXU,F ),so the two cohomolgy groups arise from the same Cech complexes. o

Theorem 14.6 Let X be a quasi-projective scheme of dimension n. Then Xadmits an open cover U consisting of at most n + 1 affine open subsets. Inparticular, it hods true that

Hp(X,F ) = 0 for p ą n

for any quasi-coherent sheaf F on X.

Proof: Let X be a quasi-projective scheme, i.e. X appears as X = YzW whereY, W Ď Pr

A are closed subschemes, and we may assume that no irreduciblecomponent of Y is contained in W, simply by discarding such components.Using induction on dim X we will prove that X is covered by n + 1 open affinesinduced from open affines in Pr

A.Consider the irreducible decomposition Y =

Ť

i Yi and observe that by primeavoidance IW Ę

Ť

IYi where IT Ď A[x0, . . . , xN ] denotes the radical homogeneousideal of a set T Ď PN . Pick a homogenous polynomial f such that f P IWz(

Ť

i IYi),and let H = V( f ). Then we infer that the set Pr

AzH = D+( f ) is affine and henceso is YzH, being a closed subscheme of an affine scheme.

cohomology of sheaves on projective space 266

By construction YzH Ď YzW = X and H Ğ Yi for any i by the choice of f .Therefore dim(Yi X H) ă dim Yi so we may use induction on the dimension tocover YXH by fewer that n open affines, all induced from the ambient projectivespace, which together with D( f ) gives a covering of X with n + 1 open affinesubsets. This shows the first claim.

For the second, note that in a Cech complex built on a covering consisting ofat most n + 1 affines open subsets, terms Cp(X,F ) with p ą n will vanish, fromwhich follows that 0 = Hp(U ,F ) = Hp(X,F ) for each F and each p ą n. o

14.3 Cohomology of sheaves on projective space

In Examples 5 and ?? we computed the sheaf cohomology of the sheaves O(d)on X = P1

k . For d ě 0, we found that H0(X,O(d)) could be identified with thespace of homogeneous polynomials of degree d, and H1(X,O(d)) = 0. On theother hand, for d ď ´2, H0(X,O(d)) = 0, while H1(X,O(d)) was non-zero.

We will now carry out a more general computation for the cohomologygroups for O(d) for any projective space Pn

A over a ring A. The strategy ishowever the same, we have a distinguished covering via the open sets D+(xi)

and we use Cech complex associated to this covering to compute the cohomology.

Theorem 14.7 Let X = PnA = Proj R where R = A[x0, . . . , xn] where A is a

ring.

i)

H0(X,OX(m)) =

#

Rm for m ě 0

0 otherwise

ii)

Hn(X,OX(m)) =

#

A for m = ń´ 1

0 for m ą ń´ 1

iii) For m ě 0, there is a perfect pairinga of A-modules

H0(X,OX(m))ˆHn(X,OX(´mń´1))Ñ Hn(X,OX(ń´1)) » A.

iv) For 0 ă i ă n and all m P Z, we have

Hi(X,OX(m)) = 0.

aRecall that a bilinear map M ˆ N Ñ A is a perfect pairing if the induced map M ÞÑ

HomA(N, A) is an isomorphism

Proof: The simplifying trick is to consider the OX-module F =À

mPZ O(m).We would like to show for instance that Hi(X,F ) = 0 for i ‰ 0, n, and sincetaking Cech cohomology commutes with forming direct sums, this is equivalent


to Hi(X,OX(m)) = 0 for all m, but F has the advantage that it is a gradedOX-algebra.

Consider the Cech complex associated with the standard covering U = tUiu

where Ui = D+(xi) = Spec R(xi), and the salient observation is that

Γ(Ui,F ) =à

mPZ

Γ(Ui,O(m)) =à

mPZ

((R(m)xi)0) = Rxi

and similar equalities hold for all the intersections among the Ui’s. This impliesthat the Cech -complex for F has the following form:

ź

i

Rxid0ÝÑ

ź

i,j

Rxixjd1ÝÑ . . . dn´1

ÝÝÑ Rx0¨¨¨xn

where the maps as usual are composed of localization maps. We have a gradedisomorphism of R-modules:

H0(X,F ) = Ker d0

= t(ri)iPI |ri P Rxi , ri = rj P Rxixju

» R.

This isomorphism preserves the grading, so we get (i).For (ii): Note that Rx0¨¨¨xn is a free graded A-module spanned by monomials

of the formxa0

0 ¨ ¨ ¨ xann

for multidegrees (a0, . . . , an) P Zn+1. The image of dn´1 is spanned by suchmonomials where at least one ai is non-negative. Hence

Hn(X,F ) = Coker dn´1

= A

xa00 ¨ ¨ ¨ x

ann

ˇ

ˇ ai ă 0@i(

Ă Rx0...xn

Hence

Hn(X,OX(m)) = Hn(X,F )m

= A!

xa00 ¨ ¨ ¨ x

ann

ˇ

ˇ ai ă 0@i,ÿ

ai = m)

Ă Rx0 ...xn

In degree ń´ 1 there is only one such monomial, namely x´10 ¨ ¨ ¨ x´1

n .(iii): If we identify Hn(X,OX(´m´ n´ 1)) with

A!

xa00 ¨ ¨ ¨ x

ann

ˇ

ˇ ai ă 0@i,ÿ

ai = m)

and H0(X,O(m)) = Rm, we can define the pairing via multiplication of Laurentpolynomials:

H0(X,O(m))ˆ Hn(X,OX(´m´ n´ 1))Ñ Rx0¨¨¨xn

(xm00 ¨ ¨ ¨ xmn

n )ˆ (xa00 ¨ ¨ ¨ x

ann ) ÞÑ xa0+m0

0 ¨ ¨ ¨ xan+mnn


Here the exponents satisfy mi ě 0, ai ă 0ř

ai = ´m´ n´ 1,ř

mi = m. Thisgives a map

H0(X,O(m))ˆ Hn(X,OX(´m´ n´ 1))Ñ Hn(X,OX(ń´ 1)) = Ax´10 ¨ ¨ ¨ x´1

n

sending (xm00 ¨ ¨ ¨ xmn

n ) ˆ (xa00 ¨ ¨ ¨ x

ann ) to zero if mi + ai ě 0 for some i. This

pairing is perfect: The dual of a monomial (xm00 ¨ ¨ ¨ xmn

n ) is represented by(x´m0´1

0 ¨ ¨ ¨ x´mn´1n ).

(iv) This point is more involved, and we proceed by induction on n. Forn = 1, there is nothing to prove. For n ą 1, let H = V(xn) » Pn´1 be thehyperplane determined by xn. We have an exact sequence

0 Ñ R(´1) ¨xnÝÑ R Ñ R/(xn)Ñ 0 (14.3)

Applying „, we find

0 Ñ OX(´1)Ñ OX Ñ i˚OH Ñ 0

where i : H Ñ X is the inclusion. If we take the direct sum of all the twists ofthis sequence, we get

0 Ñ F (´1)Ñ F Ñ i˚FH Ñ 0

where FH =À

mPZ OH(m). By induction on n, we have for 0 ă i ă n´ 1 and allm P Z: Hi(X, i˚OH(m)) = Hi(H,OH(m)) = 0. So taking the long exact seqenceof cohomology, we get isomorphisms

Hi(X,F (´1)) ¨xnÝÑ Hi(X,F )

for 1 ă i ă n ´ 1. We claim that we have isomorphisms also for i = 1 andi = n´ 1. For i = 1, this follows because the sequence

0 Ñ H0(X,F (´1))Ñ H0(X,F )Ñ H0(X, i˚FH)Ñ 0

is exact (this is the same sequence as (14.3)).For i = n´ 1 we need to show that

0 Ñ Hn´1(X, i˚FH)δÝÑ Hn(X,F (´1)) ¨xn

ÝÑ Hn(X,F )

is exact. The kernel of ¨xn, is generated by monomials xa00 ¨ ¨ ¨ x

ann with ai ă 0 for

all i. So it suffices to show that the connecting map δ is just multiplication byx´1

n . Define R1 = R/xn. Writing the arrows in the Cech complex, vertically weget the diagram

0 //

ś

i R(´1)x0¨¨¨xi¨¨¨xn

¨xn //

ś

i Rx0¨¨¨xi¨¨¨xn//

R1x0¨¨¨xn´1//

0

0 // Rx0¨¨¨xn(´1)¨xn // Rx0¨¨¨xn

// 0

extended example: plane curves 269

If xa00 ¨ ¨ ¨ x

an´1n´1 is a monomial in Hn´1(H,FH) with ai ă 0 for all 1 ď i ď n´ 1,

then it comes from an (n + 1)-tuple inś

i Rx0¨¨¨xi¨¨¨xn which maps to ˘xa00 ¨ ¨ ¨ x

an´1n´1

in Rx0¨¨¨xn , which is in turn mapped onto by the monomial xa00 ¨ ¨ ¨ x

an´1n´1 x´1

n inRx0¨¨¨xn(´1). So δ(xa0

0 ¨ ¨ ¨ xan´1n´1 ) is represented by the monomial xa0

0 ¨ ¨ ¨ xan´1n´1 x´1

n inHn(X,F (´1)).

Now we claim that we have an isomorphism H˚(X,F )xn = H˚(Un,F |Un).Indeed, since Γ(Un,F ) = F (X)xn , the Cech complex of F |Un with respect to thecovering Ui XUn is just the localization of C‚(X,F ) at xn. Localization is exact,so it preserves cohomology, which gives the claim.

We know that Hi(X,F )xn = Hi(Un,F |Ur) = 0 for all i ą 0, since Un is affine.Hence for l " 0, xl

nHi(X,F ) = 0 as an A-module. However, we have shownthat ¨xn gives an isomorphism of Hi(X,F ) for 0 ă i ă n. This implies thatHi(X,F ) = 0. o

Corollary 14.8 Let k be a field. Then for m ě 0

i) dimk H0(Pnk ,O(m)) = (m+n

n );ii) dimk Hn(Pn

k ,O(´m)) = (m´1n ).

All other cohomology groups are 0.

14.4 Extended example: Plane curves

Let X = V( f ) Ă P2k be a plane curve, defined by an homogeneous polynomial

f (x0, x1, x2) of degree d. Let us compute the groups of the structure sheafHi(X,OX). We have the ideal sheaf sequence

0 Ñ IX Ñ OP2 Ñ i˚OX Ñ 0

where the ideal sheaf IX is the kernel of the restriction OP2 Ñ i˚OX. By Section12.8, OP2(´X) » OP2(´d), and the sequence can be rewritten as

0 OP2(´d) OP2 i˚OX 0.¨ f

From the short exact sequence, we get the long exact sequence as follows:

0 H0(P2,O(´d)) H0(P2,OP2) H0(X,OX)

H1(P2,O(´d)) H1(P2,OP2) H1(X,OX)

H2(P2,O(´d)) H2(P2,OP2) 0.

Using the results on cohomology of line bundles on P2, we deduce the equalityH0(X,OX) » k and hence

H1(X,OX) » k(d´1

2 ).

extended example: the twisted cubic in3

270

The dimension of the cohomology group on the left is the genus of the curve X(it will be introduced properly in Chapter 19). So the above can be rephrased assaying the genus of a plane curve of degree d is 1

2 (d´ 1)(d´ 2).

14.5 Extended example: The twisted cubic in P3

Let k be a field and consider P3 = Proj R where R = k[x0, x1, x2, x3]. We willcontinue Example 10 and consider the twisted cubic curve C = V(I) whereI Ă R is the ideal generated by the 2ˆ 2-minors of the matrix

M =

(x0 x1 x2

x1 x2 x3

)

Let us by hand compute the group H1(X,OX). Of course we know what theanswer should be, since X » P1, and H1(P1,OP1) = 0. Indeed, S = R/I isisomorphic to the third Veronese subring k[s, t](3) = k[s3, s2t, st2, t3]; the Proj ofthis ring is P1

k .Now, to compute H1(X,OX) on X, it is convenient to relate it to a cohomol-

ogy group on P3. We have H1(X,OX) = H1(P3, i˚OX) where i : X Ñ P3 is theinclusion. The sheaf i˚OX fits into the ideal sheaf sequence

0 Ñ I Ñ OP3 Ñ i˚OX Ñ 0.

where I is the ideal sheaf of X in P3. Applying the long exact sequence incohomology, we get

¨ ¨ ¨ H1(P3, I) H1(P3,OP3) H1(P3, i˚OX)

H2(P3, I) H2(P3,OP3) ¨ ¨ ¨

By our description of sheaf cohomology on P3, H1(P3,OP3) = H2(P3,OP3) =

0, which implies that H1(X,OX) = H2(P3, I). We can compute the lattercohomology group using the exact sequence of Example 10:

0 Ñ OP3(´3)2 Ñ OP3(´2)3 Ñ I Ñ 0.

Now, taking the long exact sequence we get

¨ ¨ ¨ H2(P3,O(´3)2) H2(P3,OP3(´2)3) H2(P3, I)

H3(P3,O(´3)2) H3(P3,OP3(´2)3) H3(P3, I)

extended example: non-split locally free sheaves 271

Here H2(P3,O(´2)) = 0 and H3(P3,O(´3)) = 0 by our previsous computa-tions. Hence by exactness, we find H2(P3, I) = 0. It follows that H1(X,OX) = 0also, as expected.Exercise 14.1 Prove Lemma 14.4 in more detail. M

Exercise 14.2 Using the sequences above, show that

o H0(P3, I(2)) = k3 (find a basis!)

o H1(P3, I(m)) = 0 for all m P Z.

o H2(P3, I(´1)) = k.

M

14.6 Extended example: Non-split locally free sheaves

A locally free sheaf is said to be split if it is isomorphic to a direct sum ofinvertible sheaves. We have seen several examples of locally free sheaves thatare not free, even on affine schemes, but a priori it is not so clear whether theseare direct sums of projective modules of rank 1. In this section we will studythe sheaf E from Section 12.9 and show that it is indeed non-split.

The sheaf E is the locally free sheaf of rank n on Pnk sitting in the exact

sequence (12.6)0 Ñ OPn

k(´1)Ñ On+1

PnkÑ E Ñ 0.

Suppose that E is not split, i.e., E is not isomorphic to a direct sum of invertiblesheaves. Since Pic(Pn

k ) = Z is generated by the class of O(1), this would meanthat E » OPn

k(a1)‘ ¨ ¨ ¨ ‘OPn

k(an) for some integers a1, . . . an P Z.

Recall that for n ě 2, we have Hn´1(Pnk ,O(m)) = 0 for any m P Z. So if we

could show that Hn´1(Pnk , E ) ‰ 0, we would have a contradiction. Actually, it is

the case that Hn´1(Pnk , E ) = 0, but we can instead consider F = E (ń), which

fits into the sequence

0 Ñ OPnk(ń´ 1)Ñ OPn

k(ń)n+1 Ñ F Ñ 0.

Taking the long exact sequence in cohomology, we get

¨ ¨ ¨ Ñ Hn´1(On+1Pn

k)Ñ Hn´1(F )

δÝÑ Hn(OPn

k(ń´ 1))Ñ Hn(On+1

Pnk)Ñ ¨ ¨ ¨

Here the two outer cohomology groups are zero, by Theorem 14.6. Hence, byexactness, we find that Hn´1(Pn

k ,F (´1)) » H0(PnkOPn

k) = k. This implies that

F = E (ń), and hence E cannot be a sum of invertible sheaves, and we aredone.

The above gives an example of a non-split locally free sheaf of rank n. How-ever, coming up with examples of non-split sheaves of low rank is a notoriouslydifficult problem, even for projective space. In fact, a famous conjecture of

extended example: hyperelliptic curves 272

Hartshorne says that any rank 2 vector bundle on Pn for n ě 5 is split. (On P4

this statement does not hold, as shown by the so-called Horrocks–Mumford bun-dle). This is related to the long-standing conjecture that any smooth codimension2 subvariety of Pn is a complete intersection.

14.7 Extended example: Hyperelliptic curves

Let us recall the hyperelliptic curves defined in Chapter 3. Let k be a field. Foran integer g ě 1 we consider the scheme X glued together by the affine schemesU = Spec A and V = Spec B, where

A =k[x, y]

(ý2 + a2g+1x2g+1 + ¨ ¨ ¨+ a1x)and B =

k[u, v](´v2 + a2g+1u + ¨ ¨ ¨+ a1u2g+1)

As before, we glue D(x) to D(u) using the identifications u = x´1 and v =

x´g´1y. Figure 14.1:One of theaffine charts ofX

Let us compute the cohomology groups of OX using Cech cohomology.We will use the affine covering U = tU, Vu above. Viewing the first ring as ak[x]-module, we can write

k[x, y](ý2 + a2g+1x2g+1 + ¨ ¨ ¨+ a1x)

= k[x]‘ k[x]y

and similarly B » k[u]‘ k[u]v as a k[u]-module.Since U has only two elements, the Cech complex of OX has only two terms,

OX(U)‘OX(V) and OX(U XV) and the differential between them

d0 : (k[x]‘ k[x]y)‘(

k[x´1]‘ k[x´1]x´g´1y)Ñ k[x˘1]‘ k[x˘1]y

is given by by the assignment

d0(p(x) + q(x)y, r(x´1) + s(x´1)x´g´1y) =

= p(x)´ r(x´1) + (q(x)´ s(x´1)x´g´1)y.

Comparing monomials xmyn on each side, we deduce that

H0(X,OX) = Ker d0 = k

andH1(X,OX) = Coker d0 = ktyx´1, yx´2, . . . , yx´gu » kg.

In particular, dimk H1(X,OX) = g. The latter invariant us usually referred to asthe arithmetic genus of a curve; we have shown that the hyperelliptic curve X hasarithmtic genus g.

For g = 2, we get a particularly interesting curve – an irreducible projectivecurve which cannot be embedded in P2. Indeed, we showed that for any

extended example: hyperelliptic curves 273

irreducible curve in P2 of degree d and the corresponding arithmetic genusequals dim H1(X,OX) =

12 (d´ 1)(d´ 2). However, there is no integer solution

to 12 (d´ 1)(d´ 2) = 2. This implies the following:

Proposition 14.9 There exist non-singular projective curves which cannot beembedded in P2.

Note that we still haven’t proved that X is projective. As we have justshown, there is no closed immersion X Ñ P2 in general for g ě 2. However,it is not hard to see that X can be embedded into the weighted projective spaceP(1, 1, g + 1) = Proj k[x0, x1, w] given by the equation

w2 = a2g+1x2g+10 x1 + ¨ ¨ ¨+ a1x0x2g+1

1 (14.4)

Note that this makes sense if w has degree g + 1, but it does not define asubscheme of P2.

Exercises(14.3) Let X Ă Pn

A be a projective scheme over A and let F be a coherent sheafon X.

a) Show more generally that each cohomology group Hi(X,F ) is finitelygenerated over the ring OX(X).

b) Give an example showing that a) fails without the "projectivity" assumption.

(14.4) Let X Ă P5 denote a quadric hypersurface (i.e., X = V(q) for a homoge-neous degree 2 polynomial). Recall the exact sequence 12.7

0 Ñ OX(´1)4 Ñ O4X Ñ E Ñ 0

where E is a locally free sheaf of rank 2.(i) Use the exact sequences (12.8) to show that

Hi(X,OX(´1)) = 0

for all i ě 0.(ii) Use the exact sequence (12.7) to show that E is not split.

(14.5) Let n ą 0 be an integer and consider the integral projective schemeˇ

X = Proj(R), where R is the ring

R = k[x, y, z, w]/(x2, xy, y2, unx´ vny)

a) Show that X is irreducible, non-reduced, and of dimension 1.

b) Compute H0(X,OX) and H1(X,OX).

M

extended example: bezout’s theorem 274

14.8 Extended example: Bezout’s theorem

Let k be an algebraically closed field. Let C and D be two curves in P2k of degrees

d and e respectively. We assume here that C and D have no common component,so that Z is assumed to be 0-dimensional.

Let us compute the cohomology group H0(Z,OZ). If we assume Z =

tx1, . . . , xru is contained in D(x0) » k[x, y] (which we may arrange by a linearcoordinate change), then

OZ(Z) =rà

i=1

(k[x, y]( f , g)

)mxi

(14.5)

where f , g are the dehomogenized equations for C and D. In other words,dim H0(Z,OZ) is the sum of the multiplicities at the points xi:

dim H0(Z,OZ) =rÿ

i=1

dimk

(k[x, y]( f , g)

)mxi

On the other hand, we can compute H0(Z,OZ) using the ideal sheaf sequence

0 Ñ IZ Ñ OP2 Ñ i˚OZ Ñ 0

we deduce that dim H0(Z,OZ) = dim H1(P2, IZ)´ 1. We proceed to study thelatter cohomology group. Recall the exact sequence from Section 12.8,

0 Ñ OPnk(´d´ e)Ñ OPn

k(´d)‘OPn

k(é)Ñ IZ Ñ 0

Taking the long exact sequence of cohomology we obtain

0 Ñ H1(P2, I)Ñ H2(P2,O(´d´ e)Ñ H2(P2,O(´d))‘ H0(P2,O(é))Ñ 0

From which we get the triumphant conclusion that

dim H0(Z,OZ) = dim H1(P2k , IZ) + 1

= dim H2(O(´d´ e)´ dim H2(O(´d))´ dim H2(O(é)) + 1

=

(d + e´ 1

2

)´

(d´ 1

2

)´

(e´ 1

2

)+ 1

= de

In other words, we have proved Bezout’s theorem.

rÿ

i=1

dimk

(k[x, y]( f , g)

)mxi

= de

14.9 Cech cohomology and the Picard group

The Picard group Pic(X) is an important invariant of a scheme X, but sinceit is defined as an abstract group of invertible sheaves on X, it may not be so

cech cohomology and the picard group 275

obvious how to compute it. In this section, we remedy this, by relating it to acohomology group.

Let L be an invertible sheaf on X and let U = tUiu be a trivializing cover.Thus for each Ui there should be isomorphisms φi : L|Ui Ñ OUi such thatφi = φj agree on the overlaps Ui XUj. Note that φj ˝ φ´1

i is an isomorphismOUiXUj Ñ OUiXUj . Now, we have an isomorphism

HomOX (OUiXUj ,OUiXUj) » OX(Ui XUj) (14.6)

(given by h ÞÑ h(1) and s P Γ(Ui XUj,OX) gives a homomorphism given bymultiplication by s).

Recall the sheaf of units in OˆX ; over an open set U Ă X, these where theelements s having a multiplicative inverse s´1 P Γ(U,OX). Equivalently, OˆX (U)

consists of sections of OX(U) such that for each x P U, the germ sx does not liein the maximal ideal of OX,x.

From the correspondence described by the isomorhpism in (14.6) we inferthat the group of isomorphisms OUiXUj Ñ OUiXUj corresponds exactly to thegroup of units in OX(Ui XUj), i.e.

IsomOX (OUiXUj ,OUiXUj) » O˚X(Ui XUj).

In order to specify the invertible sheaf L, we must say how to glue together thesheaves OUi and OUj along Ui XUj — in other words, we have to specify a unitsij P Γ(Ui XUj,O˚X) for each i, j. These units sij cannot be chosen completely atrandom as we need to make sure they will be compatible on the triple overlapsUi XUj XUk. It turns out to be enough that they satisfy the one constraint thatsijsjkski = 1 on Ui XUj XUk, or in other words that

sjks´1ik sij = 1

in O˚X(Ui X Uj). Since O˚X is a sheaf of abelian groups with multiplicationbeing the groups structure, a restatement of this is that the collection tsiju P

OX(Ui XUj) forms a 1-cocyle! This means that we get a well-defined element inH1(X,O˚X). This is, as one verifies, independent of the choice of cover Ui, andthe map Pic(X)Ñ H1(X,O˚X) so obtained is in fact a group homomorphism.

Conversely, any element s P H1(X,O˚X) is represented by a cover U = tUiu

of X and cocycles sij P C1(U ,O˚X). The cocyle condition implies that the sij

define isomorphisms OUi |Uij Ñ OUi |Uij which glue the OUi together to form aninvertible sheaf. This provides an inverse to the above map. We have thereforeshown:

Theorem 14.10 Let X be a scheme. Then there is a canonical isomorphism

Pic(X) » H1(X,O˚X).

cech cohomology and the picard group 276

This result is helpful for computing Picard groups, because we can computethe H1 using the Cech -complex. One must take a little bit care, because thesheaf O˚X is usually not quasi-coherent. Here are a few examples.Example 14.11 Let X be the affine line with two origins. We will use the usualopen covering of U of X by the two open subsets U, V Ă X isomorphic to A1

k .We have

o H1(U,O˚) = H1(A1k ,O˚) = 0 because Pic(A1

k) = 0

o H1(U XV,O˚) = 0 because U XV » Spec k[x˘1] which has trivial Picardgroup.

The higher cohomology groups Hi(U,O˚), Hi(V,O˚) and Hi(U XV,O˚) alsovanish, because X has dimension 1. Hence Leray’s theorem says that H1(X,O˚X)can be computed via the Cech complex

0 O˚X(U)Ô˚X(V) O˚X(U XV) 0

k˚ ˆ k˚ k[x˘1]˚

Note that O˚X(UXV) consists of elements of the form azn (a P k˚, n P Z). HencePic(X) = Z. K

Example 14.12 Here is an example with a reducible scheme. Let k be a fieldand set

X = Spec(

k[x, y]xy(x + y + 1)

)Consider the components U = V(x), V = V(y) and W = V(x + y + 1). Asabove, we have H1(U,O˚X) = H1(A1,O˚X) = 0, and similarly for the othercomponents. Thus by Leray’s theorem, we may compute Pic(X) by the Cech-complex of the coveirng tU, V, Wu, which takes the form

0 k˚ ˆ k˚ ˆ k˚ k˚ ˆ k˚ ˆ k˚ 0ρ

where ρ(a, b, c) = (ba´1, ca´1, cb´1). From this it follows that Pic(X) » Coker ρ =

k˚. K

Chapter 15

Divisors and linear systems

When studying a scheme it is natural to ask about its closed subschemes. Wehave seen that such subschemes are in a one-to-one correspondence with quasi-coherent ideal sheaves. For (integral) subschemes of codimension one, theseideal sheaves tend to be much simpler than for higher codimension. This isessentially because of Krull’s Hauptidealsatz, which says roughly that ( andunder certain hypotheses), such ideal sheaves are generated locally by oneelement. This is the prototype of a Cartier divisor; a subscheme which is locallycut out by one equation.

The prototype example of a divisor is a hypersurface of projective space; thatis, an integral subscheme of Pn of codimension one. Each such subscheme isdefined by a homogeneous ideal a of k[x0, . . . , xn], and since the codimension isassumed to be one, and because the polynomial ring is factorial, the ideal a isprincipal, i.e., a = ( f ) for some homogeneous polynomial f P k[x0, . . . , xn]. Togive a concrete example, consider the case of P2, and the curve

D0 = V(x3 + y3 + z3) Ă P2k

This is clearly integral (if the characteristic of k is not three), since the definingequation is irreducible. Similarly, we can consider the subscheme

D1 = V(xyz)

This subscheme is reduced, but not irreducible: D1 has three irreducible compo-nents V(x),V(y) and V(z).

The main idea of divisors is that one can talk about sums and differencesof such subschemes, thereby turning them into a group. This is illustrated inthe example above, by writing D1 = V(x) + V(y) + V(z). The sum here iscompletely formal – it is an element in the free group on integral subschemes ofcodimension one. Such a sum is by definition, a Weil divisor.

There is an equivalence relation defined on such objects, designed to capturewhen two divisors belong to the same family. In the example D0 and D1 are in

278

fact connected by a family of subschemes in P2, namely

D(s:t) = V(sxyz + t(x3 + y3 + z3))

More precisely, there is a closed subscheme D Ă P1 ˆP2 defined by the aboveequation, so that the fiber over a closed point (s : t) P P1 is exactly the curveD(s:t) of P2

k . Geometrically, we have a family of ‘moving divisors’, parameterizedover the projective line. We say the two divisors are linearly equivalent. The keyfeature is that there is this morphism f : D Ñ P1, and f´1((0 : 1)) = D0 andf´1((1 : 0)) = D1. Moreover, the quotient

g =x3 + y3 + z3

xyz

defines a rational function on P2. This is the pullback of the rational functions/t on P1: g = f ˚(s/t). We can use this to define an equivalence relation ondivisors by declaring D0 „ D1 if there is a rational function g so that g haszeroes along D0 and poles along D1 (counting with multiplicities).

There is a second approach to divisors, which is perhaps more algebraic innature, namely Cartier divisors. The definition is motivated by the fact thatintegral subschemes of codimension one are typically defined by a single equa-tion locally. Note that in the above example, the special properties of projectivespace imply that D0 and D1 are defined globally by a single equation. It is nothard to come up with examples of schemes with subschemes of codimensionone that are not. In fact, for most schemes, the concept of a ‘globally definedequation’ does not make sense, since we do not have global coordinates to workwith. However, locally this concept makes sense: We can consider subschemesY Ă X so that the ideal sheaf IY is locally generated by a single element fi P Ai,on some affine covering X =

Ť

Spec Ai. In other words, the ideal sheaf IY is aninvertible sheaf.

While Weil divisors are conceptual and geometric, Cartier divisors do havesome advantages. For instance, they are very closely related to invertible sheavesand line bundles, which in turn makes computations with them easier. Forinstance, given a morphism f : X Ñ Y, we would like to define a ‘pullback’ of adivisor on Y to a divisor on X – this turns out to only be possible for Cartierdivisors. There are also other settings where the special properties of Cartierdivisors are essential, for instance defining intersection products.

That being said, for a non-singular scheme we will see that the variousnotions of a divisor are equivalent, and there are natural ways to switch betweenthe three types in the diagram

Weil divisors

Cartier divisors invertible sheaves

weil divisors 279

The main interest in these concepts is of course that they give us tools toclassify varieties and schemes. We will see in Chapter 16 that given a divisorD on X, then under certain conditions, there is a morphism f : X Ñ Pn suchthat D = f´1(H) for some hyperplane H » Pn´1 Ă Pn. We can then use thisto study the geometry of X: We can ask about the fibers of f , whether it is anclosed immersion, and if so, what the equations of the image is, and so on.

We will in this chapter assume that

All schemes are integral and noetherian.

Noetherianness is somewhat important, since we want to talk about the decom-position of a closed subschemes into its irreducible components. The assumptionof integrality, especially irreducibility, is not essential for most of the statements,but it makes the definitions more transparent, and more importantly, the proofsconsiderably simpler (e.g., not having to worry about nilpotent elements allowsus to work with meromorphic functions as elements in a fraction field, rathersome more obscure localization). Still, we remark that the theory of Cartierdivisors work in a completely general setting, although there are some subtlepoints one has to take into account. If you are curious about general backgroundon meromorphic functions on arbitrary schemes, see EGA IV4, section 20.

15.1 Weil divisors

In this section X will denote a separated, integral, normal noetherian scheme.Recall that this means that each local ring OX,x is an integral domain, which isintegrally closed in its function field K = k(X).

Definition 15.1 o A prime divisor on X is a closed integral subschemeY of codimension 1.

o A Weil divisor on X is a finite formal sum

D =ÿ

niYi (15.1)

where ni P Z and Yi are prime divisors.

o We say D is effective if all the ni are non-negative in (15.1).

o We callŤ

i Yi the support of D.

o We denote by Div(X) the group of Weil divisors; this is the free abeliangroup on prime divisors.

If Y is a prime divisor on X and U Ă X is an open set, then YXU is naturallya prime divisor on U. It follows that we obtain a presheaf U ÞÑ Div(U). This is

weil divisors 280

in fact a sheaf; it coincides withà

xPX, codim x=1jx˚(Ztxu)

where jx : txu Ñ X is the inclusion of a point x of codimension 1 in X. We willdenote this sheaf by Div.

The divisor of a rational functionThe main reason for the normality assumption is that if X is normal, thenit is regular in codimension one: This implies that if Y Ă X is a prime divisorwith generic point η P X, the local ring OX,η (which has Krull dimension 1)is a discrete valuation ring, with a corresponding valuation v : Kˆ Ñ Z (seeAppendix A). The concept of a valuation is a generalization of the ‘order’ of azero or a pole of a meromorphic function in complex analysis. Intuitively, anelement f P Kˆ has positive valuation N if it vanishes to order N along Y, andnegative valuation Ń if it has a pole of order N there.

To define this properly, let Y Ă X be a prime divisor, and let η P X be itsgeneric point. Then we define for a non-zero element f P OX,η ,

vY( f ) = n

where n is the unique non-negative integer so that f P mn ´ mn+1. In thefunction field K = k(X), an element f is represented by a fraction g/h andwe define vY( f ) = vY(g)´ vY(h). (Note that this is independent of the chosenrepresentative). With this definition, we have OX,η = v´1

Y (Zě0); OˆX,η = v´1Y (0);

and the maximal ideal is given by m = v´1Y (Zě1).

Definition 15.2 If f P Kˆ, we define its corresponding Weil divisor as

div( f ) =ÿ

Y

vY( f )Y

Divisors of the form div( f ) are called principal divisors, and they generate asubgroup Div0(X) Ď Div(X).

In the definition above, the sum runs over all prime divisors of X. This sumis well defined by the following lemma:

Lemma 15.3 Suppose that X is an integral normal noetherian scheme with fractionfield K and let f P K. Then vY( f ) = 0 for all but finitely many prime divisors Y.

Proof: We first reduce to the case when X is affine. Let U = Spec A be an openaffine subset such that f |U P Γ(U,OX). Since X is noetherian and integral, thecomplement Z = XÚ is a closed subset of X of codimension ě 1 which hasfinitely many irreducible components; in particular, only finitely many prime

weil divisors 281

divisors Y are supported in Z. So we reduce to the affine case X = Spec A andf P Γ(X,OX), by ignoring these finitely many components. Then vY( f ) ě 0automatically, and vY( f ) ą 0 if and only if Y is contained in V( f ); and sinceV( f ) has only finitely many irreducible components of codimension 1, we aredone. o

The proof of the lemma above shows where we make use of some of the finitenessassumptions on our schemes. Unfortunately, there is no getting around it, asthe next example shows.Example 15.4 Imitating the construction of the affine line with two origin, wecan construct the affine line X with infinitely many origins: this scheme is integral,normal, locally noetherian with fraction field k(t), but there are infinitely manyclosed points p P X for which vp(t) = 1. K

Example 15.5 Let k be algebraically closed, X = A1k = Spec k[x] and let K = k(x).

Here prime divisors in X correspond to closed points [a] P A1k associated to

maximal ideals (x´ a) Ă k[x]. Let f = x2(x´1)x+1 P K. Then v[a]( f ) = 0 for all a

except when a = 0,˘1, where we have

v[0]( f ) = 2, v[1]( f ) = 1, v[´1]( f ) = ´1

Hence the divisor of f is 2[0] + [1]´ [´1]. K

Example 15.6 Consider X = P1k = Proj k[x0, x1] and let K = k(t) be the fraction

field where t = x1x0

. Let f = t2

t´1 P K. We consider the two affine charts D+(x0)

and D+(x1) separately:On U = D+(x0) = Spec k[t], the function f defines an element in OU,p for

every p ‰ [1], and it is invertible for every p ‰ [0], [1]. In the point p = [0], thelocal ring equals OX,p = k[t](t), and we may write

f = t2(t´ 1)´1 = t2(unit).

This has valuation 2.At p = [1] P U, we have OX,p = k[u](u) where u = t´ 1. Here we may write

f =t2

t´ 1=

(u + 1)2

u= u´1(unit).

It follows that v[1]( f ) = ´1.

On U = D+(x1) = Spec k[u], where u = x0x1

. We write f = u´2

u´1´1 = 1uú2 .

Only non-zero valuations are: v[0] = ´1 and v[1] = ´1. Note that the point[1] P D+(x1) coincides with the point (1 : 1) which we found also in D+(x0)

above. It follows that the divisor of f is given by

div( f ) = 2(1 : 0)´ (1 : 0)´ (1 : 1).

K

Example 15.7 Let X be the curve V(y2 ´ x3 ´ 1) Ă A2k . Then x, y and x2/y

define rational functions on X. We have div x = (0, 1) + (0,´1) and div y =

weil divisors 282

(´1, 0) + (ω, 0) + (ω2, 0) where ω is a primitive third root of ´1, and

div(x2/y) = 2(0,´1) + 2(0, 1)´ (´1, 0)´ (ω, 0)´ (ω2, 0)

K

The Weil divisor of a section of an invertible sheafLet L be an invertible sheaf and let s be a rational section (a section defined overan open set V Ď X). We can define a Weil divisor div(s) as follows. For Y aprime divisor, let ηY denote its generic point. Let U Ă X be a neighbourhoodof ηY such that there is a trivialization φ : L|U Ñ OX|U . On U, φ(s) defines arational section of OX(U), and hence an element f in the function field K. Wedefine vY(s) = vY( f ) and

div(s) =ÿ

Y

vY(s)Y.

It is not hard to see that this sum is finite, and independent of the choices ofopens U.Example 15.8 Let L = O(2) on X = P1

k = Proj k[x0, x1] and let K be the fraction

field. The quotient s =x3

0x1

defines a rational section of L. Let us compute thedivisor associated to s: Let t = x0

x1be a coordinate on U = D+(x1) = Spec k[t].

OX(2)(U) = k[

x0

x1

]x2

1 = k[t]x21

So the rational function f = φ(s) is given by x30

x31= t3 which has non-zero

valuation only at the point t = 0 P U, where we have vY( f ) = 3. To computediv(s), we must also consider the point outside D+(x1). On U = D+(x0), weuse the coordinate u = x1

x0, and we have

OX(2)(U) = k[u]x20

So the rational function φ(s) is given by f = x0x1

= u´1. This has valuationvY( f ) = ´1 at t = 0 (and vY( f ) = 0 at all other points). Hence we obtain

div(s) = 3(0 : 1)´ (1 : 0)

K

The sheaf associated to a Weil divisorLet D =

ř

nYY be a Weil divisor on X. We would like to form a sheaf, denotedOX(D), which should consist of rational functions with poles ‘at worst alongD’. If f = g/h is such a rational function where g, h are coprime, we havediv( f ) = div(g)´ div(h), so if D is a prime divisor, we want the pole div(h) to

weil divisors 283

be ‘cancelled out’ by D, i.e., D´ div(h) is effective. In other words, we wantdiv( f ) + D to be an effective Weil divisor. Thus, concretely, we define the sheafOX(D) as follows:

OX(D)(U) = t f P K|(div( f ) + D)|U ě 0u Y t0u

= t f P K| vY( f ) ě ńY for all Y with ηY P Uu Y t0u.

Here Y ranges over all prime divisors in X and ηY denotes the generic point ofY.

The sheaf OX(D) is a quasi-coherent sheaf on X. (We will see soon that it isinvertible if and only if D is a Cartier divisor).Example 15.9 Let X be the projective line P1

k = Proj k[x0, x1] over k and letD = V(x1) = (1 : 0). Let U0 = Spec k[x1/x0] = Spec k[t], U1 = Spec k[x0/x1] =

Spec k[s] be the standard covering of P1k (so s = t´1 on U0 XU1). Note that the

point (1 : 0) does not lie in U1 = D+(x1). This means that a rational functionf P K such that div( f ) + D is effective on U1 must be regular on U1, i.e.,

Γ(U1,OX(D)) = k[s]

On U0, we are looking at elements f P k(t) having valuation ě ´1 at t = 0. Thisimplies that

Γ(U0,OX(D)) = k[t]‘ k[t]t´1

Now, we may think of elements in Γ(X,OX(D)) as pairs ( f , g) with f , g P Ksections of OX(D) over U0 and U1 respectively, so that f = g on U0 XU1. Hereg = g(s) is a polynomial in s, and

f (t) = p(t) + q(t)t´1 = p(s´1) + q(s´1)s.

If f = g in k[t, t´1] it is clear that we must have deg p, q ď 1. This implies that

Γ(X,OX(D)) = k‘ k t´1.

In fact, we will see in a bit that OX(D) » OP1(1). K

Exercise 15.1 Let L = OX(D) for a Weil divisor D. Let U = Spec A Ă X bean open affine subset. Show that for f P A, Γ(U,OX(D)) f = Γ(D( f ),OX(D)).Deduce that OX(D) is a quasi-coherent sheaf. M

The class group

Definition 15.10 We define the class group of X as

Cl(X) = Div(X)/ Div0(X).

Two Weil divisors D, D1 are said to be linearly equivalent (written D „ D1) ifthey have the same image in Cl(X), or equivalently, that D´D1 is principal.

weil divisors 284

Example 15.11 Any divisor on A1k for an algebraically closed field k is principal.

Indeed, if D =řn

i=1 ni[pi] where ni P Z and pi P k, then

f =nź

i=1

(t´ pi)ni

is an element of k(t)ˆ with div( f ) = D. It follows that Cl(A1k) = 0 in this case.

K

The class group and unique factorization domainsThe term ‘class group’ comes from algebraic number theory and its originsbe traced back Kummer’s work on Fermat’s last theorem. If A is a Dedekinddomain then Cl(Spec A) coincides with the class group Cl(A) of A, whichmeasures how far A is from being a unique factorization domain. So forinstance Cl(Z) = 0 and Cl(Z[

?´5] = Z/2 (the latter is not an UFD, since

2 ¨ 3 = (1´?´5)(1 +

?´5)).

To prove the result we want, we will need the following two facts fromcommutative algebra.

o Hartog’s extension theorem: Let A be an integrally closed integral domain.Then

A =č

ht(p)=1

Ap (15.2)

where the intersection is taken inside the fraction field of A.

o A noetherian integral domain A is a unique factorization domain if andonly if every prime ideal p of height 1 is principal;

The intersection in (15.2) is taken over all prime ideals of height 1. SinceAp is integrally closed, it is a discrete valuation ring; we let vp : K Ñ Z denotethe corresponding valuation, so that Ap = ta P K|vp(a) ě 0u. In that case (15.2)says that A = ta P K|vp(a) ě 0 for all pu and Aˆ = ta P K|vp(a) = 0 for all pu.Hence the following sequence is exact:

0 Ñ Aˆ Ñ Kˆ ÝÑà

ht(p)=1

Z (15.3)

where the rightmost map is a ÞÑ (vp(a)). The map to the right is not alwayssurjective; in fact the cokernel of that map is exactly the class group Cl(A) =

Cl(Spec A). Indeed, we may identifyÀ

ht(p)=1 Z with Div(Spec A), and notethat the above sequence is part of the following:

0 Ñ Aˆ Ñ Kˆ divÝÝÑ Div(Spec A)Ñ Cl(A)Ñ 0. (15.4)

which is exact by the definition of linear equivalence of Weil divisors.

weil divisors 285

Proposition 15.12 Let A be a noetherian integral domain and let X = Spec A.Then the following are equivalent:

i) Cl(X) = 0 and X is normal.ii) Every height one prime ideal in A is principal

iii) A is a unique factorization domain.

Proof: If A is a UFD, then it is integrally closed, and hence normal. Theequivalence of ii) and iii) was noted above. We now show that i)ô ii).ð: If Y Ă X is a prime divisor in Spec A, then Y = V(p) for some prime

ideal p Ă A of height 1. Hence by assumption Y = V( f ) for some f P A, i.e.,Y = div( f ), and so Cl(X) = 0.

ñ:: If Cl(X) = 0, let p be a prime of height 1, and let Y = V(p) Ă X. Byassumption, there is a f P Kˆ such that div( f ) = Y. We want to show that infact f P A and that p = ( f ). But this follows from the exact sequence (15.4), sincevq( f ) = 0 for q ‰ p and vp( f ) = 1, and so f lies in ta P Kˆ|vp(a) ě 0u = Aˆ.

To prove that f generates p: Take any g P p. Then vp(g) ě 1 and vq(g) ě 0for all q ‰ p. It follows that vq(g/ f ) ě 0 for all prime ideals q P Spec A. Henceg/ f P Aq for all q prime of height 1, and hence g/ f P A, by the same argumentas above. It follows that g P f A and so p = f A is principal. o

In particular, since A = k[x1, . . . , xn] is a unique factorization domain, we get

Corollary 15.13 Cl(Ank ) = 0.

A useful exact sequenceGiven a scheme X and an open subset U, the restriction of a prime divisor onX is a prime divisor on U, so it is natural to ask how the two class groups arerelated. The answer is given by the following theorem:

Theorem 15.14 Let X be a normal, integral scheme, let Z Ă X be a closedsubscheme and let U = X´Z. If Z1, . . . , Zr are the prime divisors correspondingto the codimension 1 components of Z, then there is an exact sequence

rà

i=1ZZi Ñ Cl(X)Ñ Cl(U)Ñ 0 (15.5)

where the map Cl(X)Ñ Cl(U) is defined by [Y] ÞÑ [YXU].

Proof: If Y is a prime divisor on U, the closure in X is a prime divisor in X, sothe map is surjective.

We just need to check exactness in the middle. Suppose Y is a prime divisorwhich is principal on U. Then Y|U = div( f ) for some f P K(U) = K = k(X).Now D = div( f ) is a divisor on X such that D|U = div( f )|U . Hence DÝ is a

weil divisors 286

Weil divisor supported in XÚ, and hence it must be a linear combination ofthe Zi’s. o

As a special case, we see that removing a codimension 2 subset does notchange the group of Weil divisors. So for instance Cl(A2 ´ 0) = Cl(A2).Example 15.15 Consider the projective line P1

k over a field k, and let P be apoint. We have the exact sequence

Z[P]Ñ Cl(P1)Ñ Cl(A1)Ñ 0

We saw that Cl(A1) = 0, so the map Z Ñ Cl(P1) is surjective. It is also injective:If [nP] = 0 in Cl(P1) for some n, then nP = div( f ) for some f P k(P1). Considerthe open set U = P1 ´ P » A1. Then nP|U = 0, so we must have div( f )|A1 = 0,and consequently f P Γ(A1,Oˆ) = k˚. Hence f is constant, and so n = 0. Itfollows that Cl(P1) = Z. K

Projective spaceWrite Pn

k = Proj R, R = k[x0, . . . , xn]. Prime divisors on Pnk correspond to height

one prime ideals in R, i.e., p = (g) where g is a homogeneous irreduciblepolynomial. We can use this to define the degree of a divisor, by taking thedegrees of the corresponding polynomials:

Div(Pn)Ñ Zÿ

niV(gi) ÞÑÿ

ni deg gi

If f P K(Pnk ) is a rational function, then f can be written as a quotient of two

homogeneous polynomials of the same degree. More precisel, if f P K(Pnk ) is

a rational function, we can write it as f =ś

i f nii where the fi are irreducible

coprime polynomials in R and ni P Z. Let us first show that

div( f ) =ÿ

ni[V( fi)]

If Y Ă Pnk is a prime divisor, let y P Y be the generic point. Since Y has

codimension 1, Y = V(g) for some irreducible polynomial g of degree d. Forany other polynomial h of degree d, the fraction g/h is a generator of myOX,y. Wecan write f = (g/h)r f 1 with r = ni if fi divides g (and 0 if no fi divides g) and f 1

a rational function not containing g in the numerator or the denominator. Thismeans that vY( f ) = r, and hence div( f ) =

ř

ni[V( fi)]. Note that deg div( f ) =ř

ni deg fi = 0. It follows that the degree map descends to a map

deg : Cl(Pn)Ñ Zÿ

niV(gi) ÞÑÿ

ni deg gi

We claim that this is an isomorphism. Note that deg H = 1, where H = V(x0)

is a hyperplane, so deg is surjective. Now, any Z =ř

ni[V( fi)] in the kernel

weil divisors 287

of deg, must haveř

ni deg fi = 0. Consider the element f =ś

i f nii , which

now defines an element of K. Using the formula for div( f ) above, we see thatZ = div( f ), and hence Z is a principal divisor, and so deg is injective.

We have thus shown:

Proposition 15.16 Cl(Pnk ) = Z.

Example 15.17 Cl(P2) is generated by the class of line L Ă P2, e.g., L = V(x0),and any two lines L, L1 are linearly equivalent. K

Example 15.18 Consider the curve X as in Figure 10, given by

X = V(y2z´ x3 ´ xz2) Ă P2.

For a line L = V(y) on P2k let L|X denote the restriction of L to X (i.e., the Weil

divisor LX X on X which is of codimension 1 since X is integral). Moreover, foranother line L1 = V(z), the two restrictions L|X and L1|X are linearly equivalentdivisors on X, since L|X ´ L1|X = div( x

z |X). This argument applies for any twolines L, L1 in P2, so we get many relations between divisors on X. The figurebelow shows one example where L|X = P + Q + R and L1 = 2S + T. K

Figure 15.1: Two linearly equivalent divisors on a plane cubic

Exercise 15.2 Let P2 = Proj k[x0, x1, x2] and let f denote an irreducible homo-geneous polynomial of degree d ě 1. f determines a prime divisor D = V( f ).Consider the open set U = P2

k ´D. Show that the above exact sequence abovetakes the form

0 Ñ Z¨dÝÑ Z Ñ Cl(U)Ñ 0

Deduce that Cl(U) = Z/d. M

cartier divisors 288

15.2 Cartier divisors

Let X be a noetherian integral scheme with function field K. Let KX denote theconstant sheaf with value K = k(X). The constant sheaf K ˆ

X with value Kˆ (thegroup of non-zero elements of K) is a subsheaf, and contains OˆX , the sheaf ofunits in OX.

Definition 15.19 A Cartier divisor D on X is an element of Γ(X, K ˆX /OˆX ).

This definition of a Cartier divisor is pretty obscure, and one rarely thinks ofa divisor in this way. We can shed some light on the definition by consideringan open cover Ui of X. The sheaf axiom sequence takes the following form

0 Ñ Γ(X, K ˆX /OˆX )Ñ

ź

i

Γ(Ui, K ˆX /OˆX )Ñ

ź

i,j

Γ(Uij, K ˆX /OˆX )

From this, we see that a Cartier divisor is therefore given as a set of sections si

of K ˆX /OˆX over the open sets Ui that agree on the overlaps Uij.

Since K ˆX Ñ K ˆ

X /OˆX is surjective as a map of sheaves, we may refinethe covering Ui in order lift the sections si to sections of K ˆ

X over Ui, that is,elements fi P Kˆ. In other words, a Cartier divisor is specified by the data(Ui, fi), where Ui is an open set and fi is a rational function.

To say what it means that two such data (Ui, fi) agree on Uij we must keepin mind that they are sections of K ˆ

X /OˆX . The condition si|Uij = sj|Uij translatesinto the statement that fi f´1

j |Uij is a section of OˆX over Uij. In other words,

there are units cij P OX(Uij) such that fi = cij f j over Uij. In Kˆ we have cij =fif j

,which implies that the cij satisfy the cocyle conditions

cik = cijcjk; cji = c´1ij ; cii = 1

The intuition behind the definition of a Cartier divisor is that the fi give the localequations for a subscheme of X. On an affine open set U where fi is regular, thezero-set V( fi) defines a codimension 1 closed subset of U. If both f j and fi areregular on U, their zero-sets must be the same since they differ by a unit.

Definition 15.20 The pairs (Ui, fi) are called the local defining data or thelocal equations for the divisor D (with respect to the covering Ui).

Such defining data are not unique: t(Ui, fi)uiPI and t(Vj, gj)uiPI give the sameCartier divisor if fig´1

j P Γ(Ui XVj,OˆX ) for all i, j.Example 15.21 If f = f (x1, . . . , xn) is a rational function on An

k , then (Ank , f )

forms a Cartier divisor. K

Example 15.22 On P1 we can use the standard covering U0 = Spec k[t] andU1 = Spec k[t´1]. Then there is a Cartier divisor D given by (U0, t) and (U1, 1).


K

Now, the set of Cartier divisors naturally form an abelian group: GivenD and D1 represented by the data t(Ui, fi)uiPI and t(Vi, gi)uiPJ , we can defineD + D1 as the Cartier divisor associated to the data

t(Ui XVj, figj)ui,j

Moreover, the inverse ´D will be defined as t(Ui, f´1i )uiPI , and the identity is

defined by the data t(Ui, fi)uiPI where fi P Γ(Ui,OˆX ). We denote the group ofCartier divisors by CaDiv(X) = Γ(X, K ˆ

X /OˆX ).

Definition 15.23 We say that a Cartier divisor is principal if it is equal (asan element of CaDiv(X)) to the Cartier divisor (X, f ) where f P Kˆ.

The principal Cartier divisors form a subgroup of CaDiv(X), which is typi-cally smaller than CaDiv(X). Note on the other hand that, by definition, anyCartier divisor is ‘locally principal’, since it becomes principal when restrictedto each Ui.

Definition 15.24 We define CaCl(X) to be the group of Cartier divisors mod-ulo principal divisors:

CaCl(X) = CaDiv(X)/t(X, f )| f P Kû

Two Cartier divisors D, D1 are said to be linearly equivalent, if D´D1 = (X, f )for some principal divisor (X, f ), or equivalently, [D] = [D1] in CaCl(X).

Cartier divisors and invertible sheavesWe saw earlier that each Weil divisor D gave rise to a quasi-coherent sheafOX(D). The same is true for Cartier divisors, and in this case the correspondingsheaf will turn out to be invertible.

Let D be a Cartier divisor on a scheme X given by the data t(Ui, fi)uiPI . Wewill associate to it an invertible sheaf, which we denote by OX(D). As in thecase of Weil divisors, the intuition is that sections of OX(D) should correspondto rational functions with at worst poles along D, e.g., if fi P OX(U) we allowrational functions like 1

fi. On Ui we therefore take the subsheaf f´1

i OUi Ă KUi

of the constant sheaf K. This subsheaf is isomorphic to OUi and has f´1i as a

local generator. So over an affine subset U = Spec A Ă Ui, the sheaf is the sheafassociated to f´1

i A Ď K(A). On the intersection, Ui XUj, we have fi = cij f j

where cij is an invertible section of OUij . This means that f´1i OUij = f´1

j OUij

as subsheaves of KUij . We have therefore constructed a sheaf on each Ui, andthe elements coincide on the intersections Uij. In order to be able to glue to asheaf, there is a cocycle condition that has to be satisfied. But since these sheaves


are all subsheaves of a fixed sheaf K , the gluing maps are actually identitymaps, and the cocycle condition is automatically satisfied. It follows that thesheaves f´1

i OUi glue to a sheaf OX(D) defined on all of X. It is by constructioninvertible, since it is invertible on each Ui.

Explicitly, OX(D) is defined as the subsheaf of KX given by

Γ(V,OX(D)) = t f P K | fi f P Γ(Ui XV,OX)@i P Iu

Two different data (Ui, fi) and (Vj, gj) for the same divisor D give rise to thesame invertible sheaf. This is because over Ui XVj, we have fi = dijgj for somesections dij P OX(Ui XVj)

ˆ. This means that f´1i OUiXVj = g´1

i OUiXVj , and sothe sheaf is uniquely determined as a subsheaf of KX.

Proposition 15.25 Let X be an integral noetherian scheme and let D and D1

be two Cartier divisors.

i) OX(D + D1) » OX(D)bOX(D1)ii) OX(D) » OX(D1) if and only if D and D1 are linearly equivalent.

Proof: We can pick a common covering Ui so that both D and D1 are bothrepresented by data (Ui, fi), (Ui, f 1i ). Then D + D1 is defined by (Ui, fi f 1i ). Lo-cally, over Ui the sheaf OX(D + D1) is defined as the subsheaf of K given by( fi f 1i )

´1OUi = f´1i f 1´1

i OUi . The tensor product is locally(!) given as f´1i OUi b

f 1i´1OUi , which is clearly isomorphic to f´1

i f 1´1i OUi via the mapping defined by

a f´1i b b f 1i

´1ÞÑ ab fi

´1 f 1j´1.

For the second claim, it suffices (by point (i)) to show that OX(D) » OX ifand only if D is a principal Cartier divisor. So suppose that OX(D) Ď KX isa sub OX-module which is isomorphic to OX. Then the image of 1 in OX willbe a section s of OX(D) which generates OX(D) everywhere. This means thatfor f = s´1, (X, f ) is the local defining data for D, and D is a principal Cartierdivisor. Conversely, if D = (X, f ), then OX(D) = f´1OX, and multiplication byf P K gives an isomorphism OX(D) » OX. o

CaCl(X) and Pic(X)

By the item (i) and (ii) in Proposition 15.13, we see that the natural map ρ :CaDiv(X)Ñ Pic(X), which sends D to the class of OX(D) in Pic(X) is additiveand has the subgroup of principal divisors CaDiv0(X) as its kernel. This meansthat the induced map ρ : CaCl(X)Ñ Pic(X) is injective. In this section we willshow that this map is also surjective, so that in fact CaCl(X) » Pic(X).

Proposition 15.26 When X is integral, the map ρ : CaCl(X) Ñ Pic(X) isan isomorphism.

Proof: We need to show that ρ is surjective. It suffices to show that anyinvertible OX-module L is isomorphic to a submodule of KX: If L Ď K , let


Ui be a trivializing cover of L and let gi be its local generators. Then we haveL|Ui = giOUi Ă KUi and the gi are rational functions on Ui. On Uij = Ui XUj,we have giOUij = L|Uij = gjOUij , and it follows that gi = cijgj for units cij P OÛij

.

Consequently, (Ui, g´1i ) forms a set of local defining data for a Cartier divisor

D, and of course we have L = OX(D).Let L be an invertible sheaf and consider the sheaf LbOX KX. Let Ui Ă X

be an open cover such that L|Ui = OX|Ui . Note that the restriction of LbOX KX

to each Ui is a constant sheaf (isomorphic to KX). Since X is irreducible, anysheaf whose restriction to opens in a covering is constant, is in fact a constantsheaf, and therefore LbOX KX » KX as sheaves on X. Now we can regard L asa rank 1 subsheaf of KX using the composition L ãÑ LbKX » KX. Hence bythe above paragraph, ρ is surjective. o

From Cartier to WeilFrom (15.4) we have for each open subset V Ă X the following exact sequence:

0 Ñ OX(V)ˆ Ñ Kˆ divÝÝÑ Div(V)

This gives rise to an exact sequence of sheaves

0 Ñ OˆX Ñ K ˆX Ñ Div (15.6)

and in turn the following injective map of sheaves

φ : K ˆX /OˆX Ñ Div

Taking global sections, we get an injective map ι : CaDiv(X)Ñ Div(X).Here is a more explicit description of ι: Let D be a Cartier divisor given by

the data (Ui, fi). If Y is a prime divisor on X, with generic point η, then sinceUi is a cover, η lies in some Ui. We can then define

vY(D) = vY( fi)

This is independent of the choice of Ui: If η P Ui XUj, then fi f´1j is an element

of OˆX (Ui XUj), and so vY( fi f´1j ) = 0, and hence vY( fi) = vY( f j). Then ι is

defined byι(D) =

ÿ

Y

vY(D)Y.

In particular, we may view Cartier divisors as a subgroup of the group of Weildivisors.

Proposition 15.27 Let X be an integral normal scheme. Then the followingare equivalent:

i) ι : CaDiv(X)Ñ Div(X) is an isomorphism;ii) The exact sequence (15.6) is exact on the right;

iii) X is locally factorial (all the local rings OX,x are UFDs).


Proof: We already noted that ι is injective, so the question is about surjectivity.ii) ñ i): If (15.6) is exact at the right, then φ is an isomorphism, and hence

so is ι (take global sections).i) ñ ii): Assume ι : Γ(X, K ˆ

X /OˆX ) Ñ Div(X) is surjective. If we viewDiv =

À

codim Y=1 jY˚Z it is easy to see that Div(X) Ñ Div(U) is surjective forany open set U Ă X. Taking the direct limit, we see that Div(X) Ñ Divx issurjective. Thus if ι is surjective, then (15.6) is exact on stalks by the followingdiagram:

0 OˆX (X) Kˆ Γ(X, K ˆX /OˆX ) 0

0 OˆX,x Kˆ Divx

ii) ô iii): Exactness of (15.6) can be checked on stalks. Let x P X and letA = OX,x be the local ring. Then the stalk of Div =

À

codim Y=1 jY˚Z equalsDiv(Spec A) and the rightmost map in (15.6) is simply

Kˆ Ñ Div(Spec A)

which sends f to the sumř

vp( f )V(p) where p is a height 1 prime ideal of A.(To check this, reduce to an affine open set containing x and compute the directlimit using distinguished opens). This is surjective if and only if every p Ă A isa principal ideal. This happens if and only if OX,x is a UFD. o

So if X is locally factorial, every Weil divisor comes from a Cartier divisor,and vica versa. The intuition is that this holds whenever X has ‘mild’ singulari-ties. For instance, regular local Noetherian rings (i.e., dimk m/m2 = dim A) arealso UFDs ([Atiyah-MacDonald Ch. 7] or [Stacks 0AG0]). So in particular, theabove applies to the main examples of interest:

Corollary 15.28 On a non-singular variety X, then the map ι : CaDiv(X) Ñ

Div(X) is an isomorphism. Moreover, this induces natural isomorphisms between thegroups of

i) Weil divisors (modulo linear equivalence)ii) Cartier divisors (modulo linear equivalence)

iii) Invertible sheaves (modulo isomorphism)

From our previous computation of Cl(Ank ), we get the following theorem:

Theorem 15.29 Let k be a field. Then Pic(Ank ) = Cl(An

k ) = CaCl(Ank ) = 0.

effective divisors and linear systems 293

Projective spaceLet us take a closer look at the projective space Pn

k over a field k. Write Pnk =

Proj R where R = k[x0, . . . , xn]. Consider the standard covering Ui = D+(xi) ofPn

k .Let F(x0, . . . , xn) P Rd denote a homogeneous polynomial of degree d. Then

F(x/xi) = F( x0xi

, . . . , xi´1xi

, 1, xi+1xi

, . . . , xnxi) defines a non-zero regular function on

Ui, and the collection(Ui, F(x/xi))

forms a Cartier divisor D on X. Indeed, on the overlap Ui XUj we have therelation

F(x/xi) =(

xj/xi)d F(x/xj)

and xj/xi is a regular and invertible function on Ui XUj. The correspondinginvertible sheaf is exactly OPn

k(d). Two homogeneous polynomials F, G of the

same degree d give linearly equivalent divisors, because the quotient F(x)/G(x)is a global rational function on Pn

k .We previously computed that Cl(Pn

k ) = Z, so Corollary 15.16 gives thefollowing:

Corollary 15.30 On Pnk any invertible sheaf is isomorphic to some OPn

k(m).

15.3 Effective divisors and linear systems

We say that a Cartier divisor D is effective (and write D ě 0) if D can berepresented by local data (Ui, fi) where the rational functions actually lie inOX(Ui). It is straightforward to verify that if this is true for one set of data, thenit holds for any other set as well.

We will write D ě D1 if D´ D1 ě 0, i.e., the difference D´ D1 is effective.Note also that if D and D1 are both effective, then so is D + D1. This means thatPic(X) is an ordered group.

Effective divisors are of paramount importance in algebraic geometry; theycarry lots of essential geometric information. With the set up above, D beingeffective is equivalent to the statement that OX(´D) (regarded as a subsheaf ofKX) is contained in OX. The OX-module OX(´D) is a thus coherent sheaf ofideals which is locally generated by one element (in other words, it is locallyprincipal). We will usually denote the corresponding closed subscheme alsoby D (this is a somewhat bad example of abusing the notation, but it has itsadvantages).

The inclusion OX(´D) Ă OX induces an exact sequence

0 Ñ OX(´D)Ñ OX Ñ OD Ñ 0

where the right hand is to be interpreted both as the cokernel of the left-mostmap and as the structure sheaf of the subscheme associated to D. Moreover, the

effective divisors and linear systems 294

support of the OD (i.e., the set of points x P X, such that OD,x ‰ 0) coincideswith underlying topological space of D.

So effective divisors give rise to closed subschemes of codimension one. Theconverse is also true: If D is a closed subscheme which is locally defined byone equation (which is not a zero-divisor), then we can find, for any x P X,an open affine neighbourhood Ux and an element fx P A = Γ(Ux,OX) sothat D = V( fx) = Spec A/( fx A). Two such elements fx, f 1x generate the sameprincipal ideal, so there must be a relation fx = cx f 1x for some cx P Aˆ. Fromthis, we see that (Ux, fx) form the defining data for a Cartier divisor, whichwe will also denote by D. Indeed, on an overlap Uxy = Ux X Uy we havefx|Uxy = cxy fy|Uxy for a section cxy of OÛxy

. We have therefore proved thefollowing theorem:

Theorem 15.31 Let X be an noetherian, integral scheme. Then there is a one-to-one correspondence between closed subschemes D Ď X locally defined by aprincipal ideal, and effective Cartier divisors on X.

The inclusion OX(´D) Ă OX can be dualized, i.e., we apply the functorHomOX (´,OX), and obtain a map α : OX Ñ OX(D). Such a map is uniquelydetermined by its value on 1, i.e., the global section σ = α(1) P Γ(X,OX(D)).

Conversely, given a global section in Γ(X,OX(D)), we dually have a mapOX(´D) Ñ OX. When X is integral, then this gives us a divisor, which is ofcourse the effective divisor D. We have therefore proved:

Theorem 15.32 Let X be an noetherian, integral scheme and let D be a Cartierdivisor on X. Then D is linearly equivalent to an effective divisor if and onlyif OX(D) has a non-zero global section. For each section σ, we get a divisordenoted by div σ. Two such sections σ, σ1 give rise to the same divisor if andonly if σ = cσ1 where c P Γ(X,OX)

ˆ.

Definition 15.33 The set of effective divisors D1 linearly equivalent to D isdenoted by |D|. This is called the complete linear system of D.

The name ‘linear system’ comes from the special case when X is a projectivevariety X over a field k (thus X is integral, separated of finite type over k). Inthis case, we have Γ(X,OX)

ˆ = kˆ, and the previous discussion shows that thelinear system |D| is given by

|D| =

D1|D1 ě 0 and D1 „ D(

= (Γ(X,OX(D))´ 0) /kˆ

= PΓ(X,OX(D))

When X is projective over k, the cohomology groups H0(X,OX(D)) is finitedimensional as k-vector spaces (we will prove this fact in Chapter 16), so the set

appendix 295

of effective divisors D1 linearly equivalent to D is (as a set) a projective spacePn

k .

Definition 15.34 A linear system of divisors is a linear subspace of a com-plete linear system |D|.

Example 15.35 Consider the case X = Pnk and D = dH, where H is the

hyperplane divisor (so H is a Cartier divisor with OX(H) » OX(1)). In this casethe linear system of D associated to OX(dH) is given by the set of homogeneouspolynomials of degree d modulo scalars, i.e.,

|D| =

$

&

%

ÿ

i0+¨¨¨+in=d

ai0,...,in xi00 ¨ ¨ ¨ x

inn

,

.

-

/kˆ » PNk

where N = (n+dd )´ 1. The points of this projective space correspond to degree d

hypersurfaces, and the coefficients ai0,...,in give homogeneous coordinates on it.K

15.4 Appendix

There is an alternative way of seeing the homomorphism CaDiv(X)Ñ Pic(X)

using the exact sequence

0 Ñ OˆX Ñ K ˆX Ñ K ˆ

X /OˆX Ñ 0.

If we take the long exact sequence, we get a map on cohomology

H0(K ˆX /OˆX )Ñ H1(X,OˆX )

If we identify CaDiv(X) with K ˆX /OˆX and recall the isomorphism Pic(X) »

H1(X,OˆX ), we obtain a map CaDiv Ñ Pic(X) which indeed is the same as themap ρ defined above. We can use sheaf cohomology to recover the statement inProposition 15.14.

Let us first show that H1(X, K ˆX ) = 0. For a covering Ui, this is the cokernel

of the right-most map in

0 Ñ Γ(X, K ˆX )Ñ

ź

i

Γ(Ui, K ˆ)Ñź

i,j

Γ(Uij, K ˆ)

However, since Γ(U, KX) = K is constant for all U, we see that the map issurjective, and so H1(X, K ˆ

X ) = 0.Now, by the long exact sequence, the map

H0(K ˆX /OˆX )Ñ H1(X,OˆX )

is an isomorphism.

quadrics 296

15.5 Quadrics

The smooth quadric surfaceLet k be a field, and let Q = P1

k ˆP1k . Recall that Q embeds as a quadric surface

in P3k via the Segre embedding. So we can view Q both as a fiber product

P1 ˆP1 and the quadric V(xy´ zw) Ă P3.

Weil divisors on Q

Since Q is a product of two P1s there are natural ways of constructing Weildivisors on Q from those on P1. For instance, we can let

L1 = (0 : 1)ˆP1 Ă Q,

which is a prime divisor on Q corresponding to the ‘vertical fiber’ of Q. Similarly,L2 = P1ˆ (0 : 1) is a Weil divisor on Q. From these we obtain an exact sequence

ZL1 ‘ZL2 Ñ Cl(Q)Ñ Cl(Q´ L1 ´ L2)Ñ 0

Here Q´ L1 ´ L2 = U11 = Spec k[x´1, y´1]. This is the Spec of a UFD, henceCl(Q´ L1 ´ L2) = 0. This shows that Cl(Q) is generated by the classes of L1

and L2. We claim that the first map is also injective, so that in fact that

Cl(Q) = ZL1 ‘ZL2.

For this, we need to analyse the two divisors L1, L2 in more detail. It turns outthat this is easier if we switch to the perspective of Cartier divisors. Cartier

divisors on Q

To study Cartier divisors on Q, we first need a covering. Q is covered by fouraffine subsets

U00 = Spec k[x, y] U10 = Spec k[x´1, y]

U01 = Spec k[x, y´1] U11 = Spec k[x´1, y´1]

Consider P1k = W0 YW1, where W0 = Spec k[t], W1 = Spec k[t´1]. The first

projection p1 : Q Ñ P1k is induced by the ring maps

k[t]Ñ k[x, y] k[t´1]Ñ k[x´1, y]

t ÞÑ x t´1 ÞÑ x´1

k[t´1]Ñ k[x, y´1]; k[t´1]Ñ k[x´1, y´1];

t ÞÑ x t´1 ÞÑ x´1

Let p = (0 : 1) be the Weil divisor on P1. The Cartier data of p is given by(W0, t), (W1, 1), so that OP1(p) » OP1(1). The pullback D = p˚1 (p) is a Cartier

quadrics 297

divisor on Q, corresponding to the Weil divisor (0 : 1)ˆP1. The correspondingCartier data is given by

(U00, x), (U10, 1)

(U01, x), (U11, 1)

Let L1 = (0 : 1)ˆP1k and L2 = P1

k ˆ (0 : 1). Consider the restriction of D toL2. L2 is covered by the two open subsets V0 = U00 X L2 = Spec k[x, y]/y =

Spec k[x], V1 = U10 X L2 = Spec k[x´1, y]/(y) = Spec k[x´1]. In terms of theseopens, the restriction D|L2 has Cartier data

(V0, x), (V1, 1)

obtained by restricting the data above. In particular, identifying L2 » P1, wesee that OQ(D)|L1 » OP1(1). In particular, since Cl(P1) = Z, no multiple nD isequivalent to 0 in Cl(Q): if that were the case, we would have OQ(nD) » OQ,and hence OQ(nD)|L2 » OQ|L2 » OP1 , a contradiction.

In all, this shows that the first map in the exact sequence (15.5) is injective.Hence

Cl(Q) » ZL1 ‘ZL2.

If D is a divisor on Q, D „ aL1 + bL2 and we call (a, b) the ‘type’ of D.A divisor of type (1, 0) or (0, 1) is a line on the quadric surface Q Ă P3. Wehave i˚OP3(1) » OQ(L1 + L2), so a (1, 1)-divisor is represented by a hyperplanesection of Q (a conic). A prime divisor of type (1, 2) or (2, 1) is a twisted cubiccurve.

The quadric coneLet X = Spec R where R = k[x, y, z]/(xy´ z2), and k has characteristic ‰ 2. LetZ = V(y, z) be the closed subscheme corresponding to the line ty = z = 0u. Notethat Z » Spec k[x, y, z]/(xy´ z2, y, z) = Spec k[x], so it is integral of codimension1.

A singular quadric surface

Note that X´ Z = X´V(y) = D(y), and the latter equals

Spec k[x, y, y´1, z]/(xy´ z2) = Spec k[y]y[t, u]/(t´ u2) = Spec k[y]y[u]

quadrics 298

which is the spectrum of a UFD. It follows that Cl(X´ Z) = 0. Recall now thesequence

Z Ñ Cl(X)Ñ Cl(X´ Z)Ñ 0

where the first map sends 1 to [Z]. Hence Cl(X) is generated by [Z].We first show that 2Z = 0 in Cl(X). This is because we can consider the

divisor of x. The rational function x is invertible in every stalk OX,p except whenp P V(x). Moreover, by the defining equation xy = z2, we see that the divisor ofx can only be non-zero along Z. The valuation at the generic point η of Z is 2:The local ring equals

OX,η = (k[x, y, z]/(xy´ z2))(y,z)

and since x is invertible here, we see that y P (z2) and that z is the uniformizer.Now we show that Z is not a principal divisor. It suffices to prove that this

is not principal in SpecOX,p where p P X is the singular point of X. The localring here is

OX,p = (k[x, y, z]/(xy´ z2))(x,y,z)

In this ring p = (x, z) is a height 1 prime ideal, but it is not principal: Letm Ă OX,x be the maximal ideal. Note that x, y P m, since x, y are not units.Moreover, it is clear that the vector space m/m2 (which is the Zariski cotangentspace at x) is 3-dimensional, spanned by tx, y, zu. Then x, y gives a 2-dimensionalsubspace of m/m2. Hence, since x and y are linearly independent in this quotient,there couldn’t be an non-constant element f P OX,x for which x = a f , y = b f .This [Z] ‰ 0 in Cl(X) and hence

Cl(X) = Z/2.

Note that the open subscheme X ´ (0, 0, 0) is factorial. Hence removing acodimension 2 subset has an effect on CaCl(X). Recall however, that the classgroup of Weil divisors Cl(X) stays unchanged under removing a codimension 2

subset.

Projective quadric cone

Let X = Proj R where R = k[x, y, z, w]/(xy ´ z2). Let H = V(w) be thehyperplane determined by w. We have

0 Ñ ZH Ñ Cl(X)Ñ Cl(X´ H)Ñ 0

(Here H is a divisor corresponding to the restriction of OP3(1), hence it is non-torsion in Cl(X), so the first map is injective). X´ H is isomorphic to the affinequadric cone from before, hence Cl(X ´ H) = Z/2. Using this sequence, wesee that Cl(X) = Z, generated by a Weil divisor D such that H = 2D. Moreprecisely, D is the divisor V(x, z) which is supported on a line on X.

The Weil divisor D is not Cartier; being Cartier is a local condition, so thisfollows from the example of the affine quadric cone above. Here is an alternative

quadrics 299

way to see it: If D = V(x, z) is Cartier, the sheaf L = OX(D) is invertible,and hence so is its restrtction to the line ` = V(x, z) » P1

k . The Picard groupof P1

k is Z, generated by OP1(1), so we have L|` » OP1(a) for some a P Z.On the other hand, we know that the divisor H = 2D is Cartier and in factOX(H) » OP3(1)|X (the local generator is given by w). Restricting further to `,we obtain OP3(1)|` » OP1(1) (as the divisor of w is just one point on `). Butthese two observations imply that 2a = 1, a contradiction. Hence D is notCartier.

Quadric hypersurfaces in higher dimensionHere is an application of the ‘useful exact sequence’ (15.5).

Lemma 15.36 (Nagata’s lemma) Let A be a noetherian integral domain, and letx P A´ 0. Suppose that (x) is prime, and that Ax is a UFD. Then A is a UFD.

Proof: We first show that A is normal. Of course Ax is normal, being a UFD.So if t P K(A) is integral in A, it lies in Ax. We need to check that if a/xn P Ax

is integral over A and x does not divide a, then n = 0. If we have an integralrelation

(a/xn)N + b1(a/xn)N´1 + ¨ ¨ ¨+ bN = 0

Multiplying by xnN we get aN P xA, so x|a, because A is an integral domain.Hence A is normal.

Now the Weil divisor D = div x is an effective divisor and so there is anexact sequence

ZD Ñ Cl(Spec A)Ñ Cl(Spec Ax) = 0 Ñ 0

The image of the left-most map is 0, so Cl(A) = 0, and so A is a UFD. o

Let A = k[x1, . . . , xn, y, z]/(x21 + ¨ ¨ ¨+ x2

m ´ yz). We will prove that A is aUFD for m ě 3. A is a domain, since the defining ideal is prime. Apply Nagata’slemma with the element y:

Ay = k[x1, . . . , xn, y, y´1, z]/(y´1(x21 + ¨ ¨ ¨+ x2

m)´ z) » k[x1, . . . , xn, y, y´1, z]

which is a UFD. We show that y is prime: Taking the quotient we get

A/y = k[x1, . . . , xn, x]/(x21 + ¨ ¨ ¨+ x2

m)

which is an integral domain, because x21 + ¨ ¨ ¨+ x2

m is irreducible (for m ě 3).Note that for m = 2, we get the quadric cone, which we have seen is not a

UFD.Applying a change of variables, we find the following description of the

class groups of quadrics in any dimension:

extended example: hirzebruch surfaces 300

Proposition 15.37 Let k be a field containing?´1 and let X = V(x2

0 + ¨ ¨ ¨+

x2m) Ă An+1

k = Spec k[x0, . . . , xn].

i) m = 2, Cl(X) = Z/2

ii) m = 3, Cl(X) = Z

iii) m ě 4, Cl(X) = 0

There is also the following statement for projective quadrics:

Proposition 15.38 Let X = V(x20 + ¨ ¨ ¨+ x2

m) Ă Pn = Proj k[x0, . . . , xn].

i) m = 2, Cl(X) = Z;ii) m = 3, Cl(X) = Z2;

iii) m ě 4, Cl(X) = Z.

Exercises(15.3) Show that for the weighted projective space P = P(1, 1, d) we haveˇ

Cl(P) = ZD and CaCl(P) = ZH where H = dD.(15.4) Let X denote the affine line over k with two origins. Compute Pic(X).ˇ

M

15.6 Extended example: Hirzebruch surfaces

Let r ě 0 be an integer and consider the scheme X which is glued together bythe four affine scheme charts

U00 = Spec k[x, y]

U01 = Spec k[x, y´1]

U10 = Spec k[x´1, xry]

U11 = Spec k[x´1, x´ry´1]

This is a non-singular, integral 2-dimensional scheme over k. When k = C, theseare the so-called r-th Hirzebruch surfaces. In many ways, these surface behave asthe ’Mübius strips’ in algebraic geometry. Note in particular, when r = 0, weget P1

k ˆP1k .

Divisors

Let us define two divisors D1, D2 on X by the following Cartier divisors. WritingK = k(x, y), the Cartier data is given by

D1 =

[(U00, x), (U01, x)(U10, 1), (U11, 1)

], D2 =

[(U00, y), (U01, 1)(U10, y), (U11, 1)

]


We will show that D1, D2 generate CaCl(X) » Pic(X). Note that both of thesedivisors are effective, since they are defined by rational functions which areregular on the Uij. In fact D1 » P1, since it is glued together by V0 = U00 X

V(x) = Spec k[y] and V1 = U01 XV(x) = Spec k[y´1]. Moreover, D2 restrictedto D1 is given by the Cartier data (V0, y), (V1, 1), which corresponds to a closedpoint on D1. Hence OX(D2)D1 » OP1(1). This shows that D2 is non-torsion inCl(X), since no multiple of it is trivial when restricted to D1. A similar argumentshows that D1 is non-torsion in Cl(X).

Let D11 be the Cartier divisor

D11 =

[(U00, 1), (U01, 1)

(U10, x´1), (U11, x´1)

],

We can compute that D11|D1 equals the divisor (V0, 1), (V1, 1) which is principal.In fact,

div x = D1 ´D11

So, D1 = D11 in Cl(X). This shows that D1 and D2 are independent in Cl(X),because OX(D11) restricts to OX on D1, but as OP1(1) on D2.

Now let U = X´D1 ´D2. This is isomorphic to U11 which is the spectrumof k[x´1, x´ry´1] which is a UFD. The exact sequence

ZD1 ‘ZD2 Ñ Cl(X)Ñ 0

and the previous analysis shows that Cl(X) = ZD1 ‘ZD2.

Sheaf cohomology

We now want to compute Hi(X,OX(D)) for a divisor D = aD1 + bD2. Asusual, we utilize a Cech complex. We want to mimick the computation for Pn.In that proof, the Z-graded polynomial ring k[x0, . . . , xn] played an importantrole, and the groups in the Cech complex corresponded to degree 0 localizationsof it. For X there is no such Z-graded ring lying around, but we can get by byintroducing the bigraded ring

R = k[x0, x1, y0, y1]

where the degrees of the variables are defined by

deg x0 = (1, 0), deg x1 = (1, 0), deg x2 = (0, 1), deg x3 = (´r, 1).

By identifying x = x1x0

, y =xr

0y1y0

, we find that the Cech complex of Uij can bewritten

R[x0x1] R[x0x1y0]

‘ ‘

R[x0x1] R[x0x1y0]

‘ Ñ ‘ Ñ R[x0x1y0y1]

R[x0x1] R[x0x1y0]

‘ ‘

R[x0x1] R[x0x1y0]


where the bracket means that we take the (0, 0)-part of the localization. So forinstance,

R[x0y0] = R[

x1

x0,

xr0y1

y0

]As in the Pn case, we now have a bigraded isomorphism

à

a,bPZ

H0(X,OX(aD1 + bD2)) »č

i,j

R[xiyj] = R

In particular, H0(X,OX(aD1 + bD2)) can be identified with polynomials ofbidegree (a, b) in R. So for example H0(X,O(D1)) corresponds to degree (1, 0)-polynomials, e.g., linear combinations of x0, x1. These two sections correspondto the sections 1, x above. Similarly, H0(X,O(D2)) is 1-dimensional.

Perhaps the most interesting divisor is E = D2 ´ rD1,which is effective.When r ą 0, this has H0(X,OX(nE)) = k for every n ě 0, but OX(nE) fi OX forn ‰ 0. Therefore E couldn’t possibly be globally generated. In fact, E » P1 and

OX(E)|E » OP1(´r)

In particular, this gives an example of an effective divisor which does not pullback (restrict) to an effective divisor.

Chapter 16

Maps to projective space

Given a scheme X it is natural to ask when there is a morphism to a projectivespace

f : X Ñ Pn,

or when there is a closed immersion X ãÑ Pn. Given such a morphism, weget geometric information about X using this map, e.g., by studying the fibersf´1(y); pulling back sheaves from Pn; or describing the equations of the image.

The corresponding question for An has already been answered. MorphismsX Ñ An are in one-to-one correspondence with elements of Γ(X,OX)

n. In fancyterms, An represents the contravariant functor

AffSch Ñ Sets

X ÞÑ Γ(X,OX)n

So which functor does projective space represent? Intuitively we would liketo associate to each morphism f : X Ñ Pn to a set of data on X. As we haveseen, there is not so much information in the space of global sections Γ(X,OX)

in the context of projective schemes. However, we do have something canonicalassociated to Pn, namely the invertible sheaf OPn(1). Taking the usual globalsections x0, . . . , xn on OPn(1), we can pull back via f ˚ to get n+ 1 global sectionssi = f ˚xi of the invertible sheaf L = f ˚O(1). Note that there is no point of Pn

where the xi simultaneously vanish. More precisely, for every y P Pn, the stalkOPn(1)y is generated by one of the xi (as an OPn -module). So by the propertiesof the pullback, we see that the same statement holds for L and the sections si

on X. We say that L is globally generated by the sections si.The main result in this chapter is that there is a way to reverse this process. In

other words, from a given invertible sheaf L and n+ 1 global sections si P Γ(X, L)with the above property, we can uniquely reconstruct a morphism f : X Ñ Pn

so that f ˚OPn(1) = L and f ˚xi = si. Thus (L, s0, . . . , sn) is the exactly the datawe are after.

globally generated sheaves 304

16.1 Globally generated sheaves

Definition 16.1 Let X be a scheme and let F be an OX-module. We say Fis globally generated (or generated by global sections) if there is a familyof sections si P F (X), i P I, such that the germs of si generate Fx as anOX,x-module.

Equivalently, F is globally generated if there is a surjection

O IX Ñ F Ñ 0

for some index set I. In particular, any quotient of a globally generated sheaf isalso globally generated.

Let us consider a few examples:Example 16.2 On an affine scheme any quasi-coherent sheaf is globally gener-ated. Indeed, if X = Spec A, F = ĂM, for some A-module M, then picking anypresentation AI Ñ M Ñ 0 for M shows that F is globally generated. K

Example 16.3 Let R be a graded ring generated in degree 1 and set X = Proj R.Then F = O(1) is globally generated. Indeed, the only way F could failto be globally generated is that there is a point x P X for which all sectionss P Γ(X,O(1)) = R1 simultaneously vanish. However, by assumption R1

generates the irrelevant ideal, so this is impossible.On the other hand, if R is not generated in degree 1, then it can happen

that the sheaf O(1) has no global sections at all. This happens for instance forthe weighted projective space P(2, 3, 4) = Proj k[x2, x3, x4] (with deg xi = i). Thesheaf O(´1) is likewise not typically globally generated (unless, say, X is apoint). K

Example 16.4 For a closed subscheme Y Ă X, the structure sheaf i˚OY is globallygenerated (generated by the section ‘1’). On the other hand the correspondingideal sheaf I is typically not globally generated. For Y a closed point in P1

k , thenIY » O(´1) has no global sections. K

Example 16.5 The locally free sheaves from Section 12.9 are both globallygenerated. For instance, the sheaf E from (12.6) admits a surjection On+1 Ñ

E Ñ 0. K

Pullbacks and pushforwardsLet us now consider how global generation behaves under pullbacks and push-forwards.

We focus on the case where F is an invertible sheaf, since these play the keyrole in this chapter. Recall from Chapter 11 that if f : X Ñ Y is a morphism ofschemes, and L is an invertible sheaf on Y, then f ˚L is invertible as well: If L istrivial on U Ă Y, then f ˚L is trivial on f´1(U).

morphisms to projective space 305

We can pull back sections s P Γ(Y, L) as well. This is because we have thecanonical map of sheaves

f 7 : L Ñ f˚( f ˚L).

Evaluating this over the open set U = Y, we see that for a global sections P Γ(Y, L), the pullback f ˚(s) = f 7(Y)(s) is a global section in f ˚L.

This pullback section is especially simple in the case where X and Y areboth affine; f is induced by a ring map φ : A Ñ B; and L = OY (which isalways the case locally). In this case, f ˚OY = OX and the pullback of a sections P A = Γ(Y,OY) is simply φ(s) P B = Γ(X,OX).

Let us fix some notation: For a section s P Γ(X, L) of an invertible sheaf L,we denote its zero set as

V(s) = tx P X | s(x) = 0u,

and the corresponding principal open set

Xs = tx P X | s(x) ‰ 0u.

(Recall that s(x) is the residue class of s in Lx/mxLx.)

Proposition 16.6 Let f : X Ñ Y be a morphism of schemes and let L be aninvertible sheaf on Y. Then

o We have f´1(V(s)) = V( f ˚s) and f´1(Xs) = X f˚s.

o If L is generated by global sections s0, . . . , sn, then f ˚L is generated bythe sections t0 = f ˚s0, . . . , tn = f ˚sn, and X is covered by the open setsXt0 , . . . , Xtn .

Proof: For each of these statements, we may reduce to the case X = Spec B;Y = Spec A and L = OY. In that case (i) follows from the fact that f´1(V(a)) =V(φ(a)), which we have seen several times before.

For (ii), we note that hypothesis gives that the sections s0, . . . , sn are elementsin A that generate the unit ideal. But then clearly the same holds for thepullbacks φ(s0), . . . , φ(sn). o

Remark 16.7 For the pushforward, f˚F is typically not globally generated even whenF is the structure sheaf. For example, if f : P1 Ñ P1 is the map induced by k[u2, v2] Ă

k[u, v], then f˚OP1 » OP1 ‘OP1(´1). The latter sheaf is not globally generated, sinceit has O(´1) as a quotient (which has no global sections).

16.2 Morphisms to projective space

Warm up: The twisted cubicConsider the ring R = k[u3, u2v, uv2, v3] with the grading such that the mono-mials have degree 1. We have seen that Proj R = P1

k (since R is the Veronese


subring k[u, v](3) of k[u, v]). R is isomorphic to the ring

k[x0, x1, x2, x3]/(x21 ´ x0x2, x0x3 ´ x1x2, x2

2 ´ x1x3).

This shows that Proj R embeds as a curve in P3k (the twisted cubic) defined by

three quadrics.If we consider these spaces as a projective varieties, we could say that there

is a morphism f : P1 Ñ P3, given in homogeneous coordinates by

(u : v) ÞÑ (u3 : u2v : uv2 : v3).

However, we need to be a little bit more rigorous here: the u, v are not regu-lar functions (not even rational functions). Note, however, that on D+(u) =

Spec R(u), the ratio t = vu is a regular section in Γ(D+(u),OX). Moreover, the

ring homomorphismφ : k[ x1

x0, x2

x0, x3

x0] Ñ k[t]

x1x0

ÞÑ tx2x0

ÞÑ t2

x3x0

ÞÑ t3

gives a morphism of affine schemes

D+(u)Ñ D+(x0) Ă P3k

Note that this corresponds to the usual parameterization of the affine twistedcubic A1 Ñ A3, t ÞÑ (t, t2, t3). Similarly, on D+(v), the ratio s = u

v definesa morphism D+(x0) Ñ D+(x3). On the overlaps, these maps are compatiblewith the standard gluing construction of P1

k , and so we get finally a morphismP1

k Ñ P3k .

Morphisms and globally generated invertible sheavesThe moral of the previous example is that the morphism is not specified usinga set of regular functions, but rather sections of an invertible sheaf. Using thesame type of gluing argument, we will prove the following general result:

Theorem 16.8 Let X be a scheme over a ring A, and let L be an invertible sheafon X with global sections s0, . . . , sn P Γ(X, L) which generate L. Then there is aunique morphism

f : X Ñ PnA = Proj A[x0, . . . , xn]

so that f ˚xi = si for i = 0, . . . , n.

First an easy lemma:


Lemma 16.9 Let X be a scheme and let L be an invertible sheaf on X. If s P Γ(X, L) isa global section, then there is an isomorphism

φ : OX|Xs Ñ L|Xs

which sends 1 to s.

Proof: The map φ is an isomorphism if and only if it is an isomorphism locally,so we may reduce to the case where X = Spec A and L = OX. In that caseXs = D(s) = Spec As, and s P A is a unit in As, so multiplication by s is anisomorphism As Ñ As. o

Proof (of the theorem): We first prove uniqueness. Let f : X Ñ PnA be a

morphism, and consider the pulled back sections si = f ˚xi for i = 0, . . . , n. Writefor simplicity Xi = Xsi for each i. From Proposition 16.2 we have f´1(D+(xi)) =

Xi for each i, so X is covered by the n + 1 subsets Xi. We can regard themorphism as glued together from the morphisms fi : Xi Ñ D+(xi) = Proj R(xi),where R = A[x0, . . . , xm]. This in turn corresponds to a morphism of A-algebras

f 7 : R(xi) Ñ Γ(Xi,OX).

Note that xi generates O(1) on D+(xi) and xj =xjxi

xi in R(xi) for j = 0, . . . , n.Similarly, pulling back via f 7 gives

sj = f ˚i (xj) = f 7i

(xj

xixi

)= f 7i

(xj

xi

)si

(Here we interpret the fraction xjxi

as a section of Γ(D+(xi),OPnA).) It follows

that from each morphism f : X Ñ PnA, we get n + 1 distinguished sections

s0, . . . , sn, from which we can determine the morphisms fi. Hence f is uniquelydetermined from the data (L, s0, . . . , sn).

To prove existence, we suppose that we are given n + 1 sections s0, . . . , sn of aglobally generated invertible sheaf L, we will construct a morphism to Pn

A, suchthat si is the pullback of xi. As in the above example, we define this morphismon an open cover. Let Xi = Xsi = tx P X|si(x) ‰ 0u. Since the si globallygenerate L, it follows from Lemma 16.5 that the Xi provide a local trivializingcover of L: namely there is an isomorphism ψi : OX|Xi Ñ L|Xi which sends 1to the section si. In particular, if we restrict the global section sj to Xi, we havesj = rijsi for some rij P Γ(Xi,OX). We denote this section rij by sj

si. These define

a map of A-algebras

R(xi) Ñ Γ(Xi,OXi)xj

xiÞÑ

sj

si

By the correspondence between ring homomorphisms and maps into affineschemes, we obtain a morphism of schemes fi : Xi Ñ D+(xi). On Xi X Xk,


the map sends xjxk

=xj/xixk/xi

to sjsk=

sj/sisk/si

. In other words, the following diagramcommutes:

R(xi) Γ(Xj,OX)

R(xixj) Γ(Xi X Xj,OX)

R(xj) Γ(Xj,OX)

That means that the morphisms glue to a morphism f : X Ñ Pn. It is clearthat f ˚O(1) » L and that the xi pull back to the si, since this is true over theprincipal opens D+(xi). o

Abusing notation, we will refer to a morphism φ : X Ñ PnA as given by the

data (L, s0, . . . , sn) and write

X Ñ PnA

x ÞÑ (s0(x) : ¨ ¨ ¨ : sn(x))

One should still keep in mind that the sections si are sections of L, not regularfunctions. In fact, from the above proof, we see that it is the ratios sj/si whichcan be interpretated as regular functions, locally on Xi = tx P X | si(x) ‰ 0u.

We also see that two sets of data (L, s0, . . . , sn), (L, t0, . . . , tn) give rise to thesame morphism f : X Ñ Pn

A if and only there is a section λ P OˆX so that ti = λsi

for each i. Thus morphisms f : X Ñ PnA are in bijective correspondence with the

data (L, s0, . . . , sn) modulo this equivalence relation. Note that when L = OX(D)

for a divisor D, this is exactly the linear system |D| from Chapter 15.Given a scheme X with s0, . . . , sn of a line bundle L, there is a maximal

open subset U such that the sections generate L for all points in U, namelyU =

Ťni=0 Xi. Not assuming that the si globally generate L, we still get a

morphism φ : U Ñ PnA. In other words, φ defines a rational map φ : X 99K Pn

A,which is a morphism when restricted to U.Example 16.10 Let X = P1

k = Proj k[s, t] and L = OP1k(2). Then L is globally

generated by s2, st, t2 and the corresponding morphism

φ : X Ñ P2k

(s : t) ÞÑ (s2 : st : t2)

has image V(x0x2 ´ x21) which is a smooth conic. K

Example 16.11 Cuspidal cubic. Let X = A1k and L = OX. Then, Γ(X, L) = k[t] is

infinite dimensional over k. Choosing the three sections 1, t2, t3, we get a map ofschemes

X Ñ P2k

t ÞÑ (1 : t2 : t3)


whose image in P2 is the cuspidal cubic minus the point at infinity. K

Example 16.12 Pn as a quotient space. Let X = Ank , and L = OX. Then,

Γ(X, L) = k[x0, . . . , xn]. If we take the sections x0, . . . , xn, then they generate Loutside V(x0, . . . , xn). Hence we get a morphism of schemes

Ank zV(x0, . . . , xn)Ñ Pn

k

(x0, . . . , xn) ÞÑ (x0 : ¨ ¨ ¨ : xn)

which is exactly the ‘quotient space’ description of Pn from Exercise 9.9. K

Example 16.13 Projection from a point. Consider the projective space X = PnA and

sections x1, . . . , xn of O(1), then these sections generate O(1) outside the pointp corresponding to I = (x1, . . . , xn) (that is, the closed point p = (1 : 0 : ¨ ¨ ¨ : 0)).The induced morphism Pn

A ´V(I)Ñ Pn´1A is the projection from p. K

Example 16.14 Cremona transformation. Consider the projective space X = P2A

and sections x0, x1, x2 of O(1), then the sections x0x1, x0x2, x1x2 generate O(2)outside V(x0x1, x0x2, x1x2) corresponding to the three points (1 : 0 : 0), (0 : 1 :0), (0 : 0 : 1). The induced rational map P2

A 99K P2A is the Cremona transformation.

K

Example 16.15 The Veronese surface. Consider X = P2, and L = OP2(2). Ifx0, x1, x2 are projective coordinates on X, then the quadratic monomials

x20, x2

1, x22, x0x1, x0x2, x1x2

form a basis for H0(X, L), and generate L at every point. The correspondingmap φ : X Ñ P5 is in fact a closed immersion; the image is the Veronese surface.It is a classical fact that the image is defined by the 2ˆ 2 minors of the matrixu0 u1 u2

u1 u3 u4

u2 u4 u5

K

Example 16.16 The quadric surface. Let us consider again the case Q = P1 ˆP1.Keeping the notation from Section 15.5, we have two divisors, L1 = (0 : 1)ˆP1,L2 = P1 ˆ (0 : 1). Note that each Li is globally generated (being the pullback ofa base point free divisor on P1). The corresponding map is of course the i-thprojection map pi : Q Ñ P1.

If x0, x1 is a basis for Γ(X, L1), and y0, y1 is a basis for Γ(X, L2), we find thatΓ(X, L1 + L2) is spanned by the sections

s0 = x0y0, s1 = x0y1, s2 = x1y0, s3 = x1y1

Moreover, these sections generate OQ(D) everywhere, and so we get a map

Q Ñ P3

This is of course nothing but the Segre embedding; note the quadratic relationbetween the four sections s0s3 ´ s1s2 = 0. K

application: automorphisms ofn

310

16.3 Application: Automorphisms of Pn

If k is a field, then any invertible (n + 1)ˆ (n + 1) matrix m acts on k[x0, . . . , xn]

and thus gives rise to a linear automorphism Pnk Ñ Pn

k . Moreover, two matricesm and m1 determine the same automorphism if and only if m = λm1 for somenon-zero scalar λ P k˚. So we are led to consider the projective linear group

PGLn(k) = GLn(k)/k˚

We will now prove that all automorphisms of Pnk are given by linear transforma-

tions.

Theorem 16.17 Autk(Pn) = PGLn(k).

Proof: The above shows that there is an injective map from the righthand sideto the left. To show the reverse inclusion, let φ : Pn

k Ñ Pnk be any automorphism.

Then we get an induced map

φ˚ : Pic(Pn)Ñ Pic(Pn)

which must also be an isomorphism. Since Pic(Pn) = Z, we must have eitherφ˚OPn(1) = OPn(1) or φ˚OPn(1) = O(´1). The latter case is impossible,since φ˚(OPn(1)) has a lot of global sections, whereas OPn(´1) has none. Soφ˚(OPn(1)) = OPn(1). In particular, taking global sections φ˚ gives a map

Γ(Pn,OPn(1))Ñ Γ(Pn,OPn(1)),

which is a isomorphism of k-vector spaces. However, we may choose tx0, . . . , xnu

as a basis for Γ(Pn,OPn(1)), and so in this basis φ˚ gives rise to an invertible (n+

1)ˆ (n + 1)-matrix m. By construction m induces the same linear transformationPn

k Ñ Pnk as φ, and so φ comes from an element of PGLn(k). o

16.4 Projective space as a functor

We now come to the question posed in the introduction of this chapter, namelywhich functor does projective space represent?. Recall that we defined a functorF : Schop Ñ Sets to be representable if there is a scheme X and an isomorphism offunctors Φ : hX » F; i.e., for each S P Schop a bijection Φ(S) : Hom(S, X)Ñ F(S).So intuitively we would like to associate to each morphism f : X Ñ Pn to a setof data on X. As we have seen, there is not so much information Γ(X,OX) inthe context of projective schemes. However, we do have something canonicalassociated to Pn, namely sections of O(1).

As a first question, we should ask what F(Spec k) should correspond to.Motivated by the very definition of projective space over a field, we would liketo say that the value of F on closed points should correspond to lines in a vectorspace. More precisely, morphisms Spec k Ñ Pn should be in correspondence

projective space as a functor 311

with lines l Ă kn+1 through the origin. This assignment should also be con-tinuous, e.g., for each open U Ď Pn, we should have a sub-line bundle LU ofthe trivial bundle Pn

k Ân+1k . By the correspondence between vector bundles

and invertible sheaves, this is equivalent to a subsheaf L Ď On+1U which is an

invertible sheaf.This motivates the following definition:

Definition 16.18 Define a functor F : Schop Ñ Sets by defining for a schemeX

F(X) =!

invertible subsheaves L Ă On+1X

)

=!

invertible sheaf quotients On+1X Ñ L Ñ 0

)

/ „

where the „ says that quotients On+1X Ñ L Ñ 0, On+1

X Ñ M Ñ 0 areequivalent if theyhave the same kernel.

Note that on projective space Pn = Proj Z[x0, . . . , xn], we do have such aquotient

On+1Pn Ñ OPn(1)Ñ 0.

Indeed, on a D+( f ) we can define it by sending an element (t0, . . . , tm) P R( f ) tox0t0 + ¨ ¨ ¨+ xntn P R(1)( f ). Of course this quotient is not canonical, as it dependson the choice x0, . . . , xn.

Theorem 16.19 The functor F is represented by the scheme Pn.

Proof: We need to define natural transformations of the two functors hPn to F.For a scheme X, we define

Φ(X) : Hom(X, Pn)Ñ F(X)

by sending a morphism f : X Ñ Pn to the (equivalence class of the) quotient

On+1X Ñ f ˚O(1)Ñ 0

which is the pullback of On+1 Ñ O(1)Ñ 0 on Pn. It is clear that this assignmentis functorial, i.e., it gives a natural transformation. Moreover, by definition Φsends the identity map idPn to the equivalence class of the sequence above.

To prove the theorem, we need to construct an inverse Ψ to Φ. In otherwords, to each quotient On+1

X Ñ L Ñ 0, we need to produce a morphism ofschemes f : X Ñ Pn so that On+1

Pn Ñ OPn(1) Ñ 0 pulls back to On+1X Ñ L Ñ 0.

However, from the quotient On+1X Ñ L Ñ we obtain n + 1 sections s0, . . . , sn

by taking the n + 1 maps OX Ñ On+1X Ñ L (where the first map corresponds

to a ‘coordinate inclusion’). We then define Ψ by associating On+1X Ñ L Ñ to

the morphism f : X Ñ Pn P Hom(S, Pn). One can check that two choices ofquotients induce the same map f : X Ñ Pn, so this is well-defined.

projective space as a functor 312

Finally, in the construction of the morphisms in Theorem 16.4, we haveL = f ˚O(1) and si = f ˚xi, and so Ψ is the inverse to Φ. o

Why should one care about such a statement? There are several good reasons,but perhaps the most basic is that it tells us how to think about points of thescheme projective space as corresponding to something geometric. Indeed, apriori, the one thing we know about Pn

Z is that it is obtained by gluing togethern + 1 coordinate affine n-spaces. The theorem above shows that in fact it doeswhat it is supposed to do; the k-points Pn(k) of Pn correspond to exactly to thelines in the vector space kn+1 for any field k.

The case of P1 is particularly vivid. The functor of points of A1 showsthat a morphism X Ñ A1 is equivalent to giving an element of Γ(X,OX). Inless precise terms this is what we think of as a ‘regular function’ on X. Thecorresponding statement for P1 is the following: A map X Ñ P1 corresponds toa section s of a line bundle L on X, or in other words a ‘meromorphic function’,which is only required to be regular on Xs.

GeneralizationsIf we modify of the functor of points of Pn, we obtain other interesting examplesof schemes parameterizing geometric objects. The following examples are onlymeant as basic illustrations of this point - they will not appear later in the notes.Example 16.20 The Grassmannian. Consider the functor

F(S) = trank r locally free quotients On Ñ Q Ñ 0u/ „

where two quotients again are defined to be equivalent if they have the samekernel. Then F is represented by a scheme, known as the Grassmannian Gr(r, n).

This is scheme can realized as the projective scheme in P(n+r

r )´1Z = Proj Z[pi1,...,ir ]

where 0 ď i1 ă i2 ă . . . ir ď n. Explicitly, Gr(r, n) is given by the Plücker equations

r+1ÿ

k=0

(´1)k pi1,i2,...,ir´1,jk ¨ pi1,..., jk ,...,jr+1= 0

(over all sequences of indices i1, . . . , ir and j1, . . . , jr+1). We can define the map

Gr(k, n)Ñ P(n+r

r )´1Z using the Yoneda lemma, by giving a natural transformation

between the functors. If S is a scheme, this transformation takes an S-valuedpoint of Gr(k, n), that is, a quotient On Ñ Q Ñ 0, to ^rOn Ñ ^rQ Ñ 0, which

since ^rQ has rank 1, defines a point in P(n+r

r )´1Z (S).

Proving that this scheme actually represents Gr(k, n) is similar to what wedid for projective n-space, although it is slightly more involved, due to thePlücker equations. K

Example 16.21 Projective bundles. Let E be a locally free sheaf on a scheme X.Consider the functor on Sch/X given by

F(h : Y Ñ X) = tinvertible sheaf quotients h˚E Ñ L Ñ 0u

projective embeddings* 313

Then F is represented by a scheme P(E )Ñ X over X called the projectivizationof E . One can think of closed points of this scheme as hyperplanes in the fibersof E . K

Example 16.22 Proj of an OX-module. Let X be a scheme and let F be a quasi-coherent sheaf of OX-modules. Consider the functor on Sch/X given by

F(h : Y Ñ X) = tinvertible sheaf quotients h˚F Ñ L Ñ 0u

Then F is represented by a scheme Proj(F ). Many geometric constructions canbe formulated as such schemes. For instance, if I is a sheaf of ideals on X, thenProj(F ) can be identified with the blow-up of X along I (see Hartshorne ([?])II.7). K

16.5 Projective embeddings*

We have seen how morphisms X Ñ Pn corresponds to invertible sheaves L plusn + 1 global sections s0, . . . , sn that generate it. Given this it is natural to askwhich data (L, si) that corresponds to special types of morphisms, e.g., closedimmersions. The following criterion tells us how to check this locally:

Proposition 16.23 Let φ : X Ñ PnA be a morphism corresponding to the data

(L, s0, . . . , sn). Then φ is a closed immersion if and only if for each i = 0, . . . , n,Xi = Xsi is affine, and the homomorphism A[y0, . . . , yn]Ñ Γ(Xi,OX) yj ÞÑ

sjsi

is surjective.

Proof: ñ: If φ is a closed immersion, we may view X as a closed subschemeof Pn

A (so it is given by some ideal in A[y0, . . . , yn]). Note that in this caseXi = X X D+(xi) is a closed subscheme of the affine scheme D+(xi). By ourresults on closed subschemes of affine schemes, Xi corresponds to an ideal inA[y0, . . . , yn](x0), and the map A[y0, . . . , yn]Ñ Γ(Xi,OX) yj ÞÑ

sjsi

is surjective.ð: Since Xi is affine, and the map to the right is surjective, we see that Xi

corresponds to a closed subscheme of D+(xi). As X is glued together by the Xi

we see that X is a closed subscheme of PnA as well. o

In practice however, this condition is not so easy to verify (that is, not mucheasier than checking directly that the induced map is an embedding). To get abetter criterion we first introduce some more notation.

projective embeddings* 314

Definition 16.24 Let X be a scheme over a field k.

i) A linear series V is a subspace V Ď Γ(X, L) where L is an invertiblesheaf on X.

ii) A linear series V separates points if for any two points x, y P Xthere is a section s P V such that s(x) = 0 but s(y) ‰ 0

iii) A linear series V separates tangent vectors if for any x P X the setts P V|s(x) ‰ 0u spans mxLx/m2

x.

The definitions here come from the following theorem

Theorem 16.25 Let X be a projective scheme over an algebraically closed field kand let φ : X Ñ Pn

k be a morphism over k corresponding to (L, s0, . . . , sn). Thenφ is a closed immersion if and only if V = spants0, . . . , snu separates points andtangent vectors.

We will not prove this theorem here, but let us at least say a few words aboutit.

If φ is a closed immersion, we can consider X Ď Pnk as closed subscheme,

and the si are simply restrictions of the xi to X. Now the theorem is essen-tially a consequence of the easy geometric fact that the sections x0, . . . , xn ofOPn

k(1)separate points and tangent vectors on Pn

k . Indeed, we can pick a linearform l = a0x0 + ¨ ¨ ¨+ anxn such that the hyperplane H = V(l) meets x but notnot y (here we are using that k is algebraically closed!). l|X is a linear combina-tion of the si, and so V separates points. Similarly, linear forms on Pn separatetangent vectors, so they also separate tangent vectors on the subscheme X.

The hard part is showing that the converse holds. If V is a linear seriesseparating points and tangent vectors, then the morphism φ is injective (sinceyou can use linear forms for distinguish the images of two points). The mainingredient we need is that φ is proper, which implies that φ is closed, andfurthermore that φ gives a homeomorphism onto a closed set in Pn. Giventhis, the rest of the proof is relatively straightforward: We need only check thatOPn Ñ φ˚OX is surjective, which can be done on stalks.

What makes this theorem more powerful than the previous is that theconditions on sections to separate points and tangent vectors can be turned into statements about restriction maps of certain sheaves being surjective. Thelatter task is something we can approach using the machinery of cohomology.And indeed, using the long exact sequence in cohomology we can turn this intosimple, computable numerical criteria that guarantee that the correspondingmap is an embedding.

ample invertible sheaves and serre’s theorems 315

16.6 Ample invertible sheaves and Serre’s theorems

The prototype of an invertible sheaf is the sheaf O(1) on X = Proj R, which isglobally generated if R is generated by elements x0, . . . , xn of degree 1. As wehave seen, this corresponds to a morphism f : X Ñ Pn, with the property thatf ˚OPn(1) = OProj R(1). In the case where f is a closed immersion, we can regardX as a subscheme of Pn, and O(1) is simply the restriction OPn(1)|X. Thisrestricted invertible sheaf has a special property: R and hence X is completelyrecovered by the global sections of it and the polynomial relations between them.This motivates the following definition:

Definition 16.26 Let X be a scheme over S. An invertible sheaf L on X is saidto be very ample over S if there is a closed immersion i : X Ñ Pn

S such thatL » i˚OPn

S(1). L is ample if LbN is very ample for some N ą 0.

Theorem 16.27 (Serre) Let X Ď PnA be a projective scheme over a noetherian

ring A and let L be an ample invertible sheaf. Let F be a coherent sheaf on X.Then there is an integer m0 such that F b Lm is globally generated (by a finiteset of global sections) for all m ě m0.

Proof: By replacing L with a multiple, we may assume that L is very ampleand even that L = i˚O(1) for a projective embedding i : X Ñ Pn. Since X isnoetherian, i is finite and F is coherent, we have that i˚F is coherent on Pn.Moreover, (i˚F )(m) = i˚(F b Lm), so F b Lm is globally generated if and onlyif (i˚F )(m) is. Hence we may assume that X = Pn

A.Write Pn

A = Proj R where R = Proj A[x0, . . . , xn]. We can cover X by open setsD+(xi) where i = 0, . . . , n, such that F |D(xi) =

ĂMi for some finitely generatedR(xi)-module Mi. For each i we can choose finitely many elements sij generatingMi. Regarding sij as sections of the sheaf Mi, we see that there are integers mij

such that xmiji sij extend to global sections of F (mij). Take m = maxij mij, then

for each i, j we have a section tij P Γ(X,F (m)).We claim that the global sections tij generate F (m) locally. It is sufficient

to prove this on the opens D+(xi) over which F (m) is isomorphic to the R(xi)-module Mi(m) » Mi bR R(m), which is isomorphic to xm

i Mi as a graded R(xi)-module. Since sij generates Mi, we see that xm

i sij generate F (m)|D+(xi).This shows that F (m0) is globally generated for the chosen m0 = m above.

Then also F (m) is globally generated for m ě m0, by multiplying the generatingsections of F (m0) by the various monomials in the xi. o

In particular, we see that any coherent sheaf can be written as a quotient of adirect sum of invertible sheaves of the form O(´m): Take any F (m) which isglobally generated; then there is a surjection O I Ñ F (m) Ñ 0 (with I finite!),


which becomesO(´m)I Ñ F Ñ 0 (16.1)

after tensoring by O(ń).

Theorem 16.28 (Serre) Let X = Proj R be a noetherian projective schemeover Spec A, where A is a noetherian ring, and let F be a coherent sheaf on X.Then:

i) The cohomology groupsHi(X,F )

are finitely generated A-modules for every i ě 0.ii) There exists an n0 ą 0 such that

Hi(X,F (n)) = 0.

for all n ě n0 and i ą 0.

Proof: As in the above proof, we immediately reduce to the case whereX = Pn

A.Note that both of the conclusions hold for the twisting sheaves OX(n). To

prove it for any coherent F , take a quotient of the form (16.1) and let K be thekernel, so that we have an exact sequence

0 Ñ K Ñ E Ñ F Ñ 0

where E = O(ń)I , with I finite. Note that K is again coherent.(i): Take the long exact sequenence of cohomology to get

Hi(X, E )Ñ Hi(X,F )Ñ Hi+1(X, K )

We can now prove the theorem by downwards induction on i: Hi+1(X, K )

and Hi(X, E ) are both finitely generated, and hence so is Hi(X,F ), since A isnoetherian.

(ii): Twist the above sequence by OX(m) and take the long exact sequence incohomology to get

Hi(X, E (m))Ñ Hi(X,F (m))Ñ Hi+1(X, K(m))

By downward induction on i, and the fact that Hi(X, E (m)) for any m, we findthat Hi(X,F (m)) = 0. o

Euler characteristicLet k be a field and consider a coherent sheaf F = ĂM on a projective schemeX Ă Pn

k . By Serre’s theorem, we know that that the cohomology groups Hi(X,F )

are finite as k-vector spaces. In particular, we can ask about their dimensions. It


turns out that the alternating sum of these dimensions has very good functorialproperties, so we make the following definition:

Definition 16.29 Let X be a projective scheme of dimension n over a field k.We define the Euler characteristic of F as

χ(F ) =ÿ

kě0

(´1)k dimk Hk(X,F )

This sum is well-defined, as there are only finitely many non-zero cohomol-ogy groups appearing on the right hand side.

Proposition 16.30 The Euler characteristic χ is additive on exact sequences,i.e., if 0 Ñ F 1 Ñ F Ñ F2 Ñ 0 is an exact sequence of coherent sheaves, then

χ(F ) = χ(F 1) + χ(F2).

Example 16.31 Let X = Pnk and F = O(d) for d ě 0. Then dimk H0(Pn

k ,F ) =

(n+dn ) and all of the higher cohomology groups are zero. In the case when

d ă 0, only Hn(X,F ) can be non-zero, and the rank is given by (n+dn ), where

we use the formula (xd) = x(x´ 1) ¨ ¨ ¨ (x´ d + 1)/d! for any x P R. In particular,

χ(OPnk(d)) = (n+d

d ) is a polynomial in d of degree n. K

More generally on Pnk we can take any coherent sheaf F and a free resolution

of it:0 Ñ En Ñ ¨ ¨ ¨ Ñ E1 Ñ E0 Ñ F Ñ 0

where the Ei are direct sums of invertible sheaves of the form OX(d). If wetensor this sequence by O(m), we get

0 Ñ En(m)Ñ ¨ ¨ ¨ Ñ E1(m)Ñ E0(m)Ñ F (m)Ñ 0

We claim that also χ(F (m)) is a polynomial in m. Note that this is true forthe terms χ(Ei(m)). Then since the Euler characteristic is additive on exactsequences, χ(F (m)) is also a polynomial in m. Moreover, again by Serre’stheorem, we have Hi(X,F (m)) = 0 for m " 0 and i ą 0, and so χ(F (m)) =

H0(F (m)) for m large.If we start with a coherent sheaf F on a X Ă Pn

k , applying the previousdiscussion to i˚F on Pn

k gives the following:

Corollary 16.32 Let X Ă Pn be a projective scheme and let O(1) be the very ampleinvertible sheaf. Then the function

PF (m) = χ(F (m))

is a polynomial in m, and for large m, H0(X,F (m)) = PF (m).


This polynomial is called the Hilbert polynomial of F . When F = ĂM for a gradedmodule M, this coincides with the usual Hilbert polynomial of M as defined incommutative algebra.

Exercises(16.1) We say that scheme X has the resolution property if any coherent sheaf is aquotient of a locally free sheaf.

i) Show that projective schemes of finite type over a field have the resolutionproperty.

Let X = A2 YU A2, where U = A2 ´ 0, be the affine 2-space with twoorigins. Show that X does not admit the resolution property, by the followingsteps:

Let p : A2 Ñ X and q : A2 Ñ X be the two inclusions, and let E be a locallyfree sheaf on X.

ii) Explain why p˚E and q˚E are free sheaves on A2.iii) p˚E and q˚E become isomorphic on U; use Hartog’s Lemma to show

that p˚E and q˚E are in fact isomorphic via the identity map. Conclude that E

is trivial.iv) Show that the exist sheaves on X which are not quotients of locally free

sheaves.v) Show that the affine line X = A1YU A1, in contrast, satisfies the resolution

property.M

Chapter 17

Differentials

So far we have defined schemes and surveyed a few of their basic properties (e.g.how to study sheaves on them). In this chapter, we introduce tangent spaces andKähler differentials, which allow us in some sense to do calculus on schemes. Thisin turn will allow us to define the most important sheaves in algebraic geometry,namely, the cotangent sheaf, the tangent sheaf, and the sheaves of n-forms.

Differentials appear prominently throughout many areas of mathematics,multivariable analysis, manifolds and differential geometry to mention a few.In algebraic geometry they are introduced algebraically using their formalproperties and are usually referred to as Kähler differentials after the Germanmathematician Erich Kähler (1906–2000).

Introduction

The tangent spaces of general schemes will necessarily be of a rather abstractnature, and as a motivation we first recall the concrete situation for affinevarieties embedded in some affine space An

k over a field k.Consider such an affine variety XĎAn

k , say X = V(I) where I = ( f1, . . . , fr).For a k-point p P X, we define the (embedded) tangent space of X at p as thesubspace of vectors v = (v1, . . . , vn) P kn satisfying the linear equations

nÿ

i=1

B f /Bxi(p) ¨ vi = 0 for i = 1, . . . , r. (17.1)

An equivalent definition, without reference to a specific generating set of I, isthe following:

nÿ

i=1

B f /Bxi(p) ¨ vi = 0 for all f P I. (17.2)

The two definitions are equivalent, for if f =ř

j gj f j, the chain rule gives

B f /Bxi(p) =ř

j f j(p)Bgj/Bxi(p) + gj(p)B f j/Bxi(p) =ř

igj(p)B f j/Bxi(p),

320

where the last equality holds since the fi’s vanish at p. Vectors satisfying (17.1)therefore also satisfy (17.2), and the reverse implication is trivial. In particularthe tangent space as defined in (17.1), is independent of the chosen set ofgenerators for the ideal.

Note that TpX by definition is a sub-k-vector space of kn; it is the null space ofthe Jacobian matrix

(Jacobi-matrisen)Jacobian matrix

J( f1, . . . , fr) =(B fi/Bxj(p)

), (17.3)

where 1 ď i ď r and 1 ď j ď n. The dimension of TpX is therefore given by

dim TpX = n´ rank J( f1, . . . , fr). (17.4)

There is an intrinsic description of the tangent space TpX, which does notrely on any specific embedding of X, and which will be the inspiration for thegeneral definition.

Suppose for simplicity that p = (0, . . . , 0) is the origin (we may alwaysarrange this by a linear change of coordinates), and write M = (x1, . . . , xn) Ă

k[x1, . . . , xn] for the maximal ideal at p. For a polynomial f P k[x1, . . . , xn], weconsider its linearization at p, given by

D f =nÿ

i=1

B f /Bxi(0)xi.

This is just the linear part of the Taylor expansion at p. Note that the coordinatesx1, . . . , xn give a basis for the dual space (kn)_ = Homk(kn, k). Hence we mayview D f as a linear functional on kn, and in this way we get a k-linear map

D : MÑ (kn)_.

It is clear that D is surjective, since D(xi) = xi. A polynomial f lies in kernel ofD precisely when all terms are of degree at least two, or phrased differently, thekernel of f equals M2. Hence D induces an isomorphism of k-vector spaces

M/M2 » (kn)_.

Returning to the variety X and the tangent space TpX, we take the dual of theinclusion TpX Ă kn, to obtain a surjection

(kn)_ Ñ (TpX)_.

zariski tangent spaces 321

Concretely, this map is given by restricting a linear functional on kn to thesubspace TpX. The composition

θ : M/M2 Ñ (kn)_ Ñ (TpX)_

is also surjective.We claim that Ker θ = M2 + I. Indeed, note that f P Ker θ if and only

if D f restricts to 0 on TpX. This happens if and only if D f = Dg for someg P I (since TpX is the zero locus of Dg for all g P I); that is, if and only iff ´ g P Ker D = M2, or equivalently, f PM2 + I.

It follows that we have isomorphisms of k-vector spaces

(TpX)_ »M/(M2 + I) » m/m2. (17.5)

where m Ă OX,p is the maximal ideal. Taking duals, we now have:

Proposition 17.1 There is a natural isomorphism

TpX » Homk(m/m2, k). (17.6)

17.1 Zariski tangent spaces

Motivated by the above discussion for embedded affine varieties, we proceed togive the general definition of tangent spaces for schemes. It was introduced forvarieties by Oscar Zariski in a fundamental paper ([?]) in 1947, and bears thename the

the Zariski tangentspace (Zariski-tangentrommet)

Zariski tangent space.Let X be a scheme and let x P X be a point. We consider the local ring

A = OX,x with maximal ideal mx. The quotient mx/m2x is in a natural way a

vector space over the residue class field k(x) = A/mx.

Definition 17.2 The Zariski tangent space TxX to X at x is the dual vectorspace of mx/m2

x. That is,

TxX = Homk(x)(mx/m2x, k(x)).

The space mx/m2x is called the Zariski cotangent space of X at x. An element

of TxX is called a tangent vector; it is a linear functional mx/m2x Ñ k(x).

The cotangent space is functorial in the following sense. Let f : X Ñ Y bea morphism and let y = f (x). Part of the structure of the morphism is a ringhomomorphism f 7 : OY Ñ f˚OX. It localizes to a homomorphism of local ringsf 7 : OY,y Ñ OX,x which takes the maximal ideal into the maximal ideal, andbeing a ring map, sends m2

y into m2x. Therefore it induces an additive map

f 7x : my/m2y Ñ mx/m2

x.


Moreover, for each morphism g composable with f one has

(g ˝ f )7x = f 7x ˝ g7f (x)

since (g ˝ f )7 = f 7 ˝ g7. The map f 7x is, however, just a map of k(y)-vector spaces.In general, there is no way to make my/m2

y a k(x)-vector space, and for thisreason the tangent spaces are not functorial in general; the required duals willbe with respect to different fields.

One exception is when X and Y are schemes over some field k, and x andy both are k-points. Then k(x) = k(y) = k, and we are permitted to take dualsto get a map d f : TxX Ñ TyY. Once the tangent-maps are defined, they behavefunctorially:

d(g ˝ f )x = dgy ˝ d fx

when g : Y Ñ Z is a map of k-schemes and g(y) is a k-point.Why is this?This onlyworks if dimk(y) TyYis finite; this subtlepoint is another reasonwhy the cotangentspace m/m2 ispreferable to thetangent space.

Zariski tangent spaces and the ring of dual numbersLet k be a field. The ring k[ε]/(ε2) is called the The ring of dual

numbers (ringen avduale tall)

ring of dual numbers over k. Weshall with a slightly abusive notation write it as k[ε] tacitly understanding thatε2 = 0. The spectrum of k[ε] is a very simple scheme: its underlying topologicalspace is a single point. However, the non-reduced structure on Spec k[ε] showsthat it is more interesting than Spec k. We picture it as a point p with a vector‘sticking out of it’.

There are other interesting tiny algebras related to k[ε]. If V any vector spaceover k, one may form the ‘infinitesimal’ k-algebra DV = k‘V where V is as amaximal ideal with square zero; that is, the multiplication is (a + w) ¨ (b + v) =ab + (aw + bv). The pertinent property of DV is that k-algebra homomorphismsDV Ñ k[ε] correspond bijectively to linear functionals on V; in other words,there is an isomorphism

HomAlgk(DV , k[ε]) » Homk(V, k).


Indeed, if α : DV Ñ k[ε] is given, the restriction α|V is k-linear and takes valuesin (ε) = k. For the inverse map, if α : V Ñ k is a given functional, the assignmenta + v ÞÑ a + α(v)ε defines a k-algebra map.

Proposition 17.3 Let X be a scheme over k. To give a k-morphismSpec(k[ε]) Ñ X is equivalent to giving a k-rational point x P X (meaningthat k(x) = k), and an element of TxX.

Proof: Fix a k-point x of X. Every map Spec k[ε]Ñ X that sends p to x, mustfactor through each open affine neighbourhood of x, and we may as well assumethat X is affine, say X = Spec A. Let m = mx. A homomorphism α : A Ñ k[ε]corresponds to a morphism Spec k[ε]Ñ X that sends p to x, precisely when thediagram* ˚where by abuse of

language x and pdenote the mapscorresponding to thek-points x and p.

A k[ε]

k

α

xp

commutes. Such maps factor in a unique manner through the canonical mapA Ñ A/m2 (since α(m)Ď (ε) and ε2 = 0). Now, the reduction map A/m2 Ñ

A/m = k splits as an algebra homomorphism, the structure map k Ñ A/m2

being a section, and A/m2 decomposes as an k-algebra into A/m2 = k‘ (m/m2);in other words, A/m2 = Dm/m2 in the terminology above. It follows that

HomAlgk(A, k[ε]) » HomAlgk(A/m2, k[ε]) » Homk(m/m2, k),

and we are through. o

Exercise 17.1 (Tangent space of a functor.) It is a remarkable fact that one may evendefine tangent spaces for many (contravariant) functors on the category Sch/kthat are not (a priori) representable. This holds true for most moduli functors,and one may e.g. compute dimensions and check regularity without knowingthe functor is representable or without having a grasp on the representing object.The actual condition is that the functor commute with fibered products.

So let F : Schk Ñ Sets be such a contravariant functor, and fix a k-pointof F; that is, an element p P F(Spec k). Denote by F0(Spec k[ε]) the subset ofF(Spec k[ε]) of elements that maps to p under the map F(η) : F(Spec k[ε]) ÑF(Spec k) induced by the structure map η : k Ñ k[ε]. The aim is to equipF0(Spec k[ε]) with a natural structure of a vector space over k.

a) Show that diagonal of Spec k[ε] induce a map

F0(Spec k[ε])ˆ F0(Spec k[ε])Ñ F0(Spec k[ε])

that is an abelian group law.

b) For each α P k let φα : k[ε] Ñ k[ε] be given as φα(a + bε) = a + αbε. Showthat φα is an k-algebra homomorphism. Let α act on F0(Spec k[ε]) through

derivations and kähler differentials 324

F(φα). Show that this action together with the addition from a) gives a vectorspace structure on F0.

M

Exercise 17.2 Let V and W be two vector spaces over k. Show that there is afunctorial isomorphism HomAlgk(DV , DW) » Homk(V, W). M

17.2 Derivations and Kähler differentials

So far we have considered a scheme X and a point x P X, and the cotangentspace m/m2 at x. We would like to globalize this construction, and instead offixing x, consider all points at once. That is, we would like to form a sheaf onX with stalks m/m2 at each point. This is what will be the cotangent sheaf on X.As usual, the sheaf will be defined first for affine schemes, by some algebraicconstruction, and a gluing process will cover general schemes X.

DerivationsWe will work over a base ring A, and B will be an A-algebra. We will also needa B-module M. The geometric picture to have in mind is that A = k, where k isa field, and X = Spec B Ñ Spec k a morphism.

Definition 17.4 An A-derivation (from B with values in M) is an A-linearmap D : B Ñ M satisfying the product rule, also called the Leibniz rule:

D(bb1) = bD(b1) + b1D(b).

Given that the product rule holds, it is easy to see that D is A-linear if andonly if it vanishes on all elements of the form a ¨ 1 with a P A; indeed, if D isA-linear, we have D(a ¨ 1) = aD(1) = 0 since D(1) = 0, which follows fromthe product rue applied to 12 = 1. If D vanishes on A, the product rule givesD(ab) = aD(b) + bD(a) = aD(b). We may therefore think of the elements in Bof the form a ¨ 1 as ‘constants’; note however, that a derivation also can vanishon other elements in B (a stupid example is the zero map, which is a derivation.For a more constructive example see Example 6 below).Example 17.5 The map of the polynomial ring B = k[x] to itself which is given byP(t) ÞÑ P1(t), is a k-derivation More generally, the partial differential operatorsB/Bx1, . . . , B/Bxn, as well as their k-linear combinations, are k-derivations on thepolynomial ring k[x1, . . . , xn]. K

A straightforward induction shows that the good old rules from calculushave analogues in the abstract situation: it holds true that D(bn) = nbn´1D(b)and, in case b is invertible in B, that D(1/b) = ´D(b)/b2. Moreover, if P(t) is apolynomial in A[t], one has the chain rule D(P(b)) = P1(t)D(b), where P1(t) isthe formal derivative defined as P1(t) =

ř

i iaiti´1 when P(t) =ř

i aiti.


The set of A-derivations D : B Ñ M is usually denoted by DerA(B, M). Thisset inherits a B-module structure from M, and it is as such naturally a submoduleof HomA(B, M). This gives rise to a covariant functor DerA(B,´) from ModB toitself. More precisely, if φ : M Ñ M1 is a B-linear map, we can map a derivationD P DerA(B, M) to φ ˝D : B Ñ M1, which is in turn an A-derivation of B withvalues in M1.

The set of derivations DerA(B, M) is also functorial in the base ring A and theA-algebra B; in both cases it is contravariant. If A Ñ A1 is a ring homomorphism,any A1-derivation B Ñ M is in turn an A-derivation. We therefore obtain aninclusion DerA1(B, M)Ď DerA(B, M).

The module of Kähler differentialsThe covariant functor DerA(B,´) on the category of B-modules is representable.This simply means that there exists a distinguished B-module ΩB/A and anisomorphism of functors

DerA(B,´) » HomB(ΩB/A,´). (17.7)

In more down-to-earth terms, this condition is equivalent to there being auniversal derivation* ˚The ring A is an

essential part of thestructure, but for thesake of a practicalnotation is not shown;when it is necessary toemphasize the basering, the notation willbe dB/A

dB : B Ñ ΩB/A that has the following property: For anyA-derivation D : B Ñ M there exists a unique B-module homomorphismα : ΩB/A Ñ M such that D = α ˝ dB. In terms of diagrams, we have

B ΩB/A

M.

dB

D α

The slogan is: each derivation is the pushout of the universal derivation dB bysome B-linear map.

To see directly why such a module exists, we can construct it via generatorsand relations. For each element b P B introduce a symbol db and consider thefree B-module G =

À

bPB Bdb they generate. Inside G we have the submoduleH generated by all expressions of one of the types

d(b + b1)´ db´ db1, or d(bb1)´ bdb1 ´ b1db, or da

for b, b1 P B and a P A. We then define ΩB/A = G/H, and the map dB : B ÑΩB/A is given as dB(b) = db. It is well-defined as a group homomorphism sinceany Z-linear relation among the db’s maps to zero in G/H by the imposedadditive constraint, and it is a derivation because all relations d(bb1) = bdb1 +b1db are forced to hold in G/H. Finally, it will be A-linear since da = 0 in G/H.

It is not hard to see that this module indeed satisfies the universal propertyabove: given an A-derivation D : B Ñ M, we define the B-homomorphism


α : ΩB/A Ñ M by α(db) = D(b) (which is well-defined precisely because D is aderivation!).

Definition 17.6 The elements of the module ΩB/A are called the Kähler dif-ferentials, or simply differentials of B over A.

Example 17.7 Change of constants. To any homomorphism of rings ρ : A Ñ A1

corresponds the natural inclusion DerA1(B, M)Ď DerA(B, M), which via theisomorphism (17.7) induces a surjective B-linear map β : ΩB/A Ñ ΩB/A1 . It isjust the B-linear map that arises from dB/A1 ˝ ρ by the universal property of dB/A,moreover, it renders the diagram below commutative

ΩB/A ΩB/A1

A A1.

β

dB/A

ρ

dB/A1

In terms of the generating sets in the construction above, the map β simplysends db to db; note that da1 ÞÑ 0 for all a1 P B coming from A1. K

Example 17.8 Let B = k[t] be the polynomial ring over the field k. ThenΩB/k is a free module over B generated by dt, i.e. ΩB/k = B ¨ dt. This isbest seen by verifying the universal property: the map dB : B Ñ B ¨ dt withdB( f (t))) = f 1(t)dt is a k-derivation, any other derivation D : B Ñ M complyto the same rule D( f (t)) = f 1(t)D(t); hence the corresponding B-linear map α

may be given as α(bdt) = bD(t). K

More generally we have:

Proposition 17.9 Let A be any ring and let B = A[x1, . . . , xn]. Then ΩB/A

is the free B-module generated by dx1, . . . , dxn and the universal derivation isgiven by

dB f =ÿ

(B f /Bxi)dxi.

Proof: The universal property follows from the general chain rule: for anyA-derivation D : B Ñ M into a B-module M, the formula

D( f ) =ÿ

i

(B f /Bxi)D(xi). (17.8)

holds true. Indeed, an easy induction, using the product rule, shows it to betrue when f is a monomial, and then A-linearity finishes he story. The B-linearmap α :

À

i Bdxi Ñ M which sends each basis element dxi to D(xi), will be thewanted factoring map; by the general chain rule (17.8), it satisfies the equalityD = α ˝ dB. o


ExamplesHere are some more explicit calculations of ΩB/A:(17.10) Localization. If B = S´1A is a localization of A, then ΩB/A = 0. Indeed,take b P B, and choose s P S so that sb P A. Then sdBb = dB(sb) = 0, whichimplies that dBb = 0 since s is invertible in B.(17.11) Surjections. More generally, if φ : A Ñ B is surjective, then ΩB/A = 0,because if b = φ(a), then dBb = a ¨ dB(1) = 0 in ΩB/A.(17.12) Separable field extensions. Let K = k(a) be a separable field extensionand let P(t) be the minimal polynomial of a. For any k-derivation D : K Ñ K itholds that 0 = D(0) = D(P(a)) = P1(a)D(a). Hence D(a) = 0 since P1(a) ‰ 0the element a being separable over k. The product rule implies that D(an) =

nan´1D(a) = 0 for each natural number n, and since the powers an generate Kas a vector space over k, it follows that D = 0.(17.13) Inseparable field extensions. Contrary to the separable ones, inseparableextensions have non-trivial derivations. Let us consider the simplest case whenK is obtained by adjoining a p-th root to a field k of characteristic p; that is,K = k(b) with bp = a, where a P k is not a p-th power. The minimal polynomialof b is P(t) = tp ´ a, and K = k[t]/(tp ´ a). The point is that P1(t) = ptp´1 = 0,so for each c P K the k-linear map k[t] Ñ K given by Q(t) ÞÑ Q1(t)c vanisheson P(t) and descends to a k-linear map Dc : K Ñ K. Leibniz’ rules immediatelyyields that Dc is a derivation, and as Dc(b) = c, the derivation Dc does notvanish. We conclude that Derk(K, K) » K and that ΩK/k » K as well; in fact, Db

serves as a universal derivation.(17.14) Polynomial rings once more. What follows, is a slight generalization ofProposition 17.6. Let AĎ B be an extension of rings. Then

ΩB[t]/A = B[t]dt‘ (ΩB/AbB B[t]).

Again, the way to see this, is to give a universal derivation. For each polynomialP(t) =

ř

i aiti P B[t] one defines

dB[t]P(t) = P1(t)dt +ÿ

i

dB(ai)b ti (17.9)

where dB : B Ñ ΩB/A is the universal derivation. It is the matter of a straightfor-ward computation to see that dB[t] so defined is a universal derivation.

In particular, if R = B[t1, . . . , tn] is the polynomial ring, it holds that

ΩR/A =À

iRdti ‘ (ΩB/AbB R)

(17.15) The differentials of a tensor product. Let B and C be two A-algebras. Thenthe map

d : BbA C Ñ (ΩB/AbA C)‘ (BbA ΩC/A)

properties of kähler differentials 328

given as bb c ÞÑ bb dCc + dBbb c on decomposable tensors and extended bybilinearity is a universal A derivation. We compute

d(b1bb c1c) = bb1b (c1dCc + cdCc1) + (b1dBb + bdBb1)b cc1 =

= b1b c1 ¨ (bb dCc + dBbb c) + bb c ¨ (b1b dCc1 + dBb1b c1)

and δ is a derivation, and which is universal in view of the formula

γ(dbb c + b1b dc1) = 1b c ¨ α(bb 1) + bb 1 ¨ β(1b c),

which defines the required map γ : (ΩB/AbA C)‘ (BbA ΩC/A) Ñ M. Hereα : ΩB/A Ñ M and β : ΩC/A Ñ M are the linear maps corresponding to thederivations D|Bb 1 : B Ñ M and D|1b C : C Ñ M and D : BbA C Ñ M is a givenA-derivation.

K

Exercise 17.3 Show that dB[t] as defined in (17.9) is a universal derivation. M

Exercise 17.4 Show that the derivation Db in Example 7 does not depend on thechoice of the field generator b; that is, if b1 is another element so that K = k(b1),then Db1 = Db. M

17.3 Properties of Kähler differentials

There are a few useful ways for computing modules of differentials whenchanging rings. The proofs of the following propositions are not difficult, andinvolve only basic commutative algebra.

Base changeThe Kähler differentials behave well with respect to base change:

Proposition 17.16 Let A be a ring and B be an A-algebra, and let A1 beanother A-algebra. Define B1 = BbA A1. Then there is a canonical isomorphism

ΩB1/A1 » ΩB/AbB B1

Proof: The universal derivation dB : B Ñ ΩB/A induces an A1-linear map

d1 = dBb idA1 : B1 Ñ ΩB/AbA A1 = ΩB/AbB B1

which clearly is a derivation. This will be the required universal derivationof ΩB1/A1 , and so the claim follows: let ι : B Ñ B1 = BbA A1 be the canonicalmap b ÞÑ bb 1. Given an A1-derivation D : B1 Ñ M into a B1-module, the mapD ˝ ι : B Ñ M will be an A-derivation, and consequently it will factor as α ˝ dB fora B-linear map α : ΩB/A Ñ M. The map αb idA1 : ΩB/AbA A1 Ñ MbA A1 = Mthen yields the desired factorization of D. o


Two exact sequencesThere are two fundamental exact sequences involving Kähler-differentials. Indifferential geometry a differentiable map between two manifolds, induces aderivative between the tangent bundles, and the first sequence is an algebraicanalogue of this, or rather of its dual, the pullback of differential one-forms.The second is a dual analogue of the sequence relating the tangent bundle of asubmanifold to the tangent bundle of the surrounding space and the normalbundle.

Let A be a ring and let ρ : B Ñ C be a homomorphism of A-algebras. Thereis natural homomorphism of C-modules

ρ˚ : ΩB/AbB C Ñ ΩC/A

defined by ρ˚(dBbb c) = cdCρ(b). The dual of ρ˚ corresponds, under the iden-tification (17.7), to the map DerA(C, N) Ñ DerA(B, N) that sends a derivationD : C Ñ N to D ˝ ρ. (Note that HomB(ΩA/B, N) = HomC(ΩA/BbB C, N) sinceN is a C-module.)

Moreover, there is a canonical ‘change-of-constants-map’

β : ΩC/A ΩC/B

as explained in Example 2 above.The next propositions describes the kernel of this ‘change-of-constants-map’,

and as one would suspect, it is generated by the elements shaped like db whereb P C comes from B:

Proposition 17.17 The following sequence of C-modules is exact

ΩB/AbB C ΩC/A ΩC/B 0ρ˚ β

Proof: That β ˝ ρ˚ = 0 is clear. Checking exactness amounts to showing thatfor any C-module N, the dual sequence

0 Ñ HomC(ΩC/B, N)Ñ HomC(ΩC/A, N)Ñ HomC(ΩB/A bB C, N)

is exact, and, as the map β is surjective (Example 2), only exactness in the middleis an issue. Note that HomC(ΩB/A bB C, N) = HomB(ΩB/A, N), so the in viewof the constituting isomorphisms (17.7), the sequence can be written as

0 DerB(C, N) DerA(C, N) DerA(B, N).

The map on the left merely considers a B-derivation to be an A-derivation,whereas the one on the right sends D : C Ñ N to the composition D ˝ ρ. Sayingthat D is mapped to zero in DerA(B, N), is saying that it vanishes on all elementsb in C coming from B, which is equivalent to saying it is a B-derivation; indeed,it will B-linear by Leibniz rule:

D(bx) = bD(x) + xD(b) = bD(x),


for x P C and b P C coming from B. o

In the next proposition we establish an exact sequence that relates thedifferentials of an A-algebra B and those of a quotient C = B/I. It involves amap δ : I/I2 Ñ ΩB/AbB C which sends the class of f P I mod I2 to dB f b 1, ormore formally, which results from applying the tensor functor ´bB C to therestriction dB|I : I Ñ ΩB/A. (Note that IbB C = I/I2 as C = B/I).

Proposition 17.18 (Conormal sequence) Suppose that B is an A-algebra.Let C = B/I for some ideal I Ă B and let α : B Ñ C = B/I be the canonicalmap. Then there is an exact sequence of C-modules

I/I2 ΩB/AbB C ΩC/A 0.δ α˚

Proof: As in the previous proposition it suffices to check that for each C-moduleN, the dual sequence

0 DerA(C, N) DerA(B, N) HomC(I/I2, N) = Hom(I, N)

is exact. In view of Proposition 17.8 and Exampe 5 the map α˚ is surjective, andhence the leftmost map is injective. The rightmost map associates to a derivationD : B Ñ N its restriction to I. (Note that this is indeed a homomorphism ofC-modules since IN = 0). If D|I = 0, clearly D passes to the quotient andyields a D1 : C = B/I Ñ N, which is a C-derivation since D is a B-derivation. Inother words, D lies in the image of DerA(C, N), and the sequence is exact in themiddle as well. o

Corollary 17.19 Let A be a ring and let B be a finitely generated A-algebra (or alocalization of such). Then ΩB/A is finitely generated over B.

Proof: Write B = A[x1, . . . , xn]/I for some variables x1, . . . , xn and applyProposition 17.6 on page 326 and the above proposition. o

Exercise 17.5 (The diagonal and ΩB/A.) Suppose that B is an A-algebra. There isˇ

an exact sequence of A-modules

0 I BbA B B 0µ

where µ is the multiplication map, which acts as b1b b2 ÞÑ b1b2 on decomposabletensors, and where I is the kernel of µ. Since BbA B/I » B, the module I/I2

has the structure of a B-module.

a) Show that I is generated by elements of the form ab 1´ 1b a;

b) Show that the two B-module structure on I/I2 induced from each factor ofthe tensor product agree; that is, bb 1 ¨ x = 1b b ¨ x for all x P I/I2;

c) Show that d : B Ñ I/I2 defined by db = bb 1´ 1b b is an A-derivation;

d) Show that d is a universal derivation so that I/I2 » ΩB/A and d = dB/A.


M

Exercise 17.6 Let A Ñ B be a map of Noetherian rings, π : X Ñ Y. Assume thatΩB/A = 0. Show that the diagonal ∆ is a connected component of X ˆY X =

Spec BbA B.Assume that IĎ A is finitely generated ideal such that I2 = I. Show that I is

a principal ideal generated by an idempotent. Hint: Let txiu generate I andwrite xi =

ř

j aijxj with aij P I. Consider the matrix Φ = (δij ´ aij). Show thatdet Φ annihilates I, and hence there is an e P I so that (1´ e)I = 0. Show thate2 = e and that I = (e). M

Kähler differentials and localizationWhen we later shall globalize the construction of the Kähler differenials, thefollowing two results about their behavior with respect to localizations areimportant. They both rely on the sequence in Proposition 17.8.

Proposition 17.20 Let S Ă A be a multiplicative subset mapping into thegroup of units in B. Then ‘change-of-constants-map’ is an isomorphism

ΩB/A » ΩB/S´1 A.

Proof: The ‘change-of-constants-map’ is the map β in the sequence

ΩS´1 A/AbS´1 A B ΩB/A ΩB/S´1 A 0,β

and by Example 4 we have ΩS´1 A/A = 0. o

Proposition 17.21 Suppose S is a multiplicative system in B and let ι : B ÑS´1B be the localization map. Then the natural map ι˚ yields an isomorphism

ι : S´1ΩB/A » ΩS´1B/A.

Proof: Note that S´1ΩB/A = ΩB/AbB S´1B, so we are in the context ofProposition 17.8 and may use the exact sequence there. We previously checkedthat ΩS´1B/B = 0 (Example 4) and hence ι˚ is surjective. Thus in view ofthe identity HomS´1B(S´1ΩB/A, M) = HomB(ΩB/A, M,) which is valid for anyS´1B-module M, it suffices to see that the map

DerA(S´1B, M) DerA(B, M)

corresponding to ι˚ is surjective. This is the case since every D : B Ñ M extendsto a derivation D1 : S´1B Ñ M by the formula

D1(bs´1) = (sDb´ bDs)s´2, (17.10)


some checking must be done, which is left to the reader. o

Exercise 17.7 Check that the expression D1(bs´1) in (17.10) does not dependˇ

on the choice of representative for bs´1 and that the resulting D1 is a derivation.M

Some explicit computationsWe can use the previous exact sequences to do explicit computations withΩB/A. If B is a finitely generated A-algebra, say B = A[x1, . . . , xn]/I whereI = ( f1, . . . , fr). Then we have

ΩA[x1,...,xr ]/AbA B »Àn

i=1 Bdxi.

The conormal sequence (Proposition 17.9), takes the form

I/I2 Àni=1 Bdxi ΩB/A 0δ

and as I/I2 is generated as a B-module by the classes of the f1, . . . , fr moduloI2 there is a g surjection Br Ñ I/I2 wich gives the exact sequence

Br Àni=1 Bdxi ΩB/A 0.δ1

Explicitly, in an appropriate basis, the map d’ is given by the nˆ r Jacobian matrixJ = (B f j/Bxi).

Theorem 17.22 Let A be a ring and let B = A[x1, . . . , xn]/( f1, . . . , fr). Then

ΩB/A =à

iBdxi/

ř

jB(ř

i(B f j/Bxi) dxi)

and the universal A-derivation is given as dB/A f =ř

i(B f /Bxi) dxi.

Examples(17.23) Non-singular plane curves. Let k be a field and let X = Spec R whereR = k[x, y]/( f ). Let us compute the module of differentials ΩR/k. Let fx, fy

denote the (images of the) partial derivatives of f in R. Then by the above

ΩR/k = Rdx‘ Rdy/( fxdx + fydy).

If the curve X is non-singular; i.e. that V( f , fx, fy) = H, the module of differen-tials ΩR/k will be a locally free R-module of rank one: local bases over the opensets D( fx) and D( fy) respectively, are given by

dyfx

and ´dxfy

.


Note, that on the overlap D( fx)X D( fy) the two agree since we have fxdx +

fydy = 0 in ΩR/k.(17.24) The nodal cubic. The curve XĎA2 with equation y2 = x2(x + 1) is theso-called nodal cubic. It has a singular point at the origin (0, 0) and is regularelsewhere. Let B = k[x, y]/(y2 ´ x2(x + 1). Then

ΩB/k = Bdx‘ Bdy/(2ydy´ (3x2 + 2x)dx)

In this case ΩB/k has rank one for every point (x, y) ‰ (0, 0), indeed, if y ‰ 0 dxwill be a basis, and dy will be one when x ‰ 0, except where 3x + 2 = 0, butthese points are covered by the first case as y ‰ 0 there.

At the origin, the relation 2ydy´ (3x2 + 2x)dx is identically zero, so ΩB/k

has rank two there.We can also view B as an algebra over A = k[x]. In that case, we get

ΩB/A = B/(2y)dy » k[x]/(x2(x + 1))dy.

(17.25) The cuspidal cubic curve. As indicated in he previous example the moduleof Käher differential will be locally free when A is the coordinate ring of aregular curve, but near singular points it can be a rather complicated module.Even non-trivial torsion elements can appear, as is the case for the coordinatering of plane curves with isolated singularities (in fact, it is a conjecture ithappens for all singular curves). Here we illustrate this by the coordinate ringA = k[x, y]/(y2´ x3) of the cuspidal cubic, and you will find a discussion of thegeneral case in Exercise 17.10.

The differentials are given as ΩA/k = Adx‘ Ady/(2ydy´ 3x2dx), and η =

3ydx´ 2xdy is a non-zero torsion element, it is in fact killed by y and x2:

x2η = 3x2ydx´ 2x3dy = y(3x2x´ 2ydy) = 0;

yη = 3y2dx´ 2xydy = x(3x2dx´ 2ydy) = 0.

And being non-zero (see Exercise 17.8 below), η generates a submodule iso-morphic to k[x, y]/(x2, y), which is supported at the singular point (the origin).

Exercise 17.8 Check that η is non-zero, and that the torsion part of ΩA/k isˇ

generated by η. M

K

Exercises(17.9) Let B = k[x, y]/(x2 + y2). Show that if k has characteristic ‰ 2, then

ΩB/k = (Bdx + Bdy) /(xdx + ydy)

If k has characteristic 2, then ΩB/k is the free B module Bdx + Bdy.(17.10) Torsion in the Kähler differentials. (This exercise requires some knowledgeˇ

of Koszul complexes and homological algebra). Let f P k[x, y] be a polynomial

the sheaf of differentials 334

without multiple factors and let A = k[x, y]/( f ). Show that the submodule oftorsion elements of ΩA/k is isomorphic the quotient

(( fx, fy) : f

)/( fx, fy) of

the transporter ideal(( fx, fy) : f

)in the polynomial ring R = k[x, y]. Show

that X = V( f ) is regular if and only if ΩA/k is torsion free. Show that, moreprecisely, the torsion is of length dimk A/( fx, fy)A. (This number is the sum ofa contribution from each singular point, often called the Tjurina number of thesingular point. The formula for the length is due to Zariski ([?])).(17.11) Let f P k[x, y] be the equation of a non-singular curve. Let A =

k[x, y]/( f ) and B = k[x, y]/( f 2). Show that ΩB/k » Bdx‘ Bdy if k is of charac-teristic two and that ΩB/k » ΩA/k if not.(17.12) Transcendental extensions. Let k be a field and K = k(x1, . . . , xn) a purelytranscendental field extension. Show that ΩK/k » Kn with dx1, . . . , dxn as a basis.Hint: Consider k[x1, . . . , xn] and use (8), then localize and use 17.12.(17.13) Assume that kĎK is a finitely generated field extension.ˇ

a) Show that dimK ΩK/k ě trdeg K/k;

b) Show that equality holds if and only if K is separably generated* ˚A field extensionkĎK is separablygenerated if there is atranscendence basisx1, . . . , xn for K over kso that K is separableover k(x1, . . . , xn). Ifin addition K isfinitely generated overk, the K will be finiteover k(x1, . . . , xn).

over k.

c) Show that if k is perfect, it holds that dimK ΩK/k = trdeg K/k, hence Kis separably generated over k. Hint: Let P(t) =

ř

i aiti be a minimalpolynomial in for x, show that dP = P1(t)dt +

ř

i dai ¨ ti P ΩK[t]/k is non zero.

M

17.4 The sheaf of differentials

For us, the primary motivation for studying ΩB/A is that it gives us an intrisicmodule ΩB/A associated to a ring homomorphism A Ñ B. By applying „, wethus get an intrinsic sheaf on X = Spec B associated to the map of affine schemesSpec B Ñ Spec A. We would like to globalize this construction to an arbitrarymorphism of schemes f : X Ñ S. This will lead us to form the sheaf of relativedifferentials ΩX/S which will be a quasi-coherent OX-module.

This sheaf is locally built out of the various ΩB/A on local affine charts. Theseare not just arbitrary modules that just happen to glue together to a sheaf; eachof them come with the universal property of classifying derivations D : B Ñ M.For this reason, we would like to say that the ΩX/Y should satisfy a similaruniversal property. We make the following definition:

Definition 17.26 Let F be a quasi-coherent (?) OX module. A morphismD : OX Ñ F of OX-modules is an S-derivation if for all open affine subsetsV Ă S and U Ă X with f (U) Ă V, the map D|U is an OS(V)-derivationof OX(U) with values in F . The set of all such S-derivations is denoted byDerS(OX,F ).


Definition 17.27 The sheaf of relative differentials is a pair (ΩX/S, dX/S)

of a quasi coherent (?) OX-module ΩX/S and a S-derivation dX/S : OX Ñ ΩX/S

that satisfies the following universal property: For each quasi-coherent (?) OX-module F , and each S-derivation D : OX Ñ F there exists a unique OX-linearmap α : ΩX/S Ñ F such that D = α ˝ dX/S.When S = Spec A, we sometimes write ΩX/A for ΩX/S.

In other words, ΩX/S is a sheaf that represents the functor of S-derivations,in the sense that there is a functorial isomorphism

HomOX (ΩX/S,´) » DerS(OX,´).

Exercise 17.14 Prove, using the universal property of differentials, that givesthat this sheaf is unique up isomorphism, if it exists. M

In the affine situation with a morphism X = Spec B Ñ S = Spec A we havethe module of K differentials ΩA/B and the corresponding sheaf ČΩA/B will serveas the sheaf of relative differential on X; this is just a consequence of „ being anequivalence of categories ModB and QCohX. In the general case, gluing the localdifferential on affine covers works well, and the main theorem of this sectionsays that sheaves of relative Kähler differentials exist unconditionally.

Theorem 17.28 Let f : X Ñ S be a morphism of schemes. Then there is a sheafof relative differentials ΩX/S, which is a quasi-coherent sheaf on X.Moreover, ΩX/S has the property that for each open affine open V = Spec A andeach open affine U = Spec B Ă f´1(V) it holds that

ΩX/S|U »ČΩB/A.

Also for each x P X, we have

(ΩX/S)x » ΩOX,x/OS, f (x).

Proof: Fix an open subset V = Spec A of S, and let U = Spec B be an affineopen subset in X so that f (U) Ă V. For these two, we define

ΩU/V = ČΩB/A

which is a sheaf on U. We first show that the different ΩU/V glue together to anO f´1V-module Ω f´1(V)/V when U runs through an open affine cover of f´1(V).This comes down to showing that if U1 = Spec B1 is a distinguished open affinesubset of U, then

ΩU/V |U1 » ΩU1/V .


But as B1 is a localization of B, Proposition 17.12, tells us that ι˚ is such anisomorphism with ι : B Ñ B1 Ñ the localization map. These maps dependfunctorially on the inclusions, so the gluing conditions are trivially fulfilled.

Then we show that the sheaves Ω f´1V/V for all affine opens V Ď S glue to aOX-module ΩX/S. This amounts to showing that for each distinguished openV1 = Spec A1 Ď V, and all open U = Spec B of f´1(V1), we have

ΩU/V = ΩU/V1

But this follows from Proposition 17.11, as A1 is a localization of A in a singleelement (which maps to an invertible element in B).

This means that we get an OX-module ΩX/S. Let us check that it satisfiesthe above universal property. So we need to define the universal derivationdX/S : OX Ñ ΩX/S.

Let V = Spec A Ď S and U = Spec B Ď X be an affine open subset such thatf (U) Ď V. Define dX/S(U) = dB/A. By the gluing construction above, this mapdoes not depend on the chosen affine open V, and it can be checked that theassignment is compatible with restriction maps. Hence this gives a morphism ofsheaves dX/S : OX Ñ ΩX/S, which by construction is an S-derivation.

To check that this is universal, we again work locally. Let d : OX Ñ F be an S-derivation, where F is an OX-module. Let U = Spec A Ď S and V = Spec B Ď Xso that f (U) Ď V. By the universal property of ΩB/A, we get an A-derivationD(V) : B Ñ F (V), and hence a unique B-linear map α(V) : ΩX/S(V) = ΩB/A Ñ

F (V) such that D(V) = α(V) ˝ dX/V(V). One has to check that these maps arecompatible with restriction maps (use the universal property of ΩB/A), but afterthat, we obtain a unique OX-linear map α : ΩX/S Ñ F so that D = α ˝ dX/S. o

Note that the sheaf ΩX|S is always quasi-coherent (it is by definition locallyof the form ĂM for some module). Moreover, when X is of finite type over a field,ΩB/A is finitely generated, and so ΩX|k is even coherent.Example 17.29 Let A be a ring and let X = An

S = Spec A[x1, . . . , xn] be affinen-space over S = Spec A. Then ΩX/S » On

X is the free OX-module generated bydx1, . . . , dxn. K

If X is a separated scheme over S then one could also define ΩX/S as follows.Let ∆ : X Ñ XˆS X be the diagonal morphism and let I∆ be the ideal sheaf ofthe image of ∆. Then ΩX/S = ∆˚(I∆/I2

∆). This does in fact give the same sheafas above, since these two definitions coincide when X and S are both affine(Exercise 17.5). This definition gives a quick way of obtaining the sheaf ΩX/S,but it is not very enlightening nor suited for computations.

The properties of the Kähler differentials ΩB/A translate into the followingresults for ΩX/Y:

the euler sequence and differentials ofna 337

Proposition 17.30 (Base change) Let f : X Ñ S be a morphism of schemesand let S1 be a S-scheme. Let X1 = XˆS S1 and let p : X1 Ñ X be the projection.Then

ΩX1/S1 » p˚ΩX/S

Proposition 17.31 Let X, Y, and Z be schemes along with maps XfÝÑ Y

gÝÑ Z.

Then there is an exact sequence of OX-modules

f ˚(ΩY/Z)Ñ ΩX/Z Ñ ΩX/Y Ñ 0. (17.11)

Proposition 17.32 (Conormal sequence) Let Y be a closed subscheme ofa scheme X over S. Let IY be the ideal sheaf of Y on X. Then there is an exactsequence of OX-modules

IY/I2Y Ñ ΩX/S bOY Ñ ΩY/S Ñ 0. (17.12)

17.5 The Euler sequence and differentials of PnA

We have seen that the cotangent sheaf of the affine spaces An is trivial, i.e. theyare isomorphic to On

An . In this section we will give a concrete description of thecotangent bundle of projective space, suitable for explicit computations. It comesas a short exact sequence, sometimes called the Euler-sequence, and involves atwist of the tautological map On+1

Pn Ñ OPn(1) that represents a point in Pn as‘the corresponding quotient of An’.

Euler’s theorem states that if f is a rational function of degree d, it holds thatř

xi fxi = d f , or, in particular, when f is of degree zero, one hasř

i xi fxi = 0.Now, the functions on open sets in projective space are all rational functions ofdegree zero, and so Euler tells us that their differentials all live in the kernel ofthe map

à

iOPn(´1)dxi Ñ OPn

that sendsř

i fidxi toř

i xi fi. This gives a strong heuristic argument for the nexttheorem:

Theorem 17.33 Let ring A and X = PnA the projective n-space over A. Then

there is an exact sequence

0 ΩX/A OX(´1)n+1 OX 0.

Proof: Choosing coordinates on PnA we have Pn

A = Proj R where R is thegraded A-algebra R = A[x0, . . . , xn]. We introduce a graded R-module M by the

the euler sequence and differentials ofna 338

exact sequence

0 MÀ

i R(´1)dxi Rη

where η is the ‘Euler map’ř

i fidxi ÞÑř

i fixi. It is homogenous of degree zerowhen we give each dxi degree one. Note that Coker η = R/(x0, . . . , xn), so thatwhen ‘tilded’ the sequence becomes

0 ĂMÀ

i OPn(´1)dxi OPn 0.rη

We begin with a swift recap of the construction of the projective space Proj R.It is covered by standard open affines D+(xi) each equal to Spec (Rxi)0, where(Rx)0 is the degree zero piece of the localization Rxi (equipped with naturalgrading). The overlaps of the standard opens are the distinguished open setsD+(xixj) = Spec (Rxixj)0.

The universal derivation dR : R Ñ ΩR/A =À

j Rdxj extends to a derivation

dRxi: Rxi Ñ ΩRxi /A =

À

jRxi dxj

by the usual rule for the derivative of a fraction, and it preserves degrees wheneach dxj is given degree one; that is (Rxi dxi)ν = (Rxi)ν´1dxi. Taking the degreezero part, yields a derivation

(Rxi)0 ÑÀ

j(Rxi(´1))0dxj;

that is, when exposed to tilde, a derivation

OD+(xi) Ñà

jOPn(´1)|D+(xi)dxj.

Since these derivations for different i originate from the same global derivationdR, they are forced to agree on the overlaps, and patch together to give aderivation

OPn Ñà

jOPn(´1)dxi.

It takes values in ĂM, and by universality there is a map ΩPn/A Ñ ĂM. The restof the proof consists of checking that this is an isomorphism, which is a localissue. Both ΩPn/A and ĂM are locally free of rank n, so it suffices to see that α issurjective.

On the open set D+(xi) the sheaf ΩPn/A = ΩD+(xi)/A originates from themodule Ω(Rxi )0/A, which has a basis formed by the d(xj/xi) for j ‰ i, andone checks without much resistance that the map α sends d(xi/xj) to (xjdxi ´

xidxj)/x2i (what else could it be?). But the kernel of the Euler map η is generated

by the elements xidxj ´ xjdxi , and so we are through. o

Since ΩPnA

injects into OPnA(´1)n+1 (which has no global sections), we get:

Corollary 17.34 Γ(PnA, ΩPn

A) = 0

Exercise 17.15 Show that the kernel of η is generated by n(n´ 1)/2 expressionsxidxj ´ xjdxi. M

relation with the zariski tangent space 339

17.6 Relation with the Zariski tangent space

The tangent space to a differentiable manifold at a point is defined at the spaceof ‘point derivations’ as the point, i.e. derivations from the ring of C8-germsnear the point to R. The analogue to this for a scheme X over a field k would bethe space of derivations Derk(OX,x, k(x)), where k(x) is the residue class field atx, and in view of the fundamental relation (17.4) and the equality

HomOX (ΩX/k, k(x)) = Homk(x)(ΩX/kb k(x), k(x)),

the cotangent space; i.e. the dual of the tangent space, will be ΩX/kbOX k(x).Another candidate is, however, the Zariski tangent space Homk(x)(m/m2, k(x)),

In contrast to the ‘point derivations’, the Zariski tangent space is not a relativenotion, it does not depend on the subfield k, and can be defined for any localring. The Zariski cotangent space will simply be the dual space m/m2.

These two possible tangent spaces give rise to two different notions, regu-larity and smoothness, which both in some sense mimic the property of being amanifold. Fortunately, in several cases the two are equivalent; one such situationis in described in the following proposition:

Proposition 17.35 Suppose (B,m) is a local ring with residue field K = B/mand assume that B contains a field k. If the extension kĎK is finite and separable,then the map from the conormal sequence

δ : m/m2 Ñ ΩB/k bB K

is an isomorphism.

Proof: The conormal sequence with A = k and C = K takes the followingshape:

m/m2 ΩB/kbB K ΩK/k 0,δ

and according to Example 6 on page 327 it holds that ΩK/k = 0, so δ is surjective.The map δ sends x P m to dx. We shall exhibit an inverse ψ : ΩB|k bB K Ñ

m/m2 to δ. Constructing such a map is equivalent to constructing a map ofB-modules ΩB|k Ñ m/m2, or equivalently, a derivation D : B Ñ m/m2.

The derivation D : B Ñ m/m2 will be the composition D ˝ π of the canonical‘reduction-mod-m2-map’ π : B Ñ B/m2 and a derivation D0 : B/m2 Ñ m/m2. Toconstruct the latter, we cite the lemma below that the k-algebra B/m2 splits as adirect sum B/m2 = K‘m/m2, and simply let D0 be the projection onto m/m2;that is

D0(a + x) = x,


where α P K and x P m/m2. The reduction map π being an algebra homomor-phism, it suffices to see that D0 is a k-derivation. To this end, we compute:

D0((a + x)(a1 + x1)) = D0(aa + (ax1 + a1x) + x1x)

= D0(aa) + D0(ax1 + a1x) + D0(x1x) = ax1 + a1x,

and we get the same answer when we expand

(a1 + x1)D0(a + x) + (a + x)D0(a1 + x1)

since xx1 = 0. Hence D0 is a derivation, and we get the desired inverse. Itis indeed an inverse to the map δ, since via the identification DerA(B, M) =

HomB(ΩB/A, A), it sends dx to x. o

Lemma 17.36 Let B be a local ring with maximal ideal I that satisfies I2 = 0. Assumethat B contains a field k and that the extension kĎK = B/I is finite and separable.Then B contains a subring isomorphic to K; so that B = K‘ I.

Proof: Since K is finite and separable over k, it is primitive. So let K = k(x)and let P being the minimal polynomial of x over k. It is separable, so P1(x) ‰ 0.We shall lift x to an element y P B/m2 such that that P(y) = 0 (meaningfulas kĎ B/m2 and P has coefficients in k). Then the the subring k(y) mapsisomorphically onto K.

Chose any lifting z of x. Then P(z) = ε P I. For any α P I Taylor’s formulayields

P(z + α) = P(z) + P1(z)α

as α2 = 0. Now P1(x) is a unit in B/I, and as units reduce to units (Lemma 17.24

below)and hence y = z + α is such that P(y) = 0. o

Recall that a Noetherian local ring B is called regular if the Krull dimensionequals the embedding dimension; or with m the maximal ideal and K = B/m, itholds that dimK m/m2 = dim B.

Lemma 17.37 Let π : B Ñ A be a surjective ring homomorphism with kernel I. As-sume that I2 = 0. Then every element in B that maps to a unit in A is invertible, andthere is an exact sequence of groups

1 1 + I B˚ A˚ 1π .

Proof: All elements in 1 + I are units, since if x2 = 0, it holds that (1 + x)(1´x) = 1. Assume that π(x)y = 1 and let z P B be so that π(z) = y´1 Thenxz P 1 + I and is therefore invertible, so a fortiori x is invertible. o

Exercise 17.16 Show that if K is a finitely generated extension of k with aseparating basis, there is a field K1Ď B mapping isomorphically to K. Hint:First treat the case that K = k(x) with x a variable; then use induction on thecardinality of a separating basis. M


Corollary 17.38 With notation as in Proposition 17.22 but additionally with B beingNoetherian, the ring B is a regular local ring if and only if

dim B = dimk ΩB/KbB K.

The separability condition in Proposition 17.22 is certainly necessary, this isalready the case for fields: fields are regular local rings of dimension zero, andfor a inseparable field extension kĎK the module of differentials ΩK/k is neverzero; for instance, if K = k(x) with xp = a, it holds that ΩK/k = K.

Smooth varietiesWe give a definition for smoothness of varieties. In general schemes can havecomponents of different dimension, so we if x P X is a point, we let dimx X bethe Krull dimension of a sufficiently small affine neighbourhood of x; if x is aclosed point it coincides with dimOX,x.

Definition 17.39 (Smoothness of fields) Let X be a (separated (?) )scheme of (essential (?) ) finite type over a field k and let x P X be a point. Wesay that X is smooth at x if ΩX/k is locally free of rank dimx X near x. Thescheme X is called smooth if it is smooth at every closed point.

Theorem 17.40 Let X be a variety (integral separated scheme of finite type)over a perfect field k (e.g. k algebraically closed or of characteristic zero) and letx P X be a closed point. Then the following are equivalent:

i) X is smooth at x;ii) (ΩX/k)x is free of rank dim X;

iii) X is non-singular at x.

Proof: i) ðñ ii) is just the definition of X being smooth together with the factthat a coherent module F over OX is locally free in near x if and only Fx is free.

ii) ùñ iii). Assuming that ΩOX,x/k is free of rank n = dimOX,x, we infer, bythe above proposition, that dimk(x) mx/m2

x = n, and so OX,x is a regular localring.

iii) ùñ ii). There are two salient points: The first is that if x is a regular point,the integer d(y) = ΩX/kbOX k(y) takes on its minimal value at x, and the secondis that d(y) can only increase upon specialization. The details are as follows:Let K be the function field of X. If the local ring OX,x is regular, it follows fromProposition 17.22 that dim (ΩX/k)b k(x) = dimk(x) mx/m2

x = dim X. From Ex-ercise 17.13 on page 334 follows that dimK ΩK/k = dimK ΩX/kb K ě trdeg K/k.The transcendence degree of the function field of a variety equals dim X, andhence (ΩX/k)x is a free OX,x-module by the general fact that a finite module


over an integral local ring having generic fibre of the larger dimension than thespecial one, is free (Exercise 17.17 below). o

Exercise 17.17 (Jumping of fibre dimension upon specialization.) Let A be a localˇ

integral domain with maximal ideal m, residue field k = A/m and fraction fieldK. Let M be a finite A-module and assume that dimK MbA K ě dimk MbA k.Then M is a free A-module. (See also Proposition ?? in CA) M

Exercise 17.18 Let X be a variety over a perfect field. Show that the functionˇ

field K of X is separably generated over k and that the smooth (hence regular)closed points of X form an open dense subset. Give a counterexample if k is notperfect. M

Definition 17.41 When X is smooth over k and ΩX/k is a locally free sheaf onX, we refer to it as the cotangent bundle or cotangent sheaf of X.The sheaf of p-forms is defined as

ΩpX/k =

Źp ΩX/k

In particular, if X has dimension n, ωX = ΩnX/k is called the canonical bundle

of X. As ΩX/k has rank n, ωX is locally free of rank one, i.e. an invertible sheaf.The tangent sheaf, or the tangent bundle is the sheaf

TX = HomOX (ΩX|k,OX)

Proposition 17.42 Let X = Pn. Then ΩnX = O(ń´ 1).

Proof: Consider the Euler sequence for the cotangent bundle of Pn

0 Ñ ΩPn Ñ O(´1)n+1 Ñ O Ñ 0

In general, if 0 Ñ E 1 Ñ E Ñ E 2 Ñ 0, we haveŹe E =

Źe1 E 1 bŹe2 E 2. Hence

we getnľ

ΩPn =n+1ľ

O(´1) = O(ń´ 1).

Note by the way that the tangent bundle TPn fits into the following sequence:

0 Ñ OPn Ñ OPn(1)n+1 Ñ TX Ñ 0

where the left-most map sends 1 to the vector (x0, . . . , xn). o

Example 17.43 For X = A1, we get ΩA1 = TA1 = OA1 . K

Example 17.44 Let A be a ring and let X = P1A. Then it holds that ΩX/A »

OP1(´2) and TX = OP1A(2). For this, we can use the standard covering of

P1 = Proj A[x0, x1], given by Ui = D+(xi). On U0, we have

ΩU0|A » A[x1

x0]d(

x1

x0

)

the conormal sheaf 343

, and similarly on U1. On the intersection D+(x0)XD+(x1), we have

x20d(

x1

x0

)= ´x2

1d(

x0

x1

)This gives a non-vanishing section of ΩX(2)bΩX|A and furthermore an isomor-phism ΩX|A » OX(´2). K

Smooth morphismsWe can also use the sheaves of differentials to define a notion of smoothnessfor morphisms. In short, we say that a morphism f : X Ñ S is smooth at a pointx P X is if ΩX|S is locally free of rank dimx X´ dim f (x) S there. If this is not thecase, we say that f is ramified at x, and that x is a ramification point. f is smoothif it is smooth at every point.Example 17.45 Let A = k[x] and let B = k[x, y]/(x´ y2) » k[y] where k is afield of characteristic not equal to 2. Let X = Spec B and let Y = Spec A. Letf : X Ñ Y be the morphism induced by the inclusion A ãÑ B (thus x ÞÑ y2).

Since B » k[y] it follows that ΩX = Bdy, the free B-module generated by dy.Similarly ΩY = Adx. The sequence (17.11) gives us

ΩY bA B Ñ ΩX Ñ ΩX/Y Ñ 0|| || o|

Bdx Ñ Bdy Ñ Bdy/B(2ydy)

The point is that ΩX/Y = (k[y]/(2y))dy is a torsion sheaf supported on theramification locus of the map f : X Ñ Y. (The only ramification point is above0.) Note that ΩX/Y is the quotient of ΩX by the submodule generated by theimage of dx in ΩX = Bdy. The image of dx is 2ydy. K

17.7 The conormal sheaf

Let Y Ď X be a closed subscheme defined by an ideal sheaf I . Then I/I2 isnaturally a OY-module via I/I2 = I bOX/I = I bOY. We call I/I2 theconormal sheaf of Y. Its dual, NY = HomOY(I/I2,OY) is the normal sheaf of Y inX.

the quadric surface 344

Proposition 17.46 When X and Y are non-singular schemes over a field k, thesheaves I/I2 and NY are locally free of rank r = codim(Y, X).

In this case, they fit into the exact sequences

0 Ñ I/I2 Ñ ΩX|k|Y Ñ ΩY|k Ñ 0

0 Ñ TY Ñ TX|Y Ñ NY Ñ 0

(This the conormal bundle sequence and its dual respectively. The first sequenceis exact on the left, because I/I2 is locally free).

TakingŹ

of these sequences we get the following result, which is very usefulfor computing canonical bundles of subvarieties:

Proposition 17.47 (Adjunction formula) If X and Y are non-singular,we have

ωY = ωX b

rľ

NY = ωX b det NY

In particular, if Y Ă X is a smooth divisor, we get

ωY = ωX bOX(Y)|Y

Example 17.48 Let X Ă P2 be a non-singular plane curve of degree d. ThenIX = OP2(´d), and so I/I2 = I b OP2 /I = OY(´d). Here

Ź1 I/I2 =

OY(´d) so the previous proposition shows that

ωY » OY(d´ 3)

For d = 1, this gives our previous computation that ΩP1|k = OP1(´2).Also for d = 2, Y » P1, and ΩY = OP2(´1)|Y. This is consistent with the

previous example, because OP2(1)|Y » OP1(2).For d ě 3, the invertible sheaf ωY has many global sections. In particular, we

recover the fact that a non-singular plane curve of degree d ě 3 is not rational(i.e., not isomorphic to P1). In fact, it will follow from the results of the nextchapter that Γ(Y, ωY) has dimension exactly 1

2 (d´ 1)(d´ 2). So for d ě 3, notwo smooth curves of different degrees can be isomorphic. K

17.8 The quadric surface

Let Q be the quadric surface P1 ˆP1 over a field k. We saw in the previoussection that Cl(Q) » Z2. Thus it makes sense to ask how to represent thecanonical divisor of Q in terms of L1, L2. Since Q = P1 ˆP1 and KP1 = ´2p, itshould perhaps not come as a big surprise that

KQ = ´2L1 ´ 2L2

the quadric surface 345

In fact, ΩQ = p˚1 ΩP1 ‘ p˚2 ΩP1 . So takingŹ2, we get

ωQ =2ľ

(p˚1 ΩP1 ‘ p˚2 ΩP1)

= p˚1 ΩP1 b p˚2 ΩP1

= OQ(´2, 0)bOQ(0,´2) = OQ(´2,´2)

A second way of seeing this, is to use the fact that Q is embedded by the Segreembedding Q Ă P3 as a smooth quadric surface. Then the Adjunction formulaof Proposition 17.31 on page 344 tells us that

ΩQ = ωP3 bOP3(Q)|Q = OP3(´4)bOP3(2)|Q = OQ(´2)

This gives the same answer as before, since OP3(1)|Q = OQ(L1 + L2).

Exercises(17.19) Let p ą 2 be a prime number and let k = Fp(t). Letˇ

X = Spec k[x, y]/(y2 ´ tp ´ t)

Show that all the local rings of X are regular, but X is not smooth over k.(17.20) Let C » P1 denote the twisted cubic curve in P3. Show that the conormalsheaf of C is isomorphic to OP1(´5)‘OP1(´5).

M

Chapter 18

Properties of morphisms

In this chapter, we will survey a few geometric properties of morphisms ofschemes f : X Ñ Y. We have already seen a few of these:

o f is dominant if f (X) is dense in Y (Chapter 6).

o f is of finite type (see below, and Chapter 3).

o f is separated if the diagonal morphism X Ñ XˆY X is a closed immersion(Chapter 8).

18.1 Finite morphisms

Recall that giving a morphism f : X Ñ S between two affine schemes S = Spec Aand X = Spec B, is equivalent to giving the ring homomorphism f 7 : A Ñ B, orsaid differently giving B the structure of an A-algebra.

Recall that a morphism f : X Ñ Y be is of locally finite type if Y has acover consisting of open affine subsets Vi = Spec Bi such that each f´1(Vi) canbe covered by affine subsets of the form Spec Aij, where each Aij is finitelygenerated as a Bi-algebra. Moreover, f is of finite type if, in i), one can do with afinite number of Spec Aij.

Again, when X = Spec B and Y = Spec A, the scheme X is of finite typeover A precisely when B = A[x1, . . . , xn]/a for an ideal a. One easily checks thatboth closed and open immersions are of finite type.

There is another related, but much stronger finiteness property a morphismcan have:

Definition 18.1 A morphism f : X Ñ Y is finite if there is a covering Vi =

Spec Ai such that each inverse image f´1(Vi) is affine, an if f´1(Vi) = Spec Bi,the Ai-algebra Bi is a finite Ai-module.

To underline the huge difference between the the two notions, note that ascheme X which is finite over a field k, in particular has a finite and discrete

finite morphisms 347

underlying topological space, whereas X being of finite type, merely means it iscovered by affine schemes of the form Spec k[x1, . . . , xr]/a.

This generalizes in the following way:

Proposition 18.2 A finite morphism has scheme-theoretical finite fibres. Inparticular, the fibres are finite discrete topological spaces.

Proof: We may certainly assume that both X and Y are affine; say X = Spec Band Y = Spec A. Any generator set of B as an A-module, persists being agenerator set of BbA K(A/p) as a vector space over K(A/p), where p P Spec Ais any point. o

Be aware that the converse is far from being true. One easily finds so-calledquasi-finite morphisms; that is, morphisms with all fibres finite, that are notfinite: every injective morphism is evidently quasi-finite, so for instance openimmersions will be, and open immersions are not finite except in trivial cases.The arch-type is the inclusion ι : D(x) ãÑ A1

k which on the ring level correspondsto the inclusion k[x] ãÑ k[x, x´1]; and k[x, x´1] is not a finite module over k[x].We’ll come back to the relation between quasi-finite and finite morphism whenhaving introduced proper morphism (in Section 18.6).

Examples(18.3) For n ě 1, the structure morphisms An

k Ñ Spec k and P1k Ñ Spec k are of

finite type, but not finite.(18.4) The morphism

š8i=1 A1

k Ñ A1k (identity on each component) is locally of

finite type, but not of finite type.(18.5) Consider the blow-up morphism π : X Ñ A2 from Example 5.6. Inthe local charts, π is given by Spec Z[x, t] Ñ Spec Z[x, y] induced by y ÞÑ xt,making Z[x, t] into a finitely generated Z[x, t]-algebra, so it is of finite type.However, it is not finite. In fact it is not even affine, since π´1(V) contains acopy of P1 for any neighbourhood V of the closed point o P A2, which is notpossible for affine schemes.(18.6) Let us revisit the example of a hyperelliptic curve X from Section 5.9. Inthe notation from that section, the curve X has an open covering consisting oftwo affine schemes U = Spec A and V = Spec B and there is a ‘double cover’morphism f : X Ñ P1

k . This is a finite morphism: Over U it is induced by theinclusion

k[x] Ăk[x, y]

(y2 ´ a2g+1x2g+1 ´ ¨ ¨ ¨ ´ a1x),

and the algebra on the right is isomorphic to k[x] ‘ k[x]y as a k[x]-module.A similar statement holds for the maorphism f |V : V Ñ A1

k , so f is a finitemorphism.

K

finite morphisms 348

Exercises(18.1) The degree of a finite map. Let Y = Spec A be an integral affine schemeˇ

and let f : X Ñ Y be a finite morphism, which entails that X = Spec B for anA-algebra B. Let d = dimK(Y) BbA K(A) be the dimension of the generic fibreas a vector space over K(A); it is called the degree of f . Show that there is anon-empty open subset UĎY such that for all y P U the fibre Xy is the spectrumof an k(y)-algebra of dimension d; in other words, dimky BbA k(y) = d for ally P U.(18.2) Distinguished properties. We shall say that a property P attributed toˇ

affine subsets of a scheme X is distinguished1, if the following two requirementsare fulfilled:

i) If U has P and f P Γ(U,OX), then D( f ) has P ;

ii) If tD( fi)u is a finite cover of U of distinguished open subsets eachhaving P , then U has P .

Show that if P is distinguished and there exists one open affine cover of X witheach member having P , then all open affines in X has P . Show that it sufficesthat ii) be satisfied for all coverings by two distinguished opens.(18.3) Affine morphisms. Another big difference between morphisms of finiteˇ

type and finite morphisms is that the latter are affine, in the following sense. Amorphism f : X Ñ Y is affine if f´1(U) is affine whenever UĎY is affine andopen. Show that being affine is a property local on the target; that is, if there isa cover of Y by open affines tUiu so that the f´1(Ui) are affine, then f is affine.(18.4) Show that both being of finite type and being finite are properties localon the target; i.e. a morphism f : X Ñ Y is of finite type (respectively finite)if there is an open cover tUiu so that f | f´1(Ui)

: f´1(Ui) Ñ Ui is of finite type(respectively finite).(18.5) Functoriality. Let f : X Ñ Y and g : Y Ñ Z be morphisms of schemes. LetP be one of the properties, locally of finite type, finite type or finite.

a) Show that if f and g are P , then g ˝ f is P ;

b) Show that if g ˝ f is P , then f is P ;

c) Give examples that g ˝ f and f are P , but g is not.

M

1Vakil in his notes calls this property a “local property”, but local being a overcharged name, we preferdistinguished.

flat morphisms 349

18.2 Flat morphisms

18.3 Smooth morphisms

18.4 Etale morphisms

18.5 Proper morphisms

Searching for a substitute in algebraic geometry for notion of the compact spacesin topology, it turns out that the topological notion of proper maps is suitable.These are the continuous maps with all preimages of compact sets being compact,or equivalently they are the universally closed maps, and the latter property is,together with a finiteness condition, the one adopted in scheme-theory:

Definition 18.7 A morphism f : X Ñ Y is universally closed if it is closedand stays closed when pulled back; that is, given any morphism T Ñ Y, the mapfT in the Cartesian diagram

XˆY T X

T Y

fT f

is closed. The morphism f is said to be proper if it is separated, of finite typeand universally closed. We say f is proper over Y, or just proper over k whenY = Spec k, the spectrum of a field k.

Example 18.8 A very simple of example of a morphism that is not proper is thestructure map π : A1

k Ñ Spec k. Pull it back along itself to obtain A1kˆk A1

k = A2k

with π pulling back to the first projection, and within A2k = Spec k[x, y] one finds

a lot of closed sets that project to sets not being closed; for instance, the graphof any rational function with a pole is such, the simplest being the “hyperbola”xy = 1. It projects to the non-closed set A1

kzt0u. K

Example 18.9 If φ : A Ñ B is a map of rings such that B is a finite A-module,then as we shall soon see, the induced map Spec B Ñ Spec A is proper. K

Proposition 18.10 The following statements hold true:

i) Closed immersions are proper;ii) Pullbacks of proper maps are proper;

iii) Compositions of composable proper maps are proper;iv) The product f ˆ g of two proper maps is proper;v) Given two composable maps f and g. If f ˝ g is proper and f is

separated then g is proper.

proper morphisms 350

The proposition holds true if ‘proper’ is replaced by ‘universally closed’.Proof: i): Closed immersions are separated and trivially of finite type, moreoverthey stay closed embeddings when pullbacked, hence they are also universallyclosed.ii): Let the involved maps be T Ñ Y and f : X Ñ Y with f proper. The pullbackfT is separated and of finite type as these properties are stable under basechanges, so let us check that fT = f ˆT idT is universally closed. Assuminggiven a morphism U Ñ T we find using transitivity of the tensor productthat ( fT Û idU) = ( f ˆT idT)Û idU = f Û idU , and the latter map is closedbecause f is universally closed.iii): This is undemanding: the composition of two closed maps being closed andthe pullback being a functor; i.e. ( f ˝ g)ˆ idT = ( f ˆ idT) ˝ (gˆ idT).iv): Direct from ii) and iii) follows that f ˆ g = ( f ˆ idX1) ˝ (idX ˆ g) is proper,where f : X Ñ Y and g : X1 Ñ Y1 are the two given proper maps.v): Recall the diagram set up in Exercise 8.8 on page 163

X XˆZ Y Y

X Z

Γg π

f

f˝g

where the square is Cartesian, and where Γg is the graph of g so that thecompositions of the two upper maps equals g. Since f ˝ g is proper and beingproper is stable under base change, the projection XˆZ Y Ñ Y is proper. Themap f is supposed to be separated so the graph Γg is a closed immersion. Itfollows that g being the composition g = π ˝ Γg of two proper maps, is proper.

o

Exercise 18.6 (Locallity on the target.) Properness is a property local on theˇ

target: Let f : X Ñ Y be a morphism and assume that there is an open coveringtUiu of Y such that each restriction f | f´1(Ui)

: f´1(Ui)Ñ Ui is proper, show thatthen f is proper. M

Exercise 18.7 Let X be a scheme, separated and of finite type over the fieldk, and assume that there is a closed immersion A1

k ãÑ X. Show that X is notproper over k. M

Projective morphisms are properWe proceed by describing one of the most useful properties projective morphismshave: they are proper maps. And in fact, most proper morphisms one meetswhen practising algebraic geometry are projective. There are others however,but to construct examples is not straight forward; the first varieties being properover C, but not projective, were found by Hironaka in 1960.

Projective morphisms are separated and of finite type, so to prove that theyare proper it remains to show that they are universally closed. This basically


relies on two facts, one is about sets closed under so-called spesializations andthe other about maps from spectra of valuation rings into projective spaces.

From topology we remember that compact subsets of a Hausdorff space areclosed. In the world of schemes almost no one is Hausdorff, so this fundamentalproperty is lost; quasi-compact subsets are mostly not closed (e.g. all openaffines are quasi-compact). But fortunately there is a nice substitute (formulatedin Proposition 18.6 below).

In point set topology taking limits of points from a subset is a populartechnique, and there is an analogue, although vague, for schemes. A point x ina scheme X is said to be a the specialization of another point y if x belongs to theclosure of y; that is, if x P ¯tyu. In case X is affine, this means that the inclusionpyĎ px holds between the corresponding prime ideals so that localizations arerelated by a map Apx Ñ Apy of local rings. When X is an integral scheme, alllocal rings OX,x lie naturally in the function field K(X), and it holds that x P ¯tyuif and only if OX,xĎOX,y and my XOX,xĎmx; in other words, if and only ifOX,y dominates OX,x.

We shall need the slightly stronger condition in the following lemma:

Lemma 18.11 For different points x and y in an affine integral scheme X = Spec A, itholds true that x P ¯tyu if and only if my XOX,xĎmx.

Proof: One of the implications is generally true, if x P ¯tyu the local ringOX,y dominates OX,x. Assume then that my XOX,xĎmx. Now OX,x = Apx andOY,y = Apy ; Since pAp X A = p for all prime ideals in A, we have the followingsequence of inclusions

py = my X A = my XOX,x X AĎmx X A = px.

Hence OX,xĎOX,y and my XOX,x = mx. o

One says that a set SĎX is closed under specialization if x P ¯tyu for y P Simplies that x P S. Clearly every closed set S is closed under specialization, butthe converse is not generally true as shows the following simple example.Example 18.12 Let S be any proper infinite subset of the affine line A1

k where kis a field of your choice. The only proper closed subsets of A1

k being finite, theclosure of S is the entire affine line, but S is certainly closed under specialization;indeed, every point in S is closed. K

Quasi-compact sets, however, behave better, and more generally, images of so-called quasi-compact morphisms. These are morphisms f : Z Ñ X so that X may becovered by open affines the inverse image of each under f being quasi-compact.

Proposition 18.13 Assume that f : Z Ñ X is a quasi-compact morphism ofschemes. Then the image f (Z) is closed if and only if f (Z) is closed underspecialization.


Proof: Replacing X by the closure Ęf (Z) we may assume that f is dominating,and the issue being topological, we may assume that both X and Z are reduced.If tUiu is the covering such that f´1(Ui) is quasi-compact, f (Z) is closed if andonly if each f (Z)XUi is closed. Thus we may replace X by one of the Ui’s andassume that X is affine, say X = Spec A, and that Z is quasi-compact. ThenZ will be covered by finitely many open affines, and because taking closurecommutes with forming finite unions, we may assume that Z is affine as well,say Z = Spec B.

Our morphism f is dominant so f 7 injective, and we may consider A as lyingin B, i.e. AĎ B. For every prime ideal pĎ A we have to exhibit a prime idealqĎ B such that p contains qX A. To that end, consider the multiplicative systemS = Azp in B. It is disjoint from pB, hence there is a prime ideal q in B maximalamong ideals containing pB and disjoint from S; and by construction, qX AĎ p.

o

Here comes the theorem

Theorem 18.14 Every projective morphism is proper.

Proof: We begin with a few reductions. Any projective morphism factors as thecomposition of a closed immersion ι : X Ñ Pn

S and a structure map π : PnS Ñ S,

so it will suffice to show that each π : PnS Ñ S is proper (closed immersions are

proper, and compositions of proper maps are proper). Since being closed is aproperty of maps local on the target, we may well assume that S is affine, sayS = Spec A.

Let ZĎPnA be closed subscheme, which we may assume is reduced and

irreducible. It is quasi-compact and the image π(Z) will be closed if and only ifit is closed under specialisation (Proposition 18.6 above). So pick a point x P Sthat specializes from a point η P π(Z); in other words, we have x P ¯tηu = W.There is a point ε in Z that maps to η. Consider the diagram:

OW,η OZ,ε K

OW,x R

¬

where the map ¬ is due to x being a specialization of η. Since π(ε) = η, themap π7 induces a local homomorphism OW,η Ñ OZ,ε which is injective becauseZ dominates W; that is the one marked . Finally, K is the fraction field ofOZ,ε. In the lower right part of the diagram, R is a valuation ring with fractionfield K that dominates OW,x; that is, it holds that mR XOW,x = mx. The relevant


geometric version of the diagram is as follows:

PnS Spec K

S SpecOW,x Spec R

πλ (18.1)

where the map Spec R Ñ S sends the generic point to η and the closed pointx0 to x; the map Spec K Ñ S sends the unique point in Spec K to η. Now, thepoint is that according to the next lemma, we may fill in diagram (18.1) with thedashed map λ. Then π(λ(x0)) = x, and x P π(Z). o

Lemma 18.15 (The Heavenly L’Hopitals rule) LetRĎK be a valuation ring andits fraction field. Every morphism Spec K Ñ Pn

K can be extended to a morphismSpec R Ñ Pn

K.Spec K Pn

K

Spec R

Proof: As explained in Example 11 on page 180 the map φ : Spec K Ñ PnK is

given by “homogeneous coordinates” (t0 : ¨ ¨ ¨ : tn) where the elements ti P Kare not all zero, moreover, they may be changed by a common non-zero factorwithout the map φ changing. Let ν denote the valuation on K correspondingto the valuation ring R, and let tr be such that ν(tr) is the smaller of the ν(ti)’s.Then v(tit´1

r ) = ν(ti)´ ν(tr) ě 0 and each tit´1r belongs to R. Which means that

(t1t´1r : ¨ ¨ ¨ : tnt´1

r ) is an R-point that coincides with φ on Spec K, hence it is thedesired extension. o

Exercises(18.8) The aim is to show that any local domain is dominated by a valuationˇ

ring. Let A be the local domain and K its fraction field.

a) (Chevalley’s lemma) Let x P K be a non-zero element such that neither x norx´1 belongs to A. Let a be an ideal in A. Show that either aA[x] or aA[x´1]

is a proper ideal.

b) Use Zorn’s lemma to show that there is valuation ring RĎK with AĎRand mR X A = mA.

c) Assume that A is Noetherian, show that A is contained in a dvr. Hint:Consider the integral closure B of A and localize B in a prime ideal of heightone.

(18.9) Let A be a domain.ˇ

a) Show that A =Ş

Ap where the intersection extends over all prime ideals inA that are associated to a principal ideal.


b) Show that if A is Noetherian and normal, every prime ideal p associated toa principal ideal is of height one, and that Ap is a dvr.

c) (Hartog’s Extension Theorem) Conclude that a Noetherian and normal domainA is the intersection of Ap with ht p = 1, and that each Ap is a dvr.

d) Show that if f is a rational function on a normal, integral and Noetherianscheme X which is defined in all points of height one, then f is definedeverywhere.

(18.10) This exercise is an application of Hartog’s Extension theorem, andˇ

may be considered to be a modest prelude to Zariski’s Main theorem. LetX = Spec B and Y = Spec A be two integral affine schemes and assume thatY is Noetherian and normal. Let f : X Ñ Y be a birational morphism that issurjective in codimension one; that is, every point y P Y such that the prime idealpy in B is of height one, lies in the image of f . Show that f is an isomorphism.(18.11) A subset SĎX of a scheme X is locally closed if it is the intersectionof an open and a closed set. Subsets that are finite unions of locally closedsets, are said to be constructible. Show that a constructible set is closed underspecialization if and only if it is closed.

M

Functions on proper schemesThe observation in Example 5 on page 349 inspires a result about a large classof schemes by way of a “hyperbola trick”—strikingly similar to the well-known“Rabinovitsch trick” from the theory of varieties. Turning the question in theexample around, one concludes that functions on a scheme proper over an affinescheme Spec A are very restricted (we saw an example already in Proposition 5.1on page 106). In fact they will be integral over A, and amazingly, the proof isalmost completely formal, just using the hyperbola-trick inspired by Example 5.

The proof is quite generally valid without any finiteness condition (althoughuniversally closed morphisms are quasi-compact, see Exercise 18.14 below),except for one thing. We need to invert function somewhere on X, so nilpotenceis an issue. Nilpotent functions per se pose no problems since they are integralover the base (they certainly fulfil an integral equation), but there are schemeshaving non-nilpotent function which are locally nilpotent, and we have to avoidthese schemes. For instance, the disjoint union X =

Ť

iPN Spec k[t]/ti maps toSpec k[t] and t is a section in OX(X) which is locally nilpotent.

In what follows we shall work in the category Sch/A of schemes over a ringA, and for simplicity we’ll drop the subscript A in products and also write An

for AnA.

The closed subscheme HĎA2 defined as H = V(xy´ 1), and which onecould be tempted to call a generalized hyperbola, will play a central role. It isisomorphic to Spec A[t, t´1] via the map A[x, y]/(xy´ 1)Ñ A[t, t´1] defined by


the assignments x ÞÑ t and y ÞÑ t´1; observe that both projection A1Â1 Ñ A1

sends H into A1zV(t). Moreover, H is also the fibre over 1 of the multiplicationmap A1 Â1 Ñ A1 which is induced by the map A[t]Ñ A[x, y] with t ÞÑ xy.

Theorem 18.16 Let X be a scheme such that every locally nilpotent globalsection of OX is nilpotent. Assume that X is universally closed over Spec A.Then Γ(X,OX) is integral over A.

Note that on both reduced or quasi-compact schemes every locally nilpotentsection of OX is nilpotent, so they comply with the first reqiurement.Proof: Let B = Γ(X,OX), and suppose that f is a non-zero element in B. Wemay certainly assume that f is not nilpotent (nilpotent elements are triviallyintegral), hence f is not locally nilpotent. According to Theorem 4.6 on page 98

we may view f as a map f : X Ñ A1. Then the distinguished open subset D( f )will be an open subset of Spec B, and Γ(U,OX) = B f , where U denotes theinverse image of D( f ) in X. We may invert f over U to get a map f´1 : U Ñ A1.Moreover, U is non-empty, since f is not locally nilpotent.

This map has a graph GĎU Â1ĎX Â1, and the crux of the proof isthat G is closed even in X Â1 (as a graph it is closed in U Â1, the affineline A1 being separated). We prove this separetely in a Lemma 18.10 below.When X is universally closed over A, the projection π : XÂ1 Ñ A1 is a closedmap. Consequently (Lemma 18.10) the image π(G) will be closed in A1 andcontained in A1zV(t). The ideal a of π(G) therefore satisfies a+ (t) = A[t], andwe may write 1 = F(t) + tG(t) where F and G are polynomials in A[t]. Now,that f´1 : U Ñ A1 factors through V(a), means that F( f´1) = 0, and It followsthat 1 = f´1G( f´1), which upon multiplication by a high power of f gives anintegral dependence relation for f . o

Lemma 18.17 The graph G is closed in XÂ1 and π(G) is contained in A1zV(t).

Proof: This is where the hyperbola enters the scene. The point is the equalityG = g´1H where g : XÂ1 Ñ A1 Â1 is given as g = f ˆ id, in other wordsin the following diagram where π2 denotes the second projection, the bottomsquare is Cartesian:

A1 A1

XÂ1 A1 Â1

G H

id

fîd

π2

h

Indeed, if S is any scheme and (a, b) : S Ñ XÂ1k and c : S Ñ H are morphisms

such that f ˆ id ˝ (a, b) = h ˝ c it holds true that ( f ˝ a, b) = (c, c´1). Hence a


takes values in U and (a, b) = (a, ( f ˝ a)´1) so (a, b) factors through G. Now,the upper square commutes and π2(H)ĎV(t) so π(G) lies in V(t) as well. o

There is a relation between finite and integral extensions: every finite ex-tension is integral, but the converse does not hold. There are even examplesof Noetherian domains A whose integral closure in its field of fraction is nota finite module, or for that matter, in any finite extension of the fraction field.These examples are all in characteristic p. Domains whose integral closure inany finite extensions of the fraction field is finite, are called Japanese rings (tohonour the mathematical entourage of Nagata, the hotbed of such eye-openingexamples). Examples of Japanese rings include domains of finite type over afield or over Dedekind rings of characteristic zero.

Direct from Theorem 18.9 we obtain the following corollary:

Corollary 18.18 Assume that X is an integral scheme that is universally closed overthe field K. Then Γ(X,OX) is an algebraic field extension of K. If X is proper, theextension is finite.

Proof: The space is Γ(X,OX) is an integral domain containing K whoseelements all are integral over K according to the theorem; hence it is an algebraicfield extension. If X is of finite type, the fraction field K(X) of X is a finitelygenerated extension of K, and as Γ(X,OX) is contained in K(X), it is finitelygenerated over K as well. Being algebraic, it is then finite. o

Proposition 18.19 Let A be a Noetherian Japanese ring, and let X be anintegral scheme proper over Spec A. Then Γ(X,OX) is a finite A-module.

Proof: We may well assume that the structure map π : X Ñ Spec A is dom-inating (if not, replace Spec A by the image of π which is closed and irre-ducible) with generic point η. Let K be the fraction field of A. The generic fibreXη = π´1(η) = XˆSpec A Spec K is proper over Spec K as properness is a prop-erty kept under base change, so the space of global sections Γ(Xη ,OXη )—whichlies contained in in the fraction field K(Xη)—is integral over K according to thetheorem.

Since X is assumed to be of finite type over A, the generic fibre Xη will be offinite type over K. The fraction field K(Xη) of Xη is therefore a finitely generatedfield extension of K, and being algebraic, it is a finite extension. By the theorem,Γ(Xη ,OXη ) is integral over A, and thus contained in the integral closure A of Ain K(Xη). Therefore it is finite, since A is finite and A is Noetherian. o

In the same assembly line, we find the corollary that if the genertic fibreXη of π satisfies Γ(Xη ,OXη ) = K and A is integrally closed, it ensues thatΓ(X,OX) = A. One way to ensure that Γ(Xη ,OXη ) = K is to require thatXη be geometrically connected; in other words, that Xη = Xη ˆSpec K Spec K is


connected where K is an algebraic closure of K. Indeed, then Γ(Xη ,OXη) = K

by the theorem, and since flat base change gives Γ(Xη ,Oη)bK K = Γ(Xη ,OXη),

we deduce that Γ(Xη ,Oη) = K. Thus we have shown

Proposition 18.20 Assume that X is proper over the integrally closed domainA and that the generic fibre is geometrically connected, then Γ(X,OX) = A.

Proposition 18.21 Let X and Y be two schemes and assume that Y integraland normal. Let f : X Ñ Y be a proper morphism with geometrically connectedgeneric fibre. Then f 7 : OY Ñ f˚OX is an isomorphism.

Proof: Since the open affines form a basis for te topology on Y, it suffices toprove that it holds that f 7U : Γ(U,OY) Ñ Γ( f´1(U),OX) is an isomorphism foreach open affine subset UĎY.

Let U = Spec A. Because Y is assumed to be normal, A is integrally closed inK. Moreover the restriction f | f´1(U) of f to the inverse image f´1(U) is properand has the same geometric fibre as f , and we may deduce citing xxxx thatf 7 : A = Γ(U,OY)Ñ Γ( f´1(U),OX) is an isomorphism. o

Exercises(18.12) If X is a scheme and f P Γ(X,OX), one says that f is locally nilpotentif for each x P X, there is an integer nx such that f nx = 0 in OX,x. It mayhappen that a global section f is not nilpotent but is locally nilpotent. Showthat if X is quasi-compact this cannot happen. Let X be the disjoint unionX =

Ť

n Spec k[t]/tn. Show that t is a locally nilpotent section of OX that is notnilpotent.(18.13) Let tKiu be an infinite collection of field extensions of a field k andˇ

let X =Ť

i Spec Ki be the disjoint union of their spectra. Show that X is notuniversally closed over k.(18.14) Let X and Y be two schemes. The aim of this exercise is to prove thatˇ

every universally closed morphism f : X Ñ Y is quasi-compact.

a) Consider, in any topological space X, the set Σ of pairs (U, Z) where UĎXis open and ZĎU is a discrete subset closed in U. Define a partial order onΣ by declaring (U, Z) ď (U1, Z1) if UĎU1 and Z = Z1 XU. Show that theset Σ has a maximal element. Hint: What else than Zorn’s lemma;

b) Assume that X is not quasi-compact. Show that there exists a closed infiniteand discrete subset Z of X;

c) If Y is affine, and f : X Ñ Y is universally closed, then X is quasi-compact;

d) Conclude that any universally closed morphism is quasi-compact.

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finite and quasi-finite morphisms 358

18.6 Finite and quasi-finite morphisms

A map f : X Ñ Y is quasi-finite if is of finite type and all its fibres are discretesubspaces.

Lemma 18.22 A morphism f : X Ñ Y is quasi-finite if and only if each of the scheme-theoretical fibres Xy = XˆY k(y) is a finite k(y)-schemes.

Proof: The fibre Xy is of finite type over the field k(y) hence it is finite if andonly it is discrete; this hinges on the algebraic fact that an algebra over a field isof finite dimension if and only if A has finitely many prime ideals which all aremaximal. o

Earlier, in Chapter 6 (Section ?? on page ??), we introduced the notion offinite morphisms, and as the names indicate, there is a relation between finiteand quasi-finite maps: finite maps are quasi-finite, which is not difficult toestablish. Finite maps are as well proper, which also is effortless to prove. Amuch deeper result, due to Chevalley, is that under certain generic conditionsthe converse holds true; that is, quasi-finite and proper morphisms are finite.Most of this section is devoted to a proof of this. This result belongs to a circleof ideas and results mostly going under the label of “Zariski’s main theorem”;we intend to present a proof of one of the more modest version, which is withinreach with the tools we have developed so far.

We begin with the effortless:

Lemma 18.23 Finite maps are quasi-finite and proper.

Proof: Recall that when f : X Ñ Y is finite, the target Y may be coveredby open affine subsets tUiu such that f´1(Ui) is affine, and f 7 makes eachBi = Γ( f´1(Ui),OX) a finite modules over Ai = Γ(Ui,OY). From this ensuesthat BibAi k(y) is a finite vector space over k(y) = Ai/py, and the schemetheoretical fibre Spec BibAi k(y) over y P Ui has only finitely many ideals allwhich are maximal.

To prove that f is proper, we may, since being proper is a property local onthe target, assume that both X and Y are affine, say Y = Spec A and X = Spec B.Let then T Ñ Y be any morphism. Replacing T by members of an open affinecover, it suffices to prove the case when T = Spec C is affine. All affine maps areseparable, and fT is obviously of finite type. It remains to see that fT is closed,for which we cite the Going-Up Theorem. Indeed, fT is induced by the ring mapf 7b idC : AbA C = C Ñ BbA C, the ring BbA C is a finite module over C sinceB is a finite module over A, and Going-Up applies. o

Zariski’s main theoremOne of the more famous theorems in in algebraic geometry, fundamental tounderstand birational morphisms, is Zariski’s Main theorem. In fact, there is


cluster of theorems, formulated and valid in different context, developed aroundthe original form. We shall present the most modest version and give a simpleand geometric proof the more elaborated versions being out of reach with thetools we have at hand.

Theorem 18.24 Let X and Y be integral schemes. Assume that f : X Ñ Yis a projective and quasi-finite morphism such that f 7 : OY Ñ f˚OX is anisomorphism. Then f is an isomorphism.

The map f need not be projective for the result to hold true, being propersuffices, but this requires technics we do not yet have. The point is we needany pair of points in a fibre to be contained in an open affine in order to applyLemma 18.5.Proof: We begin with establishing that f is birational. Let η and ε be thegeneric points of respectively Y and X, and denote by K the function fieldof Y; then K = OY,η . Since f is quasi-finite, the scheme-theoretical fibre Xη

is the finite disjoint union of schemes SpecOX,εi as εi runs through the fibreover η. Moreover each OX,εi is finite over K, and Γ(Xη ,OX,η) =

ś

i OX,εi . Butby flat base change (Exercise 7.14 on page 149), f 7 induces an isomorphismK = OY,η Ñ Γ(Xη ,OXη ) so there is only one OX,εi which must equal K; henceOX,ε = K.

Since f 7 is an isomorphism by hypothesis, our task is to establish that fis a homeomorphism. The map f being continuous and proper (in particularclosed), this will be the case if and only if it is bijective. The easy part is that f issurjective: the image of f is closed because f is proper, and it is dense becausef is birational, hence the image equals Y.

Turning to the injectivity we aim at a contradiction and assume that thepoints in the fibre over y P Y are x0, . . . , xr. The crucial idea is to exhibit a“rational function defined in all xi’s, but which does not assume the same valuein all the xi”, and then descend that function to Y.

In precise terms, we search for an element g in K that lies in all the local ringsOX,xi and belongs to one of the maximal ideals mxi but not to all. By assumptionthe fibre is discrete, so no xi is a specialization of any of the other points, hencethere is no dominance relations among the local rings OX,xi . Pick one of thepoints, say x0; then for all i ą 0 it holds that mxi XOX,x0 Ę mx0 , which meansthat we may find gi P mxi XOX,x0 , but gi R mx0 . Then the product g = g1 . . . gr

lies in mi XOX,x0 for i ą 0 (in particular it lies in each OX,xi ), but not in mx0 .The set Z of points where g is not defined is closed and does not meet thefibre over y, and since f is proper, the image f (Z) is a closed subset of Y notcontaining y. It follows that the entire fibre over y is contained in an open set ofthe form f´1(V) over which g is defined. But now Γ(V,OY) = Γ( f´1(V),OX)

(they have been identified by f 7), hence g| f´1(V) P Γ(V,OY) which gives thedesired contradiction since all the ideals mi XOY,y are equal (heuristically: the


function g assumes the same value in all points of the fibre). o

Note that we needed only one point in the fibre that was not a specializationof any other point there; in other words, if a point is isolated in the fibre, it is theonly one.

Corollary 18.25 (Modest Zariski’s main theorem) Assume that A is a normaldomain and X an integral scheme. Let f : X Ñ Spec A be a proper, quasi-finite andbirational morphism. Then f is an isomorphism.

Proof: Combine the theorem with Proposition 18.14 on page 357. o

Chevalley’s theoremThere are some notorious examples of domains whose integral closure in thefraction field is not a finite module over the domain, and, as a to tribute toNagata, Grothendieck called domains that do not suffer from this pathologyfor Japanese rings; a wide class of rings which for instance encompasses alldomains of finite type over a field or over a complete Noetherian domain. So areduced scheme Y is Japanese if there is an open affine covering tSpec Aiu ofY with the integral closure of each Ai being a finite Ai-module; or in short, thenormalization morphism rY Ñ Y is finite.

The following theorem is due to Chevalley, again the version we present isnot the most general. With some book-keeping one may discard the limitationsthat X and Y be integral, but more seriously, to discard the Japanese hypothesisheavier artillery than ours is needed.

Theorem 18.26 Let X and Y be integral schemes. Assume that X Ñ Y is aproper quasi-finite map and Y is Japanese. Then f is finite.

Proof: Being finite is a property local on the target, so we may well assumethat Y = Spec A, and we let Z = Spec Γ(X,OX).By the japanese assumptionabout A, the scheme Z is finite over A. We have a canonical factorization

X

Spec Γ(X,OX)

Y

g

Almost by definition g7 : OZ Ñ g˚OX is an isomorphism, and g is proper by xxx.Hence by xxx g is an isomorphism. o

Lemma 18.27 Let X be a quasi-compact scheme and f P Γ(X,OX) a section. Thens P Γ(X f ,OX), then sn f extends to a section for n ąą 0.

the valuative criterion* 361

Proof: Since X is quasi-compact it is covered by fintely many open affinesubsets Ui Spec Ai, and X f XUi = Spec Ai f . Hence ni f s P A. and they patch toa section in X. o

18.7 The valuative criterion*

The discrete valuation rings are of the simplest kind of rings; they are local, andthe only two prime ideals are the maximal ideal and the zero ideal, so they areone-dimensional. The spectrum Spec R of a dvr R has two points, a closed pointw, which corresponds to the maximal ideal m, and the generic point η. Thus thespectrum Spec K of the fraction field K is an open subscheme supported at η.General valuation rings can be much more complicated, but still it holds that amorphism Spec R Ñ X is determined by two points x and y with x P ¯tyu and afield extension k(y)ĎK.

The underlying heuristics behind the valuative criteria, are that Spec R is a“tiny piece of a curve” centred round the point x; and that Spec K is the “the tinypiece of curve with the closed point x removed”.

The heuristics say that if two different maps into X agree on a dense set,their difference will already be visible on “tiny curves”; i.e. there are two “tinypieces of curve near x” that coincide away from x. And for the image of a mapto be closed, it suffices that it be closed under “limits” (i.e. specialization) alongtiny curves.

The basic staging of the valuative criteria is diagrams shape like

Spec(K) X

Spec(R) S

ι

f

h

λ (18.2)

where R is a valuation ring and K its fraction field. The map f is the one underscrutiny, and ι and h are variable morphisms. The wild card in the set-up is theextension map λ; for f to be proper it must always exist unambiguously, andfor separated maps f it must be unique when it exists.

Theorem 18.28 (Valuative criterion) Let X be a quasi-compact schemeover S; that is, the structure map X Ñ S is quasi-compact.

i) X is separated over S if and only if for all set-ups as in (18.2) theextension λ is unique if it exists;

ii) Assume that X is of finite type over S. Then X is proper over S if andonly if for all set-ups as in (18.2) the extension λ exists and is unique.

When X is Noetherian, it suffices to consider discrete valuation rings.

Proof: The proof of the part of ii) that existence and uniqueness of the extensionλ implies f proper, is mutatis mutandis the same as the proof of Theorem 18.7,

the valuative criterion* 362

but the extension there guaranteed by “l’Hopitals rule”, is now guaranteed byhypothesis. The converse is left as an exercise.

Proof of i). We’ll show that the diagonal ∆X/S is closed assuming theextension λ is unique and leave the converse to the students. Temporarily write∆ for the image of the diagonal map; that is, ∆ = ∆X/S(X).

Aiming at an absurdity, assume that ∆ is not closed and pick a z P ∆ withz R ∆. The two projections from XˆS X to X then take different values at z. ByProposition 18.6 on page 351 (∆ is quasi-compact over S) we may find a pointy P ∆ with z P ¯tyu = W. Give W the reduced scheme structure and considerthe local ring OW,z. It is an integral domain and is therefore dominated by avaluation ring R in its fraction field K. The two projections πi : X ˆS X Ñ Xcomposed with the map Spec R Ñ SpecOW,z Ñ W Ñ X ˆS X give differentmaps into X, but generically on W they coincide, hence the maps restrict to thesame map Spec K Ñ X. And we have an instance of the staging (18.2) with twodifferent λ’s. o

Exercise 18.15 The notation is as in 18.2.

a) Show that if f is proper, an extension λ in (18.2) exists and is unique;

b) Show that if f is separated, the extension λ in (18.2) is unique.

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Chapter 19

Curves

19.1 Curves

We will in this chapter assume that k is an algebraically closed field. Recall thata variety over k is a separated, integral scheme of finite type over k. Throughoutthis chapter, a curve will mean a 1-dimensional variety over k.

19.2 The genus of a curve

Definition 19.1 The arithmetic genus of X is defined as the number

pa(X) = dimk H1(X,OX)

The geometric genus of X is defined as

pg(X) = dimk H0(X, ΩX)

Note that χ(OX) = dim H0(X,OX) ´ dim H1(X,OX), so we get pa(X) =

1´ χ(OX).These numbers are defined using different sheaves, and there is no a priori

reason to expect that they should have anything to do with each other. However,we shall see later in the chapter that there is a strong relation between them:pa = pg whenever X is regular. For the time being we will still refer to pa as thegenus of X.Example 19.2 When X = P1, we have H1(P1,O) = 0 and H0(P1, ΩP1) =

H0(P1,OP1(´2)) = 0, so the genus is 0. K

Example 19.3 Let X Ă P2 be a plane curve, defined by a homogeneous polyno-mial f (x0,1 , x2) of degree d. In Chapter 13, we computed that H1(X,OX) » k(

d´12 ).

Hence the genus of X is (d´1)(d´2)2 . K

divisors on curves 364

19.3 Divisors on Curves

Let X be a non-singular curve. Since X has dimension 1 and we are workingover an algebraically closed field, a Weil divisor D on X is just a finite formalcombination of closed points on X,

D =ÿ

ni pi

where pi P X(k). We say that D is effective if each ni ě 0.The degree of D is defined as the sum deg D =

ř

ni. Equivalently, if we viewD as a Cartier divisor given by the data (Ui, fi), then

deg D =ÿ

xPX

vx(D)

where vx(D) = vx( fi) is the valuation of fi along x.Recall that each Weil divisor determines an invertible sheaf OX(D), which

over an open set U takes the value

OX(D)(U) = t f P K|(div f + D)|U ě 0u

Then D is effective if and only if Γ(OX(D)) ‰ 0. In particular, if Γ(X,OX(D))

has dimension at least 2, there is a second effective divisor D1 =ř

miqi suchthat D and D1 are linearly equivalent.

The canonical divisorWhen X is a non-singular curve, the sheaf of differentials ΩX is a locally freesheaf of rank 1, i.e., an invertible sheaf. Thus ΩX gives rise to a divisor, whichwe denote by KX. We can describe KX as a Cartier divisor as follows. A localsection of ΩX is of the form ω = fUdx where fU P OX(U). We can then define aCartier divisor with the data

(U, fU)

This is well-defined, because on the overlaps UXV, two sections fUdx, fVdy arerelated by fU = dy

dx fV , and dydx is a unit in OX(U XV).

Any other section ω1 of ΩX has the form f ω for some f P Kˆ. Therefore,

div(ω1) = div(ω) + div( f )

so the divisor class [KX] := [div(ω)] P Cl(X) associated to ω is independent ofthe chosen ω. We call this the canonical class of X.

Definition 19.4 A canonical divisor of X is a divisor KX, so that OX(KX) »

ΩX.

Example 19.5 Let X = P1k , and consider the open set U0 = D+(x0). On U0 we

have a local coordinate t = x1x0

, and we consider the differential form dt = d(

x1x0

).

morphisms of curves 365

On the overlap U0 XU1, we have

d(

x0

x1

)= d(t´1) = ´t´2dt = ´

(x0

x1

)2

d(

x1

x0

)It follows that ΩX » OP1(´2). Thus a canonical divisor is given by D = ´2pwhere p is a point. K

19.4 Morphisms of curves

For a closed point x P X, the local ring OX,x is a discrete valuation ring andthere is an element t P OX,x generating the maximal ideal m. We say that t is alocal parameter or a uniformizing parameter if vx(t) = 1. Note that we can alwaysnormalize the valuation so that a generator of m has valuation 1.

Proposition 19.6 Let X be a non-singular projective curve over k and let Y beany curve. Let f : X Ñ Y be a morphism. Then either

i) f (X) is a point in Y; orii) f (X) = Y;

In the case (ii) K(X) Ě K(Y) is a finite extension, f is a finite morphism and Yis also projective.

Proof: Since X is projective, the image f (X) must be closed in Y. On the otherhand X is irreducible, and hence so is f (X). Hence f (X) is either a point or allof Y.

If f (X) = Y, then f is dominant, so we get an inclusion of function fieldsK(Y) Ď K(X). Both of these fields are finitely generated over k of the sametranscendence degree (i.e., 1), so the extension is finite algebraic. Let V =

Spec B Ă Y be an open set of Y and let A denote the integral closure of B inK(X). Then by [Hartshorne I.6.7] we have Spec A = f´1(V). So since A is afinite B-module, we have that f is finite. o

We also have the following theorem of [Hartshorne ([?]) Ch. I. 6.12]:

Theorem 19.7 There is an equivalence of categories

i) Non-singular projective curves, and dominant morphismsii) Function fields K of dimension 1 over k and k-homomorphisms.

The equivalence is defined as follows: From a non-constant morphismf : X Ñ Y we obtain an induced morphism f 7 : K(Y) Ñ K(X) between thefunction fields. Conversely, for each function field K, one considers the set S ofvaluations v : Kˆ Ñ Z and shows that it admits a topology and furthermore ascheme structure, making it into a 1-dimensional scheme over k.

morphisms of curves 366

Pullbacks of divisorsRecall that given a morphism f : X Ñ Y we have a well-defined pullback mapon Picard groups

f ˚ : Pic(Y)Ñ Pic(X)

sending the class of an invertible sheaf L on X to that of f ˚L. We also defined apullback map on Cartier divisors f ˚ : CaDiv(Y)Ñ CaDiv(X), by pulling backthe local equations from Y to X. In the setting of curves, we can make this pullback a little bit more explicit.

Let t P OY,y denote the uniformizing parameter. We consider t as an elementof K(X) via the field extension K(X) Ě K(Y). In particular, we can talk aboutthe valuations of t in each local ring OX,x and define the Weil divisor

f ˚(y) =ÿ

xP f´1(y)

vx(t)x

(Since f is finite, there are only finitely many preimages of y, and so the sum isfinite). The expression is also independent of the choice of t: If t1 is a differentuniformizing parameter, we can in any case write t1 = ut where u is a unit, sothat vx(t) = vx(t1).

Thus we get a well defined map

f ˚ : Div(Y)Ñ Div(X)

which descends to the above map f ˚ : Pic(Y)Ñ Pic(X).

Lemma 19.8 We have deg f ˚D = deg f ¨ deg D.

Proof: TODO. o

Definition 19.9 For a morphism f : X Ñ Y, we call the number ex = vx(t)the ramification index of f at x. If ex ą 1, we say that f is ramified at x.

Example 19.10 Let A = k[x] and let B = k[x, y]/(x´ y2) » k[y] where k is afield. Let X = Spec B and let Y = Spec A. Let f : X Ñ Y be the morphisminduced by the inclusion A ãÑ B (thus x ÞÑ y2).

The morphism f is ramified only at the origin (0, 0), and here the ramificationindex is two. Indeed, x is a uniformizing parameter of OY,y = k[x](x), while y isthe uniformizer of OX,x = B(x,y). Then we have v(0,0)(x) = v(0,0)(y2) = 2. K

The reader might notice a resemblance between the previous example andExample 16, where ramification was defined in terms of the relative sheaf ofdifferentials ΩX|Y. In that example, ΩX|Y was a torsion sheaf supported on thesingle point (0, 0). This correspondence between the two notions of ramificationis a general fact (at least in characteristic 0), and we have the useful formula forthe ramification indexes of curves:

ep = length(ΩX|Y)p + 1


Morphisms to P1

Let X be a non-singular curve, and let f P K. Then f induces a morphism φ :X Ñ P1 in the following way. Let U = X´ Supp f´1(8) and V = X´ f´1(0),so that f P OX(U) and 1/ f P OX(V).

Write P1 = Proj k[x0, x1]. On D+(x0), the map k[x1/x0] Ñ OX(U) sendingx1/x0 ÞÑ f , induces a map φU : U Ñ D+(x0) Ď P1. Similarly, on D+(x1), weget a map φV : V Ñ P1. These maps coincide on U XV, and therefore we get amorphism

φ : X Ñ P1

This morphism is non-constant, hence finite.

19.5 Hyperelliptic curves

Let us recall the hyperelliptic curves defined in Chapter 3. For an integer g ě 1we consider the scheme X glued together by the affine schemes U = Spec A andV = Spec B, where

A =k[x, y]

(ý2 + a2g+1x2g+1 + ¨ ¨ ¨+ a1x)and B =

k[u, v](´v2 + a2g+1u + ¨ ¨ ¨+ a1u2g+1)

As before, we glue D(x) to D(u) using the identifications u = x´1 and v =

x´g´1y.In Chapter ?? we showed that the genus of X was g and claimed that X was

actually projective.Let us examine the last point in more detail, and give a new projective

embedding of X. To do this, we will need to work out the groups Γ(X,OX(nP))for a point p P X.

Let us for simplicity assume that a2g+1 = 1. Let p be the unique closed pointgiven by V(u, v) in X. In the local ring at p, we have

u = v2(1 + a2gu + ¨ ¨ ¨+ a1u2g)´1 = v2(unit),

and hence vOp generates mp. We compute the some valuations of elements inOp:

νp(v) = 1, νp(u) = 2, νp(x) = ´2, νp(y) = 1 + (g + 1)(´2) = ´(2g + 1)

We’ve seen that Γ(X,OX) = k, which agrees with our expectation that thereare no non-constant regular function on a projective curve. Let us consider thecase where the rational functions are allowed to have poles at p (and only at p).In other words, we are interested in elements s P Γ(X,OX(p)). Note that thepoint p does not lie in U; this means that s is regular there, and hence can be


viewed as a polynomial in x, y. Now, as A = k[x]‘ k[x]y as a k[x]-module, we canexpress any element s can be expressed as f (x) + h(x)y. We can then calculate

νp( f (x) + h(x)y) = mintνp( f (x)), νp(h(x))νp(y)u

= mint´2 deg f ,´(2 deg h + 2g + 1)u

Thus, since we assume g ě 1, any non-constant rational function with a pole atp must have valuation ď ´2 there, and hence we have Γ(X,OX(p)) = k.

On the other hand for the divisor 2p we gain an extra section, correspondingto the rational function x:

Γ(X,OX(2p)) = kt1, xu

Note that O(2p)p = Op ¨ x. The section x P Γ(X,OX(2p)) is nonvanishing atp, while the section 1 P Γ(X,OX(2p)) is vanishing at p, since 1 = u ¨ x andu P m Ă Op. Note that the linear series generated by 1, x generates OX(2p)everywhere, inducing the morphism

XϕÝÑ P1

(x, y) ÞÑ (1 : x)

This morphism is exactly the double cover above.It get’s even more interesting if we allow even higher order poles at p. The

computation above shows that

Γ(X,O(3p)) =

#

kt1, x, yu if g = 1

kt1, xu if g ą 1

If we try g = 1, we can who, using the embedding criterion of Chapter 16, thatthe sections x0 = 1, x1 = x, x2 = y give an embedding

X ãÑ P2k

(x, y) ÞÑ [1 : x : y]

The image is even seen to be a non-singular cubic curve: One computes thatΓ(X,O(6p)) is 6-dimensional, but we have 7 global sections: 1, x, y, x2, xy, x3, y2.That means that there must be some relation between them. But of course this isthe relation

y2 = a3x3 + a2x2 + a1x

giving the following relation in P2:

x22x0 = a3x3

1 + a2x0x21 + a1x2

0x1

For g = 2, 3p does not give a projective embedding. However, the map given by5p gives something interesting: We obtain

Γ(X,OX(5p)) = kt1, x, x2, yu


These sections generate OX(5p), so we obtain a morphism

φ : X Ñ P3

given by the sections u0 = 1, u1 = x, u2 = x2, u3 = y of L = OX(5p). Notice thatu0u2 ´ u2

1 = 0, so X lies on a quadric surface. In fact, the image of φ is preciselythe relations between the sections:

C = V(u2

1 ´ u0u2, u20u2 + u3

2 ´ u1u23, u2

0u1 + u1u22 ´ u0u2

3)Ă P3

k

The map φ is in this case a closed immersion, showing that X is projective.

Chapter 20

The Riemann–Roch theorem

When X is a projective curve over a field k, the cohomology groups Hi(X, F) arefinite-dimensional k-vector spaces and we define

hi(X,F ) := dimk Hi(X,F )

Note that in this case, hi(X,F ) = 0 for all i ě 2, so we have two cohomologygroups h0(X,F ) and h1(X,F ) to work with.

Recall, that we defined for a sheaf F , the Euler characteristic χ(F ) as thealternating sum of the hi(X,F ). One useful property of χ(X,´) is that it isadditive on exact sequences:

Lemma 20.1 Let 0 Ñ F 1 Ñ F Ñ F2 Ñ 0 be an exact sequence of sheaves. Then

χ(F ) = χ(F 1) + χ(F2)

This follows because if 0 Ñ V0 Ñ V1 Ñ ¨ ¨ ¨ Ñ Vn Ñ 0 is an exact sequenceof k-vector spaces, then

ř

i(´1)i dimk V = 0. Applying this to the long exactsequence in cohomology gives the claim.

The most important sequence we will encounter is the ideal sheaf sequenceof a point p P X, which takes the form

0 Ñ OX(´p)Ñ OX Ñ k(p)Ñ 0 (20.1)

where the first map is the inclusion and the second is evaluation at p. Here wehave identified the ideal sheaf mp Ă OX by the invertible sheaf OX(´p), and thesheaf i˚Op with the skyscraper sheaf with value k(p) at p. If L is an invertiblesheaf, we can tensor (20.1) by L and get

0 Ñ L(´p)Ñ L Ñ k(p)Ñ 0 (20.2)

where L(´p) is the invertible sheaf of sections of L vanishing at p. Taking χ, weget

χ(L(´p)) = χ(L)´ χ(k(´p)) = χ(L)´ 1.

371

Theorem 20.2 (Easy Riemann–Roch) Let X be a smooth projective curve ofgenus g and let D be a Cartier divisor on X. Then

χ(OX(D)) = h0(OX(D))´ h1(X,OX(D)) = deg D + 1´ g

Proof: Let p P X be a point and consider the sequence (20.2) with L = OX(D +

p). Then, as we just saw, χ(OX(D + p)) = χ(OX(D)) + 1. Also the right-handside of the equation above increases by 1 by adding p to D (since deg(D + p) =deg D + 1). This means that the theorem holds for a divisor D if and only if itholds for D + p for any closed point p. So by adding and subtracting points,we can reduce to the case when D = 0. But in that case, the left hand side ofthe formula is by definition dimk H0(X,OX)´ dimk H1(X,OX) = 1´ g, whichequals the right hand side. o

The formula above is extremely useful because the right hand side is soeasy to compute. The number we are really after is the number h0(X,OX(D)),since this is the dimension of global sections of OX(D). This in turn wouldhelp us to study X geometrically, since we could use sections of OX(D) todefine rational maps X 99K Pn. So if we, for some reason, could argue that say,H1(X,OX(D)) = 0, we would have a formula for the dimension of the space ofglobal sections of OX(D).

In any case, we can certainly say that h1(X,OX(D)) ě 0, so we get thefollowing bound on h0(X,OX(D)):

Corollary 20.3 h0(X,OX(D)) ě deg D + 1´ g

Example 20.4 A typical feature is that H1(X,OX(D)) = 0 provided that thedegree deg D is large enough. This is essentially a consequence of Serre’stheorem. To give an example, consider again the case where X is a hyperellipticcurve of genus 2, as in the introduction. We have the following table of thevarious cohomology groups Hi(X,OX(np)) for the point p = (u, v):

D 0 1p 2p 3p 4p 5p 6p 7pχ(OX(D)) -1 0 1 2 3 4 5 6

H0(X,OX(D)) 1 1 2 2 3 4 5 6

H1(X,OX(D)) 2 1 1 0 0 0 0 0

and it is not so hard to prove directly using the Cech complex that H1(X,OX(np)) =0 for all n ě 4. K

Fortunately, there are more general results which tell us when H1(X,OX(D)) =

0. This is due to the following fundamental theorem:

372

Theorem 20.5 (Serre duality) Let X be a smooth projective variety of dimen-sion n and let D be a Cartier divisor on X. Then for each 0 ď p ď n,

dimk Hp(X,OX(D)) = dimk Hn´p(X,OX(KX ´D))

So if X is a curve, we get that h1(X,OX(D)) = h0(X,OX(KX ´D)) and theRiemann–Roch theorem takes the following form:

Theorem 20.6 (Riemann–Roch) Let X be a projective curve of genus g andlet D P Div(X) be a divisor. Then

h0(X,OX(D))´ h0(X,OX(KX ´D)) = deg D + 1´ g

This is a much stronger statement than the Riemann–Roch formula we hadbefore, since group H0(X,OX(KX ´D)) is easier to interpret: it is the space ofglobal sections of the sheaf associated to the divisor KX ´ D, or equivalentlyΩX(´D). It is usually easier to argue that there can be no such global sectionsof this sheaf. For instance, in the case deg D ą dim KX then KX ´D cannot beeffective: effective divisors

ř

ni pi have non-negative degree.So what is this degree of the canonical divisor KX? From Serre duality,

we get that H0(X,OX(KX)) and H1(X,OX) have the same dimension, so thegeometric genus and arithmetic genus agree:

pg = pa = g.

Then applying the Riemann–Roch formula to D = KX, we get

g´ 1 = dimk H0(X,OX(K))´ dimk H0(X,OX(KX ´ K)) = deg K + 1´ g

and so deg KX = 2g´ 2. This observation gives us

Corollary 20.7 Suppose that D is a Cartier divisor of degree ą 2g ´ 2. ThenH1(X,OX(D)) = 0, and

dimk H0(X,OX(D)) = deg D + 1´ g

Moreover, if deg D = 2g´ 2, then H1(X,OX(D)) ‰ 0 only if D „ KX.

Chapter 21

The Serre duality theorem

The aim of this chapter is to prove the following:

Theorem 21.1 (Serre duality) Let X be a smooth projective curve over analgebraically closed field k. Then there is an invertible sheaf ωX on X, togetherwith an isomorphism t : H1(X, ωX)Ñ k, such that for any coherent sheaf F onX, there is a perfect pairing

H0(X,F )ˆ H1(X, ωX bF_)Ñ H1(X, ωX) » k (21.1)

In particular, H1(X, ωX bF_) » H0(X,F )_.

The sheaf ωX is called a dualizing sheaf. We prove the existence first forX = P1 (in which case ωP1 = OP1(´2)), and then for a general curve using aNoether normalization f : X Ñ P1. The fact that P1, and hence X, can be acovered by two affine open sets simplifies things a lot. In particular, we have aconcrete interpretation of the first cohomology group H1 of a sheaf, in terms ofthe Cech complex.

The existence of the dualizing sheaf ωX is usually not enough for applicationsor explicit computations. The important point is that, in the smooth case, thecotangent sheaf, which is easier to study (e.g., using the adjunction formula).

Theorem 21.2 If X is a non-singular, projective curve, the dualizing sheaf ωX

is isomorphic to the cotangent sheaf ΩX.

There are of course several known proofs of this result [?], [?], [?], [?]. Ourproof is quite elementary, in the sense that it requires no derived functors, Ext-sheaves, residues, adeles, etc. The ad hoc approach here is however much lessconceptual than the standard proofs, and give essentially no information aboutthe isomorphism H1(X, ΩX) » k.

proof of serre duality for x = 1374

21.1 Proof of Serre duality for X = P1

Lemma 21.3 Serre duality holds for P1 with ωP1 = OP1(´2) and F is an invertiblesheaf.

Proof: Legge til dette? o

Proposition 21.4 Serre duality holds for P1 with ωP1 = OP1(´2) for anycoherent sheaf F .

Proof: Let F be a coherent sheaf, and pick a surjection E Ñ F Ñ 0, where E

is a direct sum of invertible sheaves of the form OP1(a). Letting K denote thekernel, there is an exact sequence

0 Ñ K Ñ E Ñ F Ñ 0 (21.2)

Taking duals and tensoring with OP1(´2), we also have a sequence

0 Ñ F_(´2)Ñ E _(´2)Ñ F_(´2)Ñ 0 (21.3)

Taking the two long exact sequences of (21.2) and (21.3), we get a diagram

H1(X, K ) H1(X, E ) H1(X,F ) 0

H0(X, K _(´2))_ H0(X, E _(´2))_ H0(X,F_(´2))_ 0

νK νE νF

By assumption, the middle arrow is an isomorphism. Hence νF is surjective forevery coherent sheaf F . Applying this to K , we see that νK is also surjective,and thus νF is also an isomorphism by the 5-lemma. o

21.2 Two cohomological lemmas

Lemma 21.5 Let π : X Ñ Y be an affine morphism of varieties. Then for each coherentsheaf F on X, and i ě 0, we have a canonical isomorphism

Hi(X,F ) = Hi(Y, π˚F ).

Proof: Let U = tUiu be a finite affine covering of Y such that Hi(X, π˚F ) iscomputed by the Cech complex C‚(Ui, π˚F ). The hypotheses give that X iscovered by the affine subsets π´1(Ui). The lemma follows simply because theCech complexes of F and π˚F with respect to the respective coverings are thesame. o

Lemma 21.6 ("Kunneth formula") Let V and X be varieties over k with V affine.Let F denote a coherent OV-module and let G denote a coherent OX-module and writep, q : XˆX Ñ X for the two projections on XˆX. Then there is a natural isomorphism

Hi(V ˆ X, p˚F b q˚G) = Γ(V,F )bk Hi(X,G) (21.4)

schemes obtained by gluing two affines 375

Proof: Let U = tUiu denote an open affine covering of X so that C‚(U ,G)computes the cohomology group Hi(X,G). Tensoring C‚(U ,G) with the moduleM = Γ(V,F ) gives a complex C‚(U ,G)bk M which is easily seen to computethe cohomology of both sides of (21.4). o

21.3 Schemes obtained by gluing two affines

If X is a non-singular projective curve over k, we can pick a Noether normaliza-tion π : X Ñ P1, which is affine, finite and flat.

Recall the standard gluing construction of P1 as U YU1 where U = Spec A,and U1 = Spec A1, and A = k[a] and A1 = k[a1]. The gluing is defined bythe isomorphism D(a) = Spec Aa » Spec A1a1 = D(a1), using the isomorphismτ : Aa Ñ A1a1 given by τ(a) = a1´1.

Because the morphism π is affine, we find that also X can be covered bytwo affine subsets π´1(U), π´1(U1). We write V = Spec B and V1 = Spec B1 forthese subsets. Note that π|V (resp. π|V1) is induced by a ring map u : A Ñ B(resp. u1 : A1 Ñ B1), so that b = u(a) (resp. b1 = u1(a1)). Thus X is obtained bygluing V and V1 along Spec Bb and Spec B1b1 using an isomorphism σ : Bb Ñ Bb1 ,which is compatible with π, in the sense that the diagram below commutes:

Bb B1b1

Aa A1a1

σ

ua

τ

u1b1 .

Gluing sheavesGiven a quasi-coherent sheaf G on P1, we get an A-module N = Γ(U,G), andan A1-module N1 = Γ(U1,G). On D(a) = Spec Aa » Spec A1a1 = D(a1), these arerelated by an isomorphism of Aa1-modules

µ : N1a1 Ñ Na

(where we view Na as an Aa1-module using the isomorphism τ). Conversely, bythe tilde-construction and the Gluing lemma for sheaves, given modules N, N1

and an isomorphism µ as above, we can construct a quasi-coherent sheaf G onP1.

Similarly, giving a quasi-coherent sheaf F on X is equivalent to giving: AB-module M; A B1-module; and an isomorphism of Bb1-modules

ν : Mb1 Ñ Mb

F is coherent if and only if M and M1 are finitely generated.

the dualizing sheaf 376

21.4 The dualizing sheaf

We will use the gluing construction of the previous section to define a sheaf ωX

on X, starting from ωP1 = OP1(´2) on P1. To define it, we need to define twomodules on each affine open and check that they glue over the intersection.

The general construction goes as follows. Start with an A-module N andconsider the A-module

M = HomA(B, N).

The crucial point is that M can be viewed as a B-module, via the rule

b ¨ φ(y) := φ(b ¨ y), y P B

for each A-linear map φ : B Ñ N. Likewise, for an A1-module N1, the A1-moduleM1 = HomA1(B1, N1) can be viewed as a B1-module.

If N and N1 arise from a sheaf G on P1 in the construction above, there is anatural isomorphism

HomA1a1(B1b1 , N1

a1)Ñ HomAa(Bb, Na)

sending φ : B1b1 Ñ N1a1 to µ´1 ˝ φ ˝ σ : Bb Ñ Na. One checks that this is an

isomorphism of Bb-modules. Thus from any sheaf G on P1, we obtain a sheaf,denoted by π!G, on X. In fact, the map G ÞÑ π!G defines a functor from thecategory of coherent OP1-modules to OX-modules, but we will not need thisfact here.

The crucial ingredient we need is that there is a canonical isomorphism

π˚HomX(F , π!G) » HomP1(π˚F ,G). (21.5)

We first prove this locally:

Lemma 21.7 For a finitely generated B-module L, there is a natural isomorphism (ofA-modules)

HomB(L, HomA(B, N))Ñ HomA(L, N) (21.6)

Proof: The map is defined by sending φ : L Ñ HomA(B, N) to ` ÞÑ φ(`)(1).The map (21.6) is clearly an isomorphism for L = B‘n. To prove it in general,

pick a presentationB‘m Ñ B‘n Ñ L Ñ 0.

Applying HomB(´, HomA(B, N)), we get a diagram

0 HomB(L, HomA(B, N)) HomB(B, HomA(B, N))‘n HomB(B, HomA(B, N))‘m

0 HomB(L, N) HomB(B, N)‘n HomB(B, N)‘m

the dualizing sheaf equals the canonical sheaf 377

Then (21.6) is the left-most vertical map, and this is an isomorphism by the5-Lemma. o

Inspecting the proof of Lemma 21.6, we note that the isomorphism in (21.6)is compatible with localizations. Thus the isomorphisms sheafify, and we getthe sheaf isomorphism (21.5).

Definition 21.8 We define the dualizing sheaf of X as the sheaf ωX = π!ωP1 .

From here on, we can finish the proof of Serre duality on X:

Hi(X,F_ bOX ωX) = Hi(P1, π˚(F_ bOX ωX)) (Lemma 21.5)

= Hi(P1, π˚Hom(F , ωX))

= Hi(P1, Hom(π˚F , ωP1)) (by (21.5))

= Hi(P1, (π˚F )_ bOP1 ωP1)

= H1í(P1, π˚F )_ (Serre duality on P1)

= H1í(X,F )_. (Lemma 21.5)

So far we haven’t used the fact that X is non-singular; any projective curveadmits a dualizing sheaf ωX which is a coherent OX-module. In the non-singularcase we can say more:

Corollary 21.9 Let X be a non-singular projective curve. Then ωX is an invertiblesheaf.

Proof: Since X is a non-singular curve, ω is locally free if and only if it istorsion free. Write ωX = T ‘E where T is the torsion part and E is the free part.Then applying formula (21.5), shows that π˚ωX = Hom(π˚OX, ωP1), and theright side here is locally free. But this sheaf contains π˚T as a direct summand,so π˚T = 0. On a curve, the only torsion sheaf with no global sections is thezero sheaf, so T = 0 as well.

Finally, to compute the rank of ωX, we use the fact that the formation ofπ!G behaves well with localization. This implies that ωX,η at the generic pointη = Spec k(X) coincides with Homk(P1)(k(X), k(P1)). The latter is a k(P1)-vectorspace of dimension equal to the degree of X Ñ P1. Hence, as a k(X)-vectorspace it has dimension 1. o

21.5 The dualizing sheaf equals the canonical sheaf

The goal of this section is to show that the dualizing sheaf ωX is isomorphic tothe cotangent sheaf ΩX. Note that both of these sheaves are locally free: the firstby Corollary 21.9, and ΩX because X is smooth.

We will work with the self-product X ˆ X with the two projections p, q :Xˆ X Ñ X.

the dualizing sheaf equals the canonical sheaf 378

Consider the diagonal embedding i : ∆ Ñ X ˆ X. Recall that the normalbundle of ∆ in Xˆ X is isomorphic to the tangent bundle TX. We thus have anexact sequence

0 Ñ OXˆX Ñ OXˆX(∆)Ñ i˚TX Ñ 0. (21.7)

Tensoring this by q˚ωX, we get a sequence

0 Ñ q˚ωX Ñ q˚ωX(∆)Ñ i˚(ωX b TX)Ñ 0 (21.8)

Restricting the sequence (21.8) to the open set VˆX, where V = Spec R is affine,and taking the long exact sequence in cohomology, gives

Γ(V ˆ X, i˚(ωX b TX))Ñ H1(V ˆ X, q˚ωX)Ñ H1(V ˆ X, q˚ωX(∆)) (21.9)

Here we may identify the first group with Γ(V, ωX b TX) (using Lemma 21.5)and the second with Γ(V,OX)bk H1(X, ωX) = Γ(V,OX) (using Lemma 21.6; wealso use the isomorphism H1(X, ωX) = k). These identifications are compatiblewith restriction maps, so we get a map of sheaves

ρ : ωX b TX Ñ OX.

We claim that ρ is surjective. This will imply that there is a surjection ωX Ñ ΩX,which must be an isomorphism (because the kernel is locally free of rank 0).Hence ωX » ΩX.

To conclude, it suffices to prove that the group H1(V ˆX, q˚ωX(∆)) in (21.9)vanishes for each affine V Ă X. This follows by the following computation:

H1(V ˆ X, q˚ωX(∆)) = H1(X, ωX b q˚OVˆX(∆)) (Lemma 21.5)

= H0(X, HomX(q˚OVˆX(∆),OX))_ (Duality property of ωX)

= H0(X, q˚HomX(OVˆX(∆),OVˆX))_ (Change-of rings formula of Hom)

= H0(X, q˚OVˆX(´∆))_ (the dual of OVˆX is OVˆX(´∆))

= H0(V ˆ X,OVˆX(´∆))_ (Lemma 21.5)

= 0.

Why the last group is zero: sections of OVˆX(´∆) » OVˆX b I∆ correspondto sections of OVˆX vanishing along the diagonal. However, since Γ(V ˆX,OVˆX) = Γ(V,OV)b Γ(X,OX), there are no non-zero sections like this.

Chapter 22

Applications of the Riemann–Roch theorem

In this chapter we give a few of the (many) consequences of the Riemann–Rochformula. We start by translating the results of Chapter 16 into concrete numericalcriteria for a divisor D to be base point free or very ample. Then we use theseresults to classify all curves of all genus ď 4.

22.1 Very ampleness criteria

Recall the criterion of Theorem 16.11, that an invertible sheaf L is very ampleif and only if its linear system separates points and tangent vectors. UsingRiemann–Roch we can translate that result into a very simple, numerical criterionfor very ampleness on a curve:

Theorem 22.1 Let X be a non-singular projective curve and let D be a divisoron X. Then

i) |D| is base point free if and only if

h0(D´ P) = h0(D)´ 1 for every point P P X.

ii) D is very ample if and only if

h0(D´ P´Q) = h0(D)´ 2 for every two points P, Q P X

(including the case P = Q)iii) A divisor D is ample iff deg D ą 0

Proof: (i) We take the cohomology of the following exact sequence

0 Ñ OX(D´ P)Ñ OX(D)Ñ k(P)Ñ 0

and get0 Ñ H0(X,OX(D´ P))Ñ H0(X,OX(D))Ñ k

From this sequence, we get h0(D)´ 1 ď h0(D´ P) ď h0(D).

curves on1 ˆ 1

380

The right-most map takes a global section of OX(D) and evaluates it at P. Toprove that |D| is base point free, we must prove that there is a section s P OX(D)

which does not vanish at P, or equivalently, that the map H0(X,OX(D))Ñ k issurjective. But this happens if and only if h0(D´ P) = h0(D)´ 1.

(ii) If the above inequality is satisfied, we see in particular that |D| is basepoint free. So it determines a morphism φ : X Ñ Pn. We can use Theorem 16.11

that ensure that φ is an embedding. That is, we need to check that φ separates(a) points and (b) tangent vectors.

For (a), we are assuming that h0(D ´ P´ Q) = h0(D)´ 2, so the divisorD´ P is effective and does not have Q as a base point (by (i)). But this meansthat there is a section of H0(X,OX(D´ P)) which doesn’t vanish at Q. We haveH0(X, D´ P) Ď H0(X, D), so we get a section of OX(D) which vanishes at P,but not at Q. Hence |D| separates points.

For (b), we need to show that |D| separates tangent vectors, i.e., the elementsof H0(X,OX(D)) should generate the k-vector space mPOX(D)/m2

POX(D) atevery point P P X. This condition is equivalent to saying that there is a divisorD1 P |D| with multiplicity 1 at P: Note that dim TP(X) = 1, dim TPD1 = 0 if Phas multiplicity 1 in D1 and dim Tp(D1) = 1 if P has higher multiplicity. But thisis equivalent to P not being a base point of D´ P. Applying (i) again, we seethat h0(D´ 2P) = h0(D)´ 2, so we are done.

(iii) By definition, D is ample if mD is very ample for m " 0. So the resultfollows by the next result, since any divisor of degree ě 2g + 1 is very ample. o

Proposition 22.2 Let X be a non-singular projective curve and let D be adivisor on X. Then

i) If deg D ě 2g, then D is base point free.ii) If deg D ě 2g + 1, then D is very ample.

Proof: By Serre duality, h1(D) = h0(K ´ D) = 0 because deg D ą deg K =

2g´ 2. Similarly, h1(D´ P) = 0.(i) Applying Riemann–Roch, we find that h0(D ´ p) = h0(D)´ 1 for any

P P X, so we are done by the above theorem.(ii) In this case we also get h1(D´ P´Q) = 0, so Riemann–Roch shows that

h0(D´ P´Q) = h0(D)´ 2, which is the conclusion we want. o

22.2 Curves on P1 ˆ P1

Let us consider one central example, namely curves on the quadric surfaceQ = P1 ˆ P1. Recall that Cl(Q) = ZL1 ‘ZL2 where L1 = [0 : 1] ˆ P1 andL2 = P1 ˆ [0 : 1].

curves on1 ˆ 1

381

We can use this to prove that Q contains non-singular curves of any genusg ě 0. (This is in contrast with the case of P2, where only genera of the form(d´1

2 ) were allowed).To prove this, consider the divisor D = aL1 + bL2 where a, b ě 1. D is

effective, so let C P |D| be a generic element.

Lemma 22.3 C is non-singular.

Proof: D is defined by a bihomogeneous equationÿ

i+j=a,l+k=b

cij,klxi0xj

1yl0yk

1 = 0

On the open set D+(x0)XD+(y0) » A2 = Spec k[x, y] this is given byÿ

i+j=a,l+k=b

cij,klxjyk = 0

and it is clear that if the coefficients cij,kl are general, this is non-singular. Bysymmetry this also happens in the other charts, so C is non-singular. o

To compute the genus of C, we use the formula 2g´ 2 = deg ΩC. So we needto find ΩC and compute its degree. This is best computed using the Adjunctionformula of Proposition 17.31:

ΩC = ωQ bOQ(C)|X= OQ(´2L1 ´ 2L2)bO(aL1 + bL2)|C

= OC((a´ 2)L1 + (b´ 2)L2)

To compute the degree of this, we consider the degrees of L1|C and L2|C sepa-rately. Note that the degree deg L1|C is invariant under linear equivalence, so wecan compute the degree of any [s : t]ˆP1 for a general point [s : t]ˆP1. Thepoint is that as a Weil divisor, L1|X is obtained by intersecting [s : t]ˆP1 with X.When [s : t] P P1 is a general point, the intersection XX [s : t]ˆP1 is a reducedsubscheme of X, consisting of b points (as C Ă Q = P1 ˆP1 is a divisor of typeaL1 + bL2). Hence deg L1|C = b and deg L2|C = a. It follows that

2g´ 2 = deg ΩC = (a´ 2)b + (b´ 2)a = 2ab´ 2a´ 2b

Solving for g gives us the following theorem:

Theorem 22.4 Let Q = P1 ˆP1. Then a generic divisor C in |aL1 + bL2| is asmooth projective curve of genus

g = (a´ 1)(b´ 1).

In particular, Q contains non-singular curves of any genus g ě 0.

curves of genus 0 382

22.3 Curves of genus 0

The results of the previous results are particularly strong when the genus issmall. For instance, when g = 0, any divisor of positive degree is very ample!We can use this to classify all curves of genus 0. First a simple lemma:

Lemma 22.5 Let X be a non-singular curve. Then X » P1 if and only if there existsa Cartier divisor D such that deg D = 1 and h0(X,OX(D)) ě 2. In this case, thedivisor D is even very ample.

Proof:Let g P H0(X,OX(D)). Then D1 „ div g + D ě 0, so replacing D by D1 we

may assume that D is effective. Since deg D = 1, we must have D = p for somepoint p P X. Now take f P H0(X,OX(D))´ k. As above, f induces a morphismφ : X Ñ P1. This morphism has degree equal to 1, so it is birational, and henceX is isomorphic to P1. o

Proposition 22.6 A non-singular curve X is isomorphic to P1 if and only ifCl(X) » Z.

Proof: We have seen that the Picard group of any Pnk is isomorphic to Z via

the degree map deg : Pic(Pnk )Ñ Z.

Conversely, suppose X is a curve with Cl(X) » Z. Let p, q be two distinctpoints on X. By assumption, p and q are linearly equivalent, so the linear system|p| = PH0(X,OX(D)) is at least 1-dimensional. Then X » P1

k by the previouslemma. o

Theorem 22.7 Any curve of genus 0 over an algebraically closed field is isomor-phic to P1.

Proof: Let p P X be a point and consider the divisor D = p. If X has genus0, then 1 = deg D ą 2g´ 2 = ´2, so H1(X,OX(D)) = 0. Then Riemann-Rochtells us that

dim H0(X,OX(D)) = 1 + 1´ 0 = 2

Hence X » P1k by Lemma 22.5. o

We conclude by yet another characterisation of P1:

Lemma 22.8 Let C be a non-singular projective curve and D any divisor of degreed ą 0. Then

dim |D| ď deg D

with equality if and only if C » P1.


Proof: This is (IV, Ex. 1.5) in Hartshorne. Although one might guess that thislemma follows directly from Riemann-Roch, this does not seem to be the case:Riemann-Roch gives a different sort of relationship between the dimension anddegree of a divisor.

We may assume that D is effective, i.e., D = P1 + ¨ ¨ ¨+ Pd for some pointsP1, . . . , Pk P C (possibly equal) (otherwise replace D by some different effectivedivisor D1 P |D|). We induct on d.

First suppose d = 1. There is an exact sequence

0 Ñ OC Ñ OC(P)Ñ k(P)Ñ 0.

Now h0(OC) = 1 and h0(k(P)) = 1 therefore h0(OC(P)) ď 2 so dim |P| ď 1. Ifdim |P| = 1 then |P| has no base points so we obtain a morphism C Ñ P1 ofdegree deg P = 1 which must be an isomorphism, and so C » P1 is rational.

Next suppose D = P1 + ¨ ¨ ¨+ Pd. Let D1 = P1 + ¨ ¨ ¨+ Pd´1. There is an exactsequence

0 Ñ OC(D1)Ñ OC(D)Ñ k(Pd)Ñ 0.

Now h0(OC(D1)) ď d by induction and h0(k(Pd)) = 1 so h0(OC(D)) ď d +

1, therefore dim |D| ď d with equality iff h0(OC(D1)) = d. By inductionh0(OC(D1)) = d iff C is rational. o


A plane curve X Ă P2k of degree 3 has genus 1. This follows from our earlier

work on the canonical divisor, which showed ωX » OP2k(d´ 3)|X » OX, and so

g = h0(ωX) = h0(OX) = 1. In this section, we show that in fact every curve ofgenus 1 arises this way:

Theorem 22.9 Any projective curve of genus 1 can be embedded as a planecubic curve in P2

k .

Proof: Pick a point P P X and consider the divisor D = 3P. D has degree3 ě 2g + 1, so it is very ample. Furthermore, by Riemann–Roch, h0(3P) = 3, sothere is a projective embedding φ : X Ñ P2

k . The image φ(X) is a smooth curveof degree equal to deg φ˚OP2(1) = deg D = 3. o

In contrast to the g = 0 case however, there are many non-isomorphic genus1 curves. For instance, in the Legendgre family of curves in Xλ Ă P2 given by

y2z = x(x´ z)(x´ λz)

where λ P k, each Xλ is isomorphic to at most a finite number of other Xλ1 ’s.


Divisors on XLet X be a curve of genus 1. We will study the divisors on X. To make thediscussion a bit more concrete, let X Ă P2 be the curve given by y2z = x3 ´ xz2.We claim that there is an exact sequence

0 Ñ X(k)Ñ Cl(X)degÝÝÑ Z Ñ 0

This means that the class group Cl(X) is very big – its elements are in bijectionwith the k-points of X, of which there might be uncountably many. (In particular,this is another reason why X cannot be isomorphic to P1.)

If L Ă P2 is a line, we get a divisor L|X: That is, we take a section s P OP2(1)defining L and restrict it to X. The divisor of s P OX(1) consists of three pointsP, Q, R (counted with multiplicity). In particular, since any two lines are linearlyequivalent on P2, we get for every pair of lines L, L1 and corresponding triplesP, Q, R, a relation

P + Q + R „ P1 + Q1 + R1

(where „ denotes linear equivalence).Let us consider the point O = [0, 1, 0] on X. This is a special point on X: it

is an inflection point, in the sense that there is a line L = V(z) Ă P2 which hasmultiplicity three at O, so that L restricts to 3O on X. This has the property thatany three collinear points P, Q, R in X satisfy

P + Q + R „ 3O

We will use these observations to define a group structure on the set of closedpoints X(k), using the point O as the identity. The group structure will beinduced from that in Cl(X).

Consider the subgroup Cl0(X) Ă Cl(X) consisting of degree 0. This fits intothe exact sequence

0 Ñ Cl0(X)Ñ Cl(X)degÝÝÑ Z Ñ 0

We now define a map

ξ : X(k) Ñ Cl0(X)

P ÞÑ [PÓ]

Lemma 22.10 ξ is a bijection.

Proof: ξ is injective: ξ(P) = ξ(Q) implies that P „ Q. Then P = Q (otherwiseX would be rational, by Proposition 22.6). (Alternatively, it follows becauseh0(X,OX(P)) = 1).

ξ is surjective: Take a divisor D =ř

niPi P Div(X) of degree 0. ThenD1 = D+O has degree 1, so by Riemann–Roch, H0(X,OC(D1)) is 1-dimensional.


Hence there exists an effective divisor of degree 1 in |D1|, which must then be ofthe form D1 = Q. But that means that D + O „ Q, or, D „ QÓ, as desired. o

Using this bijection, we can put a group structure on the set X(k):

Theorem 22.11 The set of k-points X(k) on a genus 1 form a group.

The group law has the following famous geometric interpretation. Giventwo points p1, p2 P X, we draw the line L they span (see the figure below). Thisintersects X in one more point, say p3. In the group Cl0(X) we have

p1 + p2 + p3 = 3O

To define the ‘sum’ p1 + p2 (which should be a new k-point of X), we then lookfor a point p4 so that

p4 Ó = (p1 Ó) + (p2 Ó)

or in other words, p4 + O = p1 + p2. By the above, this becomes p4 + O =

3O´ p3 or, p3 + p4 + O = 3O. This tells us that we should define p4 as follows:We draw the line L1 from O to p3 (shown as the dotted line in the figure), anddefine p4 to be the third intersection point of L1 with X. By construction, we get(p1 Ó) + (p2 Ó) = (p4 Ó) in Cl0(X).

Given the equation of X in P2, and coordinates for the points p1, p2, we canof course write down explicit formulas for the coordinates of p4, and theyare rational functions in the coordinates of p1, p2. This is almost enough tojustify that X is a group variety, i.e., it is an algebraic variety equipped with amultiplication map m : Xˆ X Ñ X satisfying the usual group axioms, and m isa morphism of algebraic varieties.


Let X be a non-singular projective curve of genus 2. We saw one example ofsuch a curve earlier in this chapter, namely the curve obtained by gluing two


copies of the affine curve y2 = p(x) where p(x) is a polynomial of degree five.The condition that X is non-singular implies that p has distinct roots.

We already saw in Chapter XX that a genus 2 curve cannot be embedded inthe projective plane P2

k (since 2 is not a triagonal number). However, we showthe following:

Theorem 22.12 Any curve of genus 2 is isomorphic to a hyperelliptic curve

Here, a curve C is said to be hyperelliptic if there is a degree 2 map X Ñ P1.Equivalently, there is a base point free linear system of degree 2 and dimension1. Equivalently again, there exists points P, Q P X so that the invertible sheafL = OX(P + Q) is globally generated and by two global sections.

It is classical notation that a base point free linear system of degree d anddimension r is called a gr

d. So to say that a curve is hyperelliptic is to say that ithas a g1

2.Example 22.13 If g = 0, then X » P1. Let D = 2P, then H0(D) = kx2

0 + kx0x1 +

x21, so |D| » P2 is identified with the space of quadratic polynomials up to

scaling. If we take two quadratic polynomials q0, q1 with no common zeroes, weget a base point free linear system g1

2 Ă |D|. K

Example 22.14 If g = 1 any divisor of degree 2 gives a g12 by Riemann-Roch.

Indeed, if D has degree 2 then

h0(D)´ h0(K´D) = 2 + 1´ g = 2

and deg(K´D) = ´2 so h0(D) = 2 and hence dim |D| = 1. This D is base pointfree, since D´ p has degree 1, and hence since X is not rational, h0(D´ p) =1 = h0(D)´ 1. K

Example 22.15 Let X Ă P1ˆP1 be a smooth divisor of bidegree (2, g+ 1). ThenKX » OP1ˆP1(0, g´ 1) and X has genus g. Moreover, the projection p2 : X Ñ P1

is finite of degree 2, which shows that X is hyperelliptic.The projections p1, p2 : Q Ñ P1 give rise to a degree 2 and a degree g + 1

morphism of X to P1. Thus there exists a 2:1 morphism f : X Ñ P1. fcorresponds to a base point free linear system on X of degree 2 and dimension1. Thus X is hyperelliptic.

In this example, ΩX = OQ(X) b ωQ|X = OQ(2, g + 1) b OQ(´2,´2) =

OX(0, g´ 1). The latter invertible sheaf has g independent global sections soX has genus g. Moreover KX is base point free, but not very ample, since thecorresponding morphism X Ñ Pg´1 is not an embedding (it maps X onto aconic). K

To prove the theorem, we must produce a degree two map φ : X Ñ P1. Wehave a natural candidate: the canonical divisor KX, which has degree 2g´ 2 = 2.We claim that KX is base point free.

Note that we cannot apply Proposition 22.2 directly to prove this, since thedegree is too small. However, we can use Riemann–Roch to check directly that


the conditions in Theorem 22.1 apply. That is, we need to show that for everypoint P P X, we have

h0(X, KX ´ P) = h0(X, KX)´ 1 = 2´ 1 = 1

Applying Riemann–Roch to the divisor D = P, we also get h0(P)´ h0(KX´ P) =1 + 1´ 2 = 0. As P is effective, and X is not rational, we have h0(P) = 1, and soalso h0(X, KX ´ P) = 1, as we want.


The case of curves of genus 3 is especially interesting. We have seen twoexamples of curves of genus 3 so far:Example 22.16 A plane curve X Ă P2 of degree d = 4 has genus 1

2 (d´ 1)(d´2) = 3.

Notice thatΩX = OP2(d´ 3)|X = OX(1)

so ΩX is very ample, since it is the restriction of the very ample invertible sheafOP2(1) on P2. Hence KX is very ample, and the corresponding morphism isexactly the given embedding X ãÑ P2. K

Example 22.17 The curves in Section 19.5 on page 367 can be chosen to havegenus g = 3. In this case, X admits a 2:1 map to P1, and thus X is hyperelliptic.

K

Example 22.18 A curve X on the quadric surface Q » P1 ˆP1 in P3 of type(2,4) is hyperelliptic. It is a curve of degree 6 and genus 3. K

Thus these examples are a bit different. The curves in the first example havevery ample canonical divisor KX (they are ‘canonical’) whereas the two others todo not (‘hyperelliptic’). We show that this distinction is a general phenomenonfor curves of genus three:

Proposition 22.19 Let X be a curve of genus 3. Then there are two possibilities:

i) KX is very ample. Then X embeds as a plane curve of degree 4.ii) KX is not very ample. Then X is a hyperelliptic curve, and it embeds

as a (2, 4) divisor in P1 ˆP1. Moreover, KX „ 2F, where F = L1|X.

We willl deduce this from a more general result:

Theorem 22.20 Let X be a curve of genus ě 2. Then K is very ample if andonly if X is not hyperelliptic.

Proof: K is very ample if and only if h0(K´ P´Q) = h0(K)´ 2 = g´ 2 forevery P, Q P X. By Riemann–Roch, we compute

h0(P + Q)´ h0(K´ P´Q) = 2 + 1´ g = 3´ g


Hence K is very ample if and only if h0(P + Q) = 1 for every P, Q.If X is hyperelliptic, then there is a map φ : X Ñ P1, so that φ˚([1 : 0]) =

P + Q for some points P, Q P X (possibly equal). Here the linear system |P + Q|is 1-dimensional, so h0(X, P + Q) = 2, and hence KX is not very ample.

If X is not hyperelliptic, we have h0(X, P + Q) = 1 for any P, Q (otherwise itis ě 2, and P + Q induces a map X Ñ P1 of degree two), and hence KX is veryample.

We still need to check the last part of the above theorem, namely that everyhyperelliptic curve arises as a curve of type (2,4) on Q Ă P3.

We proceed as follows. Let D = P1 + ¨ ¨ ¨+ P4 denote a generic degree 4

divisor on X (so P1, . . . , P4 are general points of X). By Riemann–Roch, we get

h0(D)´ h0(K´D) = 4 + 1´ 3 = 2

We claim that h0(K ´ D) = 0, so that h0(D) = 2. Note that K ´ D has degree2g´ 2´ 4 = 0, so K ´ D is a divisor of degree 0. This is effective if and onlyif K „ D. However, there is a 4-dimensional family of divisors of the formP1 + ¨ ¨ ¨+ P4, wheras the space of effective canonical divisors has dimensiondim |K| = 2. Hence if the points Pi are chosen generically, K ´ D will not beeffective, and hence the claim holds.

From this, we obtain a linear system |D| of dimension 1. We claim that D isbase point free. We need to show that

h0(D´ P) = h0(D)´ 1 = deg D + 1´ 3)´ 1 = 1

for every point P. Suppose not, and let P be a base point of D. Since D =

P1 + P2 + P3 + P4 we may suppose that P = P4.By Riemann–Roch, we are done if we can show h0(K´D + P) = 0. However,

K´D+ P = K´P1´P2´P3. There is a 3-dimensional space of effective divisorsof the form P1 + P2 + P3 for points Pi P X, but only a 2-dimensional linear systemof effective canonical divisors |K|. Hence K´D + P is not effective.

We therefore have two morphisms from our hyperelliptic curve X; f : X ÑP1 (induced by the g1

2) and g : X Ñ P1 (induced by D). By the universalproperty of the fiber product, this gives a morphism

φ = ( f ˆ g) : X Ñ P1 ˆk P1

We claim that this is a closed immersion. Let F = P + Q P |g12|. The map D + F

induces the map F : X Ñ P3, which coincides with j ˝ φ where j : P1 ˆP1 is theSegre embedding. To prove the claim, it suffices to show that F is an embedding,or equivalently that D + F is very ample.

First claim that K „ 2F. Since both of these divisors have degree 4 itsuffices to show that K ´ 2F is effective. Note that in any case h0(X, 2F) ě 3,since if H0(X, F) = xx, yy, then x2, xy, y2 are linearly independent in H0(X, 2F)(understand why!). Now applying Riemann–Roch to D = 2F, we get

h0(2F)´ h0(K´ 2F) = 4 + 1´ 3 = 2


so h0(K´ 2F) ě 1, and K „ 2F as we want.Now, to show that D + F is very ample, we need to show that

h0(X, D + F´ p´ q) = h0(D + F)´ 2

for any pair of points p, q P X. By Riemann–Roch again, we can conclude if weknow that h0(K´D´ F + p + q) = 0. But since K „ 2F, we have

K´D´ F + p + q „ F´D + p + q

These are divisors of degree 0, so if this is effective, we must have D „ F + p + q.However, the space of effective divisors of the form D1 + p + q with D1 „ Fis 3-dimensional (since |F| has dimension 1, and p and q can be chosen freelyon X). On the other hand, as we have seen, the space of divisors of the formD = P1 + ¨ ¨ ¨ + P4 is of dimension 4, so choosing D generically means thatthisF´D + p + q is not effective. It follows that h1(D´ p´ q) = h0(D + F)´ 2and hence D is very ample. o

22.7 Curves of Genus 4

Recall that curves of genus g ě 2 split up into two disjoint classes.

i) Hyperelliptic curves: X admits a 2:1 to P1

ii) Canonical curves: KX is very ample

Here’s an example of a genus 4 curve in P1 ˆP1:Example 22.21 Consider a type (2, 5) curve C on Q Ă P3. Then C has degree7 = 2 + 5 and C is hyperelliptic (because of the degree 2 map coming fromprojection onto the first fact p1 : Q Ñ P1). A type (3, 3) curve on Q is also ofgenus 4. It is a degree 6 complete intersection of Q and a cubic surface. Curvesof type (3, 3) have at least two g1

3’s. K

In fact, using the same strategy as for g = 3, one can show that any hyperel-liptic curve of genus 4 arises this way.

Classifying curves of genus 4We start with an abstract curve X of genus 4. We may assume that X is nothyperelliptic (since in that case it embeds as a (2, 5)-divisor on P1 ˆP1). Sowe assume that KX is very ample. Therefore we have the canonical embeddingX ãÑ Pg´1 = P3. The degree of the embedded curve is deg ωX = 2g´ 2 = 6.Thus we can view X as a degree 6 genus 4 curve in P3.

What are the equations of X in P3? To answer this question we use a verypowerful technique in curve theory, namely we combine Riemann–Roch withthe sheaf cohomology on Pn. Twisting the ideal sheaf sequence of X by OP3(2)and taking cohomology gives the exact sequence

0 Ñ H0(P3, IX(2))Ñ H0(P3,OP3(2))Ñ H0(X,OX(2))Ñ ¨ ¨ ¨


Note that OP3(1)|X = KX. Applying Riemann-Roch states to the divisor D =

2KX, we get

h0(OX(2)) = deg 2KX + 1´ g + h1(OX(D)) = 12 + 1´ 4 + 0 = 9.

(Note that h1(OX(2)) = h0(KX ´ 2KX) = h0(´KX) = 0 since KX is effective).Since h0(P3,OP3(2)) = 10 it follows that the map H0(OP3(2)) Ñ H0(OX(2))must have a nontrivial kernel so h0(P3, IX(2)) ą 0.

The upshot of this is that we now know that X lies in some surface of degree2. Since X is integral, this surface cannot be a union of hyperplanes. So X lieson either a singular quadric cone Q0 = V(xy´ z2) or the nonsingular quadricsurface Q = V(xy´ zw).

If C lies on Q then it must have a type (a, b) which must satisfy a+ b = 6 and(a´ 1)(b´ 1) = 4. The only solution is a = b = 3. Since OQ(3, 3) » OP3(3)|Q,this implies that C is the restriction of a divisor on P3, that is, C = QX S for adegree 3 surface S.

The other possibility is that C lies on Q0. Computing as before, we obtain

0 Ñ H0(OX(3))Ñ H0(OP3(3))Ñ H0(OX(3))Ñ ¨ ¨ ¨

As before one sees that h0(OX(3)) = 15 and h0(OP3(3)) = 20. Thus h0(OC(3)) ě5. Let q P H0(OC(2)) be the defining equation of Q0. Then xq, yq, zq, wq PH0(OC(3)). But h0(OC(3)) ě 5 so there exists an f P H0(OC(3)) so that theglobal sections xq, yq, zq, wq, f are independent. Thus there is an f not in(q). Since f R (q) we see that S = Z( f ) Č Q so C1 = S X Q is a degree 6not necessarily nonsingular or irreducible curve. Since C Ă S and C Ă Q itfollows that C Ă C1. Since these are both integral curves of the same degreedeg C = 6 = deg C1, we must have C = C1. Thus in the case that C lies on Q0 wesee that C is also a complete intersection C = Q0 X S for some cubic surface S.

This proves the following theorem:

Theorem 22.22 Let X be a non-singular curve of genus 4. Then either

i) X is hyperelliptic (in which case X embeds as a (2, 5)-divisor inP1 ˆP1); or

ii) X = QX S is the intersection of a quadric surface and a cubic surfacein P3.

Chapter 23

More on vector bundles

23.1 The vector bundle associated to a locally free sheaf*

Locally free sheaves are in many respects the most basic sheaves on a scheme X;they are obtained by locally gluing together copies of the structure sheaf OX invarious ways. What makes these sheaves particularly interesting is the link tothe theory of vector bundles.

To explain how this works, let X be a scheme and let F be a locally free sheafof rank r on X. We want to associate to F another scheme V(F ), equippedwith a morphism π : V(F ) Ñ X, whose fibers V(F )x over points x of X arer-affine spaces. When X is a scheme over a field k, and x = Spec k is a closedpoint, we thus want V(F )x to be isomorphic to Ar

k. So we may visualize V(F )

geometrically as a family of k-vector spaces parameterized by X; X is a so-calledvector bundle.

To define V(F ) and π : V(F ) Ñ X, we start by choosing an affine opencover tUiuiPI of X and isomorphisms fi : F |Ui Ñ Or

Uifor each i. We also

assume that the overlaps Uij are affine for each i, j. If we take two isomorphisms,fi : F |Ui Ñ Or

Uiand f j : F |Uj Ñ Or

Uiand restrict to Uij = Ui XUj, we get

two different isomorphisms gi, gj : F |Uij Ñ OrUij

. Let gij = gj ˝ g´1i ; this is an

isomorphism OrUijÑ Or

Uij. Write Uij = Spec Bij for each i, j P I. Over the open

set Uij, gij(Uij) is represented by an rˆ r matrix with entries in Bij.Now gij defines a ‘linear change of coordinates’ map Ar

UijÑ Ar

Uijwhich is

induced by the isomorphism of Bij-algebras

gij : Bij[v1, . . . , vr]Ñ Bij[v1, . . . , vr]

sending (v1, . . . , vr) to gij(Uij) ¨ (v1, . . . , vr)t. It follows that we may glue ArUi

toAr

Ujalong Ar

Uij. The resulting scheme which we denote by V(F ) comes with

a morphism π : V(F )Ñ X (obtained by gluing all the projections ArUiÑ Ui),

whose fibers are affine spaces.The morphism π : V(F )Ñ X is quite special, in the following sense: Over

each Ui Ă X the maps gi give isomorphisms from π´1(Ui) to the product

vector bundles in general* 392

ArUi

= Ui Âr. When this happens, we say that π is locally trivial. Moreover, oneach intersection Uij, the isomorphism gj ˝ g´1

i : Uij Âr Ñ Uij Âr is ‘linear’in the fibers. A general morphism of schemes π : E Ñ X with these propertiesis called a vector bundle.

23.2 Vector bundles in general*

A vector bundle is supposed to be thought of as a family of vector spacesparameterized over a base space X. In the scheme setting, this means that wehave a morphism π : E Ñ X, such that the scheme theoretic fibers π´1(x) areisomorphic to an affine space Ar. The prototype example of a vector bundleis obtained by simply taking the product E = X ˆk Ar

k and letting π be thefirst projection. This is the so called trivial bundle on X. A vector bundle ismore generally a scheme together with a morphism π : E Ñ X which is locallyisomorphic to the trivial bundle. More precisely:

Definition 23.1 Let X be a scheme. A vector bundle E of rank r on X is ascheme with a morphism π : E Ñ X and an open cover U = tUiu of X suchthat for each i, there is an isomorphism φi : π´1(Ui) Ñ Ui Âr making thefollowing diagram commutative:

π´1(Ui)φi

//

π|π´1(Ui) ((

Ui Âr

pr1

Ui

such that for each affine V = Spec A Ď Uij the automorphism φj ˝ φ´1i of

V Âr is linear; i.e., induced by an automorphism θ : A[x1, . . . , xn] Ñ

A[x1, . . . , xn] such that θ(a) = a,@a P A, and θ(xi) =ř

aijxj for aij P A.

So a vector bundle is obtained by gluing together trivial bundles Ui Âr vialinear gluing maps. A convenient way of rephrasing this is in terms of transitionfunctions: Given a vector bundle π : E Ñ X defined by the data (Ui, φi) we have

vector bundles in general* 393

for each i, j an element gij P GLr(Γ(Uij,OX)) such that the diagram

Uij Âr

π´1(Uij)φj|Uij

%%

φi|Uij99

Uij Âr

idˆgij

OO

commutes.

Definition 23.2 The elements gij are the transition functions of E .

The gluing axioms for E shows that the transition functions satisfy thefollowing compatibility conditions (or ‘cocycle conditions’)

gik = gij ˝ gjk on Uijk (23.1)

gij = g´1ji on Uij

Proposition 23.3 Let X be a scheme with an open cover tUiu and assume thatgij is a collection of elements of GLr(Γ(Uij,OX)) satisfying the compatibilityconditions (23.1). Then there is a vector bundle π : E Ñ X, unique up toisomorphism, whose transition functions are the gij.

Proof: The compatibility conditions ensure that the maps (id ˆ gij) gluethe affine schemes Ui Âr along (Ui XUj) Âr to a scheme E . Moreover,the projection maps Ui Âr Ñ Ui glue to give a morphism π : E Ñ X. Byconstruction, the open set of E corresponding to Ui Âr is identified withπ´1(Ui), which gives an isomorphism φi : π´1(Ui) Ñ Ui Âr. Hence E is avector bundle with transition functions gij. o

The sheaf of sections of a vector bundleA section of a vector bundle π : E Ñ X over U Ă X is a morphism s : U Ñ E

such that π ˝ s = idU . So one imagines that s picks out a single vector in eachfiber π´1(x) for x P X a closed point.

Lemma 23.4 Let π : E Ñ X be a vector bundle given by the data (Ui, φi, gij). Asection s : X Ñ E is determined uniquely by a collection of r-tuples si P Or

X such thatfor each i, j

si|Uij = gijsj|Uij .

Proof: Given s : X Ñ E , φi ˝ s|Ui is a section of Ui Âr Ñ Ui. Henceφi ˝ s|Ui = (idU ˆ si) for some si P O(Ui)

r. By construction, these satisfy the

extended example: the tautological bundle onnk * 394

given compatibility conditions si|Uij = gijsj|Uij . Conversely, any such sections : X Ñ E defines a set of such sections si = s|Ui satisfying this condition. o

Given E , we can define a sheaf OX(E ) by defining

OX(E )(U) = tsections s : U Ñ E u

The above proposition shows that this set is naturally a group: If s : U Ñ E ,t : U Ñ E are two sections given by the data si and ti respectively, we can defines + t, by si + ti P Or

X. In fact, by multiplying si with elements of OX(U), we seethat OX(E )(U) has the structure of an OX(U)-module. This shows that OX(E )

is a sheaf of OX-modules, locally free of rank r.

Equivalence between vector bundles and locally free sheaves*To each locally free sheaf F , we associated a vector bundle π : F Ñ X bygluing together the local Ar

Ui’s together on the overlaps using the isomorphisms

fi : F |Ui Ñ OrUi

. Conversely, to each vector bundle π : E Ñ X, we associated asheaf of sections, OX(E ), which was locally free.

It is just a matter of checking that these two assignments are inverses to eachother (so that the sheaf of sections of V(F ) is exactly F ). In particular, thereis a correspondence between vector bundles and locally free sheaves of finiterank. Under this correspondence the trivial bundle XÂ1 corresponds to thestructure sheaf OX.

Proposition 23.5 The assignments F ÞÑ V(F ) and E ÞÑ OX(E ) are mutu-ally inverse, and give an equivalence between the category of locally free sheafand the category of vector bundles.

Exercise 23.1 Fill in the details for the proof of Proposition 23.5. M

23.3 Extended example: The tautological bundle on Pnk *

Let k be a field, and let Pnk denote the projective n-space over k. The k-points of

Pnk parameterizes lines through the origin in An+1(k). We define the tautological

sheaf, denoted by O(´1), but we begin by a geometric construct, of a tautologicalsubvariety L of the product Pn ˆk An+1; being tautologial means that the fibreof the projection L Ñ Pn over a k-point [l] corresponding to a line lĎAn+1 isprecisely that line.

Heuristically L is the subscheme L Ă Pn ˆk An+1 whose k-points are thepairs ([l], v), where l Ă An+1 is a line, and v P l. In terms of equations L isdefined by the 2ˆ 2 minors of the matrix(

x0 x1 . . . xn

y0 y1 . . . yn

)


where x0, . . . , xn are homogeneous coordinates on P1 and y0, . . . , yn are affinecoordinates on An+1.

Let π : L Ñ Pnk be the projection to the first factor. Then for a closed point

[a] = (a0 : ¨ ¨ ¨ : an) P Pnk , the preimage π´1([l]) is exactly the line l Ă An+1

corresponding to [a]. We claim that L is a rank one vector bundle on Pn.Over the open distinguished Ui = D+(xi) the equations for L become

xj

xiyi = yj

for j ‰ i, and hence the assignment (x0 : ¨ ¨ ¨ : xn; y0, . . . , yn) ÞÑ (x0 : . . . , xn; yi)

defines an isomorphism

φi : π´1(Ui)Ñ Ui Â1

In other words, yi is a local coordinate on the A1 on the right hand side. On theintersection Ui XUj, yi is related to yj via

xi

xjyj = yi

This means that the transition function

Uij Â1 Ñ Uij Â1

is given by idUij ˆ gij where gij =xixj

. Hence π is a vector bundle. We denote thecorresponding sheaf of sections by O(´1).

What are the sections of O(´1) ? Note that a section s : Pn Ñ L is given bya set of n + 1 sections si, such that for each i, j we have

si|Uij = gijsj|Uij =xi

xjsj|Uij (23.2)

The maps si : D+(xi)Ñ L can be represented by polynomials in x0xi

, . . . , xnxi

. Notethat on the right hand side of (23.2), the Laurent polynomial xi

xjsj has only xj in

the denominator, whereas si, on the left, has only xi’s. It follows that the onlyway this equation in polynomials can hold is that both sides are identically 0. Inparticular,

Proposition 23.6 Γ(Pnk ,O(´1)) = 0.

In particular, the sheaf O(´1) is invertible, but not isomorphic to OPnk.

Consider now the vector bundle σ : L˚ Ñ X with transition functions

gij =xj

xi

Note that these transition functions are almost as before, only that we haveinverted the gij. The corresponding vector bundle is the dual bundle of π : L Ñ

X; the fiber σ´1(x) is identified with the dual vector space (π´1(x))˚.


In this case we find that the bundle does in fact have global sections: Asection s : Pn Ñ L˚ is given by a set of sections si : Ui Ñ W such that

si|Uij = gijsj|Uij =xj

xisj|Uij (23.3)

As before, the left-hand side is a a polynomial in x0xi

, . . . , xnxi

- in particular it is aLaurent polynomial with xi in the denominators. This is ok with respect to theright-hand side, as long as si has degree 1, that is it has a pole of order at mostone at xi = 0. Conversely, you can start with any section s0 which is linear inx1x0

, . . . , xnx0

, and use (23.2) to define s1, . . . , sn as Laurent polynomials. These gluetogether to a global section s : Pn Ñ L˚.

Proposition 23.7 The space of global sections of L˚ can be identified with thevector space of linear forms in n + 1 variables.

As an instructive and not too complicated example of how cohomologygroups can give important results, we give a proof that all locally free sheaveson the projective line P1

k decompose as direct sums of invertible sheaves, andas all invertible sheaves are of the form OP1

k(n) for some integer n, we have the

following theorem:

Theorem 23.8 Assume that k is a field and that E is a locally free sheaf of rankr on the projective line P1

k . Then there is an isomorphism

E » OP1k(α1)‘ ¨ ¨ ¨ ‘OP1

k(αr),

where the αi are integers, uniquely defined up to ordering.

We need some preperatory results about zeros of sections of locally freesheaves on curves. Assume for a moment that C is a non-singular curve andthat E is locally free sheaf of finite rank on C. Let σ : OC Ñ E be an injectivemap of sheaves on C; such maps correspond bijectively to sections of E, that weas well denote by σ (the correspondence is σ Ø σ(1)). The dualized the mapσ_, sits in an exact sequence

E˚ // OC // OZ(σ)// 0 (23.4)

where Z(σ) is a closed subscheme of C. Generically σ is injective and thereforeσ_ is generically surjective, and hence Z is not equal to the whole curve C.It follows that Z(σ) is finite C being a curve. The scheme Z(σ) is called thezero-scheme of φ.

As C is non-singular, any non-zero coherent ideal IĎOC is an invertiblesheaf. The ideal sheaf of Z(σ) – or the image of σ_, if you want – is therefore aninvertible sheaf Lσ, and we may tensor the sequence (23.4) by Lσ

´1 to obtain the


following factorization of σ_

E˚b Lσ´1 σZ // // OC

// Lσ´1.

For later references we formulate this as a proposition:

Proposition 23.9 Let C be a non-singular curve over a field k and E a locallyfree sheaf of finite rank on C. Assume that σ is a non-zero section of E. Then

there is a finite subscheme Zσ in C and a surjective map E˚b Lσ´1 σZ // // OC

where Ls denotes the invertible sheaf being the ideal of Z(σ), and a factorizationσ_ = (ιb Lσ

´1) ˝ σZ where ι denotes the inclusion of the ideal Lσ in OC.

Proof (Proof of the theorem):The first step in the proof relies on two fundamental facts about locally free

sheaves of finite rank on projective spaces. Firstly, sufficiently high positivetwists have global sections; stated slightly differently Γ(P1

k , E(n)) ‰ 0 if n " 0(this property locally free sheaves of finite rank share with all coherent sheaves).

Secondly, sufficiently negative twists do not have global sections, i.e. it holdstrue that Γ(P1

k , E(ń)) = 0 for n ąą 0. The third salient point is that anyinvertible sheaf on the projective line is isomorphic to OP1

k(α) for some integer

α, in other words, the theorem is true in rank one. The proof goes by inductionon the rank of E, and this will be the start of the induction.

We now let n0 be the greatest integer such that Γ(P1k , E(ń0)) ‰ 0, and

after having replaced E by E(ń0), we may thus assume that Γ(P1k , E) ‰ 0, but

Γ(P1k , E(´α)) = 0 for all α ą 0. Let σ be a global section of E (it is called a

minimal section).

Lemma 23.10 The minimal section σ does not vanish anywhere.

Proof: Assume that Zσ ‰ 0; then its sheaf of ideals equals OP1k(´α) for some

α ą 0, and by (23.9) there is a surjective map E˚(α) Ñ OP1k, whose dual is a

section of E(´α). This contradicts the fact that σ is a minimal section. o

To prove the theorem we proceed as announced by induction on the rank rof E, the r = 1 being taken care of by the fact that every invertible sheaf on P1

k isisomorphic to OP1

k(α) for some α P Z. Choose a minimal section σ of E. Then

there is an exact sequence

0 // F // E˚ // OP1k

// 0 (23.5)

where the kernel F is locally free of rank r´1. By induction F »À

1ďiďr´1 OP1k(αi).

Lemma 23.11 H1(P1k , F) = 0

Proof: Taking the dual of (23.5) and twisting by O(´1) gives:

0 // OP1k(´1) // E(´1) // F˚(´1) // 0


Then the long exact sequence of cohomology shows H0(F˚(´1)) = 0. But thenalso H0(F˚(´2)) = 0 = H0(F˚ΩP1), so H1(F) = 0 by Serre duality. (Note thatwe only need Serre duality for a direct sum of invertible sheaves of the formOP1

k(αi) in this proof). o

Since H1(P1k , F) = 0, the long exact sequence associated to 23.5 shows that

we have a surjection

Γ(P1k , E˚) // Γ(P1

k ,OP1k) // 0

and the section 1 can be lifted to a section τ of E˚. This is translates into thediagram

0 // F // E˚ // OP1k

// 0

OP1k

τ

OO

=

==

It follows that the sequence (23.5) is split, and as a consequence we have

E˚ » OP1k‘ F,

that is E » OP1k‘À

1ďiďr´1 OP1k(´αi) and the proof is complete. o

Chapter 24

Further constructions and examples

24.1 Grassmannians

Grassmannian varieties are varieties that parameterize linear subspaces of afixed vector space. More precisely, for natural numbers r and n, there is a varietyGr(r, n) whose k-points [W] are in bijection with the set of linear subspacesW Ă kn. For r = 1, this is simply the projective space Pn´1

k . More generally,Gr(r, n) is a non-singular projective variety over k of dimension r(n´ k).

The basic idea is that a linear subspace W Ă kn is determined by a basis ofr vectors w1, . . . , wr P W. We encode these as a rˆ n matrix M with wi as rowvectors. Of course this matrix is not unique in determining W; each choice ofbasis gives a matrix with the same property. However, by Gaussian elimination,we can find a more canonical representative, by putting M in reduced echelonform. In other words, we may represent each W by a matrix M with a r ˆ ridentity matrix as a submatrix. Conversely, any such matrix M determinesa subspace W of kn and now different matrices M, M1 give rise to differentsubspaces W, W1. Note that matrices M with a fixed rˆ r identity submatrix areparameterized by an affine space of dimension nr´ r2 = r(n´ r). It thereforemakes sense to try to construct the variety Gr(r, n) by gluing together these (n

r)

affine spaces.We translate this into the language of schemes as follows. Let k be a field

and let

Mat(r, n) = Spec k

x11 x12 ¨ ¨ ¨ x1n...

. . ....

xr1 ¨ ¨ ¨ xrn

» Arnk

denote the affine space of dimension rn. We write the variables in a rˆ n array,because we would like to think of the k-points of Mat(r, n) as rˆ n-matrices.

Write M = (xij) for the matrix of indeterminates. For each ordered tupleI = (i1, . . . , ir) with 1 ď i1 ă ¨ ¨ ¨ ă ir ď n, let MI denote the submatrix of Mgiven by the columns in I. Also, let MI be M with the columns indexed by I

grassmannians 400

replaced by the identity matrix.For each I, consider the closed subset UI Ă Mat(r, n) of matrices m where the

submatrix mI is the identity matrix. As these are defined by r2 linear equations,we have

UI » Ar(n´r)k

for each I. We will for simplicity write UI = Spec k[MI ], where k[MI ] denotesthe polynomial ring with variables in the columns indexed by Ic.Example 24.1 For n = 4, r = 2, there are 6 such affine spaces A4:

U12 = Spec k

[1 0 x13 x14

0 1 x23 x24

], U13 = Spec k

[1 x12 0 x14

0 x22 1 x24

],

U14 = Spec k

[1 x12 x13 00 x22 x23 1

], U23 = Spec k

[x11 1 0 x14

x21 0 1 x24

],

U24 = Spec k

[x11 1 x13 0x21 0 x23 1

], U34 = Spec k

[x11 x12 1 0x21 x22 0 1

].

K

Next, for two ordered pairs I, J, let define the open set of UI

UI,J = D(det MIJ) Ă UI ,

corresponding to matrices m P UI so that the submatrix mJ is also invertible.Note that if m is a matrix with det mI ‰ 0, then m´1

I m has the identity matrix incolums I, so it belongs to UI . This observation gives an isomorphism

τJ,I : UI,J Ñ UJ,I

sending MJ to (MIJ)´1MI . It is induced from the ring map

φI J : k[MJ ]det MJIÑ k[MI ]det MI

J

sending x Jpq to the (p, q)-th entry of (MI

J)´1MI . Here φJ I(det MJ

I) = 1/ det MIJ ,

so the map is well defined on the localization. For instance, in the exampleabove, with I = (1, 3) and J = (1, 2), (MI

J)´1MI equals(

1 x12

0 x22

)´1(1 x12 0 x14

0 x22 1 x24

)=

(1 0 ´

x12x22

´x12x24+x14x22x22

0 1 1x22

x24x22

)so the ring map φ12,13 : k[x13, x14, x23, x24]x23 Ñ k[x12, x14, x22, x24]x22 sends(

x13 x14

x23 x24

)ÞÑ

(´

x12x22

´x12x24+x14x22x22

1x22

x24x22

)These morphisms satisfy the cocycle conditions of Chapter 3:

φK,IφI,J(MJ) = φK,I((MIJ)´1MI) = (((MK

I )´1MK)J)

´1(MKI )´1MK

= ((MKI )´1MK

J )´1(MK

I )´1MK = (MK

J )´1(MK

I )(MKI )´1MK

= (MKJ )´1MK = φK,J(MJ)

grassmannians 401

We therefore obtain a gluing diagram

UI » Ar(n´r)k UJ » A

r(n´r)k

UI,J UJ,IτJ I

The resulting scheme, denoted Gr(r, n) is called the Grassmannian of r-planes inkn.

The same formulas for the gluing of affine spaces can be done over any ringR. In fact, the construction works over any scheme: For any scheme S, there is arelative scheme Gr(r, n)Ñ S which is obtained by gluing copies of A

r(n´r)S .

The Plücker embeddingConsider the morphism

Φ : Mat(r, n)Ñ A(nr) = Spec k[yI ]

which is induced by the rˆ r determinants, Φ7(yI) = det MI . This induces amorphism

φ : U Ñ P(nr)´1

where U = Mat(r, n)´V(det MI) is the open subset of matrices of full rank.If we consider UI Ă U, we have an induced morphism φI : UI Ñ P(n

r)´1.

Proposition 24.2 The morphisms φI glue to a morphism φ : Gr(r, n) ÑP(n

r)´1, which is a closed embedding.

In particular, Gr(r, n) is a projective variety.

Gr(2, 4)Let us prove the previous proposition in the case n = 4, r = 2. In this case, theimage of φ is the quadric

Q = V(y12y34 ´ y13y24 + y14y23) Ă P5

This follows by the relation between the minors of M:∣∣∣∣∣x11 x12

x21 x22

∣∣∣∣∣∣∣∣∣∣x13 x14

x23 x24

∣∣∣∣∣´∣∣∣∣∣x11 x13

x21 x23

∣∣∣∣∣∣∣∣∣∣x12 x14

x22 x24

∣∣∣∣∣+∣∣∣∣∣x11 x14

x21 x24

∣∣∣∣∣∣∣∣∣∣x12 x13

x22 x23

∣∣∣∣∣ = 0

To conclude the proof, it suffices to show that φI restricts to an isomorphismUI » D(yI). By symmetry, we may consider the case I = (3, 4). So we con-sider the chart D(y34). In this case, we have D(y34) » Spec k[y13, y14, y23, y24],since inverting y34 allows us to eliminate y12 by the above equation. The corre-sponding map U34 Ñ D(y34) is induced by the map taking k[y13, y14, y23, y24]Ñ

k[x11, x12, x21, x22] taking (y13, y14, y23, y24) to (´x21, x11,´x22, x12). This is clearlyan isomorphism.

some explicit blow-ups 402

The universal subbundleOn G = Gr(2, 4) there is an exact sequence

0 Ñ S Ñ O4G Ñ Q Ñ 0

where S and Q are locally free sheaves of rank 2. These are called the universalsubbundle and the universal quotient bundle respectively.

To define S and the map S Ñ O4G, we work over the affine open sets

UI = D(det MI). Here the rows of the 2ˆ 4 matrix MI define a sub module ofk[MI ]4. In other words, there is a map

k[MI ]2Mt

IÝÝÑ k[MI ]4

which gives an inclusion O2UIĂ O4

UI. In the example above, k[MI ]2 maps onto

the submodule of k[MI ] given by

k[MI ]

1

x12

0x14

‘ k[MI ]

0

x22

1x24

Ă k[MI ]4

It follows almost immediately by the ‘linear algebraic’ nature of the constructionthat these inclusions glue together to a subbundle of O4

G. Formally, we canconstruct gluing maps φJ I : O2

UI JÑ O2

UJ Iusing left multiplication by the 2ˆ 2-

matrix (MJI)´t (which is invertible on the overlaps UI J). This satisfies the cocycle

relations, and is compatible with the various inclusions.

24.2 Some explicit blow-ups

Let X be a noetherian integral scheme and let I be a quasi-coherent idealsheaf on X corresponding to a closed subscheme Y Ă X. We will associateto this data a new scheme rX, called the blow-up of X along Y and a morphismπ : rX Ñ X, called the blow-up morphism. This will have the main property thatthe scheme theoretic preimage E = π´1(Y) is a Cartier divisor, and that π isan isomorphism outside E. Moreover, rX will be universal with respect to theseproperties in the following sense:


Theorem 24.3 Let X and I be as above. There is a scheme rX admitting amorphism π : rX Ñ X so that

i) The inverse image π´1(Y) of Y is an effective Cartier divisor on rX.ii) π : rX Ñ X is an isomorphism outside the support of E.

Moreover, for any morphism g : Z Ñ X from an integral scheme with theproperty that g´1(Y) is an effective Cartier divisor, there is a unique rg makingthe following diagram commute

Z rX

X

g

rg

π

The scheme rX is constructed via a gluing operation that we now explain.Given X and the ideal sheaf I , we form the Rees algebra of I given by

R(I) =à

dě0Idtd = OX ‘ I t‘ I2t2 ‘ ¨ ¨ ¨

This is again a quasi-coherent OX-module, and as such it is a graded OX-algebra.For an affine open set U = Spec A Ă X, I = I(U) is an ideal of A, andR(I)(U) =

À

dě0 Id is a graded A-algebra. It follows that we obtain a projectivescheme Proj(R(I)) which is a scheme over Spec A. Moreover, it is not so hard tocheck that the natural morphisms Proj R(I)Ñ Spec A glue to a morphism

π : rX Ñ X

Let us now consider the scheme theoretic image E of Y. Over the open setU = Spec A, this is defined by the fibre product Proj(R(I))Û (Y XU), or inother words Proj of the associated graded ring

R(I)bA A/I =à

dě0Id/Id+1td = A/I ‘ I/I2 ‘ I2/I3 ‘ ¨ ¨ ¨

Let us compute these rings in a few examples.Example 24.4 Consider A2

Z = Spec A where A = Z[x, y]. The ideal m = (x, y)corresponds to the origin in A2

k . The Rees algebra is given by

R(m) = A‘mt‘m2t2 ‘ ¨ ¨ ¨

Consider the homomorphism of graded A-modules

ψ : A[U, V] Ñ R(m)

U ÞÑ tx

V ÞÑ ty


This is clearly surjective, since R(m) is generated in degree 1. The kernel of ψ isgenerated by xV ´ yU, giving a graded isomorphism

R(I) » A[U, V]/(xV ´ yU)

We consider the blow-up X = Proj(R(I)) which is a projective scheme over A.Moreover, the graded surjection A[U, V]Ñ R(I) induces a closed immersion

X Ă PA1 = P1 Â2

Here the section projection gives the blow-up morphism π : X Ñ A2. Note thatthe first projection p : X Ñ P1 has fibers isomorphic to a line in A2. In fact, p isexactly the tautological bundle O(´1). K

Exercise 24.1 Consider A2 = Spec A as above. Show that the blow-up of theideal (x, y)d is isomorphic to the blow-up X in Example 2 (i.e., with d = 1). M

Example 24.5 We now projectivize the previous example. Consider P2 = Proj Awhere A = Z[x0, x1, x2]. The ideal m = (x0, x1) corresponds to the point(0 : 0 : 1) P P2. As above, there is a surjection of graded A-modules

ψ : A[U, V] Ñ R(m)

U ÞÑ tx0

V ÞÑ tx1

giving us an isomorphism of graded rings

R(m) » A[U, V]/(x0V ´ x1U)

This in turn gives a closed embedding of the blow-up X

X Ă P1P2 = P1 ˆP2

Here the section projection gives the blow-up up morphism π : X Ñ P2. Thefirst projection p : X Ñ P1 has fibers isomorphic to P1. K

Example 24.6 If we in Example 2 consider instead the ideal I = (x2, y2), weget a singularity on the blow-up X. Note that I corresponds to a non-reducedsubscheme of A2 supported at the origin. Since I is generated by two elements,we can still carry out the same trick and obtain an isomorphism of gradedA-modules

R(m) » A[U, V]/(x2V ´ y2U)

Note that in the chart D+(U), this is isomorphic to the affine scheme

Spec Z[x, y, v]/(x2v´ y2)

This scheme is not regular: It is singular along the v-axis V(x, y) (in other words,the preimage of the origin). The hypersurface y2 = x2v is known as Whitney’sumbrella. K


Figure 24.1: The Whitney Umbrella

Exercise 24.2 Show that also the blow-up of A2 along the ideal (x2, y) is singularand describe the singular locus. M

Example 24.7 Blow-ups of Complete intersections. Let us generalize the previousexamples, and consider An = Spec A, A = Z[x1, . . . , xn] and let f1, . . . , fr P R bea regular sequence for elements (i.e., such that the image of each fi is a non-zerodivisor in A/( f1, . . . , fi´1). Then the Rees Algebra of I = ( f1, . . . , fr) is given by

R(I) = A[w1, . . . , wr]/J

where J is the ideal generated by the 2ˆ 2-minors of the matrix(w1 w2 . . . wr

f1 f2 . . . fr

)In particular, the blow-up Proj R(I) embeds into An ˆPr´1. K

Example 24.8 A line in P3. Consider P3k = Proj R where R = k[x0, x1, x2, x3] and

let ` denote the line V(x0, x1) Ă P3. The blow-up of P3k along ` is the closed

subscheme of P3 ˆ P1 defined by the bigraded polynomial x0V ´ x1U = 0.Note that the second projection q : X Ñ P1 has fibers of dimension 2: Theycorrespond to planes H Ă P3

k containing `. K

Example 24.9 The quadric cone again. Consider the quadratic cone Q = Spec(S)where S = k[x, y, z]/(xz´ y2)). We saw in Section 15.5 that the line ` = V(x, y)defined a Weil divisor which was not Cartier. What happens if we blow up Qalong this line? Let π : X Ñ Q denote this blow-up. By the universal propertyof blowing up, the inverse image π´1(`) must be a Cartier divisor on X – so π

transforms a non-Cartier divisor into a Cartier divisor. Let us verify this claimdirectly.

Consider the graded ideal I Ă S. The Rees algebra of ` is given by

R = S[w0, w1]/(yw0 ´ zw1, xw0 ´ yw1)

Let us check that X is in fact regular. Note that X is covered by the two affineopen subsets D+(w0) and D+(x1)).

On the open set D+(w0), we have

R(w0) = S[

w1

w0

]/(y´

w1

w0z, x´ y

w1

w0) » k[z]

[w1

w0

]


So Spec R(w0) » A2k . Thus X is regular in this chart. The same happens in the

other chart D+(w1). Thus D = π´1(`), being a subscheme of codimension 1 ina non-singular variety, should be a Cartier divisor.

Note that D = π˚(`) corresponds to the ideal generated by x, y in R. In thering R, this ideal decomposes as

(x, y) = (x, y, z)X (x, y, w1)

So that D has two components: D1 = V(x, y, z) and D2 = V(x, y, w1). Note that

π(D1) is the point (0, 0, 0) P Q, whereas π1(D2) = `. To show that D is Cartier,it suffices to show that the ideals of D1 and D2 are both locally generated by oneelement. Using the description of D+(w0) above, we see that D1 is described byz = 0, and D2 by w1

w0= 0, so they are both principal. On D+(w1) = Spec k[x][w0

w1],

D1 = V(x) and D2 = 0 (since w1 = 0 defines the empty scheme on D+(w0)).Hence D is a Cartier divisor.

In this example, we could also blow up the ideal m = (x, y, z) correspondingto the origin o = (0, 0, 0). The resulting blow-up X1 would again be a regularscheme, with the property that π´1(o) is Cartier. In fact, using the universalproperty of blowing up, one can show that X and X1 are isomorphic (howeverthe two Rees algebras R(I) and R(m) are very different). K

Example 24.10 The weighted projective space P(1, 1, p). Let S = k[x0, x1, x2] wherethe variables x0, x1, x2 have degrees 1, 1, p respectively. In Chapter 9, we definedthe weighted projective plane P = P(1, 1, p) as Proj S. To study P more explicitly,it will be conventient to study the Veronese subring S(p), which is generated indegree 1 by the elements

xp0 , xp´1

0 x1, . . . , xp1 , x2 (24.1)

Using these as a basis for Γ(P,O(p)), we get a rational map φ : P 99K Pp+1.These sections are clearly base point free, so the map φ is a morphism. Further-more, working locally, one can check that φ is in fact an embedding.

The image of φ is defined by the kernel of the corresponding surjectionk[u0, . . . , up, up+1]Ñ S, or in other words, the relations between the monomials(24.1). Note that the first p + 1 monomials define the rational normal curve C inPd. Realizing this, we see that φ embeds P as the projective cone over the curve

resolution of some surface singularities 407

C. In particular, P is singular at the vertex of the cone, corresponding to the theclosed point P = V(x0, x1).

We will now consider the blow-up X of P at the point P. We claim that X isin fact regular.

K

Example 24.11 The Twisted Cubic. Let k be a field and consider the twisted cubiccurve C Ă P3

k . Recall that C is defined by three quadrics(x0 x1 x2

x1 x2 x3

)From the resolution of the ideal in Section xxx, it is not so hard to show that theRees algebra is isomorphic to

R(I) =A [w0, w1, w2]

(x1 w0 ´ x2 w1 + x3 w2, x0 w0 ´ x1 w1 + x2 w2)

This presentation shows that the blow-up X of P3k along C embeds into P3

k ˆP2k .

Here the first projection is the blow-up morphism π : X Ñ P3.The blow-up has the following geometric description. Consider a0, . . . , a3 P k

and the corresponding closed point a = (a0 : a1 : a2 : a3) in P3. If a R C, the

matrix M =

(a0 a1 a2

a1 a2 a3

)has rank 2 (since some 2ˆ 2-minor does not vanish

at a). This means that the equation M ¨w = 0 has only one non-trivial solutionup to scaling. The fiber of π : X Ñ P3 is exactly the corresponding point(w0 : w1 : w2) in P2. On the other hand, if a P C, then the two rows of M areproportional and there is a 2-dimensional null space V Ă k3 of solutions. Thefiber of π over a is exactly the projectivization of V, i.e., a line in P2. In fact,using the equations above, it is possible to see that the restriction of π to E Ă Xgives a ‘P1-bundle’ over C (i.e., π is Zariski-locally isomorphic to CˆP1 Ñ C).

The other projection q : X Ñ P2 is also interesting. Since the relations in theRees algebra are linear in the xi, it means that also q is a ‘P1-bundle’ over P2, i.e.,every closed fiber is isomorphic to P1, and it is locally trivial. If we start withO(1) on P3, we get an invertible sheaf L = q˚O(1) on X, and the pushforwardq˚L is a locally free sheaf on P2, called the Bordiga bundle. K

24.3 Resolution of some surface singularities

The quadratic cone we studied on page 5.8, is the simples member of a seriesof singular surfaces in A3, which also are prototypical examples of a class ofsurface singularities called An-singularities. These again form one series amongthe so-called Du-Val singularities (also called the rational double points). Thereis another infinite series Dn and three exceptional ones, E6, E7 and E8, but wewill only study the An’s, which are the simplest.


Generally speaking a resolution of the singularities of a variety X is a smoothvariety rX and a birational morphism π : rX Ñ X. One may specify differentadditional desired properties of π, the strongest being that π is the compositionof blow ups with smooth centers contained in singular part of X. It is naturalto include at least one example of this important technique in a section aboutblow-ups.

Du Val singularitiesThe du Val singularities are isolated surface singularities characterized by thethe configuration of the exceptional divisor. Best stated in terms of th dualgraphs where each node stand for an irreducible component of E and twonodes are connected by an edge precisely when the correponding componentsmeet. All components are isomorphic to P1, and when meeting they meettransversally. Finally, their self-intersection is ´2; meaning that the restrictionOX(E)|E » OP1(´2).

The An singularityFor each natural number n, we consider the surface Yn = V(xz´ yn+1)ĎA3. Bythe Jacobian criterion one easily checks it is singular at the origin P = (0, 0, 0),but smooth elsewhere. The surface Yn is a normal as is any surface in A3 whosesingularities are isolated, but it is not factorial. The elements x, y and z areirreducible members of the coordinate ring An = k[x, y, z]/(xz´ yn+1) so by thevery definition of An unique factorization does not hold.

Unlike what is true for the the quadratic cone, when n ě 2 lines through theorigin, apart from the two lines lx = V(x, y) and lz = V(z, y), are not containedin Yn; they meet Yn in at most n´ 1 other points. The two lines are Weil divisorson Yn that are not Cartier; however their union is given by the single equation


y = 0, and is Caartier. We’ll need to refer to the two lines ly and lz later, so we’llcall them the special lines.

For notational reasons, we extend the class An and include the two surfacesY0 and Y´1, which both are non-singular surfaces, the equations being xz´ 1and xz´ y.

We aim at describing such a resolution of the singularities of the surface Yn,and will prove the following proposition.

Proposition 24.12 There is a nonsingular surface Wn and a birational mor-phism π : Wn Ñ Yn which is the succession of blow-ups of single points.

i) The number of blow-ups is n/2 blow-ups when n is even and (n +

1)/2 if n is odd.ii) The exceptional divisor E is

E = F1 + ¨ ¨ ¨+ Fn

with each Fi » P1 and two components Fi and Fj are disjoint unless|i´ j| = 1 and in that case Fi and Fj meet transversally in one point.Further, it holds that O(Fi)|Fi = OP1(´2).

The exceptional divisor is the scheme theoretical inverse image of the singularpoint in Yn, and π induces an isomorphism between WnzE and YnzP.

Blow-up of a point in P3

To begin with we describe the blow-up π : rYn Ñ Yn of origin Pn in detail. Apractical way of doing that is to use that rYn equals the proper transform of Yn

in the blow-up ĂA3 of the origin. This blow-up of A3 is given as the closedsubscheme

rA3ĎA3 ˆP2,

where P2 = Proj k[u, v, w], and which is given by the vanishing of the 2ˆ 2-minors of the matrix (

u v wx y z

).

In other words the equations of rA3 in A3 ˆP2 are the equations

uy´ vx = uz´wx = vz´wy = 0. (24.2)

We let p : rA3 Ñ A3 denote the blow-up map, which is just the restriction ofthe first projection, and further we let q : ĂA3 Ñ P2 denote the restriction of thesecond projection.

The blow-up rA3 is covered by the three distinguished open affine subsetsD+(u), D+(v) and D+(w). Each is isomorphic to A3, the isomorphism isobtained from (24.2) after inverting the actual variable; for instance, if v ‰ 0,


solving (24.2) for x and z yields x = uv´1y and z = wv´1y. In order to simplifythe notation (and in accordance with standard dehomogenization principles), weset the inverted variable, in this case v, equal to one. Then x = uy and z = wyand D+(v) = Spec k[u, y, w]. It may be described as the subset of A3 ˆ P2

consisting of the points(uy, y, wy)ˆ (u : 1 : w)

with y, v and w varying freely.The exceptional fibre EXD+(v) is given as (x, y, z)k[u, y, w] which in view

of the relations x = uy and z = wy becomes

(x, y, z)k[u, y, w] = (uy, y, wz) = (y),

and the restriction of q to EXD+(v) yields an isomorphism with the standarddistinguished set D+(v)ĎP2.

The two other open affines D+(u) and D+(w) have similar properties. Itholds that D+(u) = Spec k[x, v, w] with y = vx and z = wx whilst D+(w) =

Spec k[u, v, z] with x = uz and y = vz. The exceptional divisor satisfies E XD+(u) = V(x) and EXD+(w) = V(z), and they are mapped isomorphically byq onto the standard distinguished sets D+(u) and D+(w) in P2.

The blowing up the singular point of rYn

Let us trace what happens to the blow-up of Yn in each of the open sets in theprevious paragtraph, and we begin with most interesting one, namely D+(v).Here the affine coordinates are y, u and w and x = uy and z = wy. With thesesubstitutions the equation xz´ yn+1 of p´1(Yn)XD+(v) takes the form

xz´ yn+1 = yu ¨ yw´ yn+1 = y2(u ¨w´ yn´1).

Discarding y2 we find that the equation becomes uw´ yn´1, which describesthat part of the proper transform rYn lying in D+(v). We note that rYn XD+(v) isjust the surface Yn´2 (and this is a crucial observation for the later iteration).

It is straightforward to determine the equation for Yn in the two remainingdistinguished opens and verify that Yn has no singularities in either: In D+(u),which is an A3 with coordinates x, v and w and transition formulas y = vx andz = wx, the equation of rYn XD+(u) takes the form

xz´ yn+1 = wx2 ´ vn+1xn+1 = x2(w´ vn+1xn´1).

Discarding x2 leaves us with the equation w´ vn+1xn+1 = 0, which describesa smooth surface. Now, x and z appears in a symmetric way, so by symmetryĂYn XD+(w) is smooth as well. We have proven:

Lemma 24.13 When n ě 3, the blow-up of Yn in the singular point has just onesingular point, which has an affine open neighbourhood isomorphic to Yn´2. Theblow-up of Y1 and Y2 are smooth.


The exceptional divisor is an important part of the description, so we notethat EX D+(v) is the union of the special lines lx and ly in Yn´2, and closingthem up, we see that E is the union of two P1’s meeting in the point Pn´2. Thetwo original special lines lx and ly in Yn are split apart, and their inverse imageseach meet one of the P1’s in one point.

lx ly

The final iterationTo arrive at the nonsingular surface Wn, we shall iterate the blow-up procedureand for each natural number r ď (n + 1)/2 recursively construct a tower

Zn´2r Zn´2r+2 . . . . . . Zn´2 = rYn Zn = Yn

of birational maps each being the blow-up of a singular point of type An´2i.It begins with the blow-up Zn = ĂYn of Yn in the singular point. As we saw,

it is covered by three open affines and unless n = 1 or n = 2, has one singularpoint Pn´2 lying on an affine piece isomorphic to Yn´2 (In case n = 1 or n = 2the surface Yn´2 is non-singular). The same holds for the blow-up Zn´4 of rYn´2

in the singular point, it is cover by five affines of which four are smooth and thefifth is isomorphic to Yn´4.

Each Zn´2r is covered by 2r + 1 affine opens, and has just one singular pointlying in an affine piece isomorphic to Yn´2r, unless of course n ´ 2r = 0 orn´ 2r = ´1, and those cases apart we apply Lemma 24.4 and let Zn´2r´2 be theblow-up of Zn´2r in the singular point.

Finally, when r = n/2 or r = (n + 1)/2 according to n being even or odd,the top surface Zn´2r will be smooth; indeed, the critical open affine, i.e. the onewhich is not a priori smooth, will either be isomorphic to Y0 or to Y´1, and eitherone is smooth. Letting Wn = Zn´2[(n+1)/2] and π the composition of the maps inthe tower (24.3), we arrive at the first part of Proposition 24.3.

The exceptional divuisorPulling back L0 = V(y) in Yn to rYn results in the divisor L1 whose ideal is y.The affine coordinates in D+(v) = A3 are u, y, w and y ‘persist being y’, thatis L1 X D+(y) are exactly the two special lines through the singular point. Soiterating this, at stage r, we we find the expression for the exceptional divisor:

En = F1 + ¨ ¨ ¨+ (F2r+1 + F2r) + ¨ ¨ ¨+ F2

andEn + lx + ly = 0

a scheme without closed points 412

When reaching the final stage, this gives ii). Using 24.3 we find

O = O(En + lx + ly)|Fi =

$

’

’

&

’

’

%

O(Fi + Fi + Fi´1)|Fi 1 ă i ă n

O(lx + Fn + Fn´1)|Fn

O(F2 + F1 + lz)|F1

and the ii) follows in view of Fi intersecting Fi+1 transversally in one point andlikewise lx (resp ly) meets Fn (resp lz) transversally in one point.Example 24.14 We underline that the surface xz + yn´1 is a prototypical An-singularity. In general an An-singularity is one with a resolution as in theproposition, and in general they are not even locally isomorphic. For instancea surface Y in A3 with equation xz f + yn+1, where f is function that does notvanish at the origin, has an An singularity at the origin and will be resolved inthe same manner as the one given by xz + yn+1, but it requiers a little additionalwork. Of course, the resolution is local and only affects the singularity at theorigin, any other singularities Y might have remains unaltered. K

Exercise 24.3 Resolve the singulary in ther example by mimicking what we didfor xy + zn+1. M

24.4 A scheme without closed points

Just to illustrate what strange existences you may come across at the fringeof the land of schemes, we furnish you with an example of a scheme withoutclosed points! It is of course counterintuitive, and meant as a warning to becareful with your arguments when you deal with general schemes. To restoreyour peace of mind we advice you to prove the following proposition:

Proposition 24.15 If X is a quasi-compact scheme, then X has a closed point.

The construction is in the end very simple. Basically, it is a gluing process,but not one like the one we did in Chapter 3 where two schemes were gluestogether along two open sets, but a process where a closed set is glued to anopen.

We shall construct an increasing sequence Xn of irreducible affine schemes,each having just one closed point. Each scheme Xn will be an open subschemeof the next one Xn+1 and will equal the complement of the unique closed point.Finally, the scheme X will be the union of the Xn’s (or rather the inductive limit).

To construct Xn+1 from Xn, we attach the spectrum Spec An+1 of a discretevaluation ring An+1 to Xn by gluing the unique closed point xn in Xn to theopen point in Spec An+1. Of course, for this to be feasible the fraction field ofAn+1 and the residue field k(xn) must be isomorphic, and we call it Kn.


Letting Xn = Spec Bn, we may describe the attaching process is by a push-out-diagram

txnu Xn

Spec An+1 Xn+1,

where txnu stands for Spec Kn. The upper horizontal map is the inclusion of theclosed point in Xn (and Kn plays the role of the residue field k(xn)), where asthe left vertical one is the inclusion of the open point in Spec An+1 (and Kn takeson the role as the fraction field of An+1). The dual diagram looks as follows

Kn Bn

Kn+1 An+1 Bn+1.

φn

π ψ

where Bn is the coordinate ring of Xn; that is, Xn = Spec Bn. The square tothe right is cartesian, i.e. corresponding to the push-out-diagram of schemesabove, and the unique closed points of Xn and Xn+1 are induced by φn andφn+1 = π ˝ ψ.

We’ll give a direct ad hoc construction of the scheme X by first defining theunderlying topological space and subsequently constructing an appropriatesheaf of rings on it. However, to understand what happens one should have thegluing process in mind.

To begin with, we describe the underlying topological space of X. Taking alook at the picture above, it is not hard to convince oneself that a model for Xwould just be the set of non negative integers X = t0, 1, 2, . . . u equipped withthe terminal topology; that is, the closed sets are sets:

Zn = tn, n + 1, n + 2, . . . u.

They obviously form a decreasing sequence, and if IĎN is any set of indices,one easily verifies that

č

iPI

Zi =

#

H if I is infinite ,

Zn if I is finite and n = max I .

Thus the sets Zn together with the empty set satisfy the axioms for the closedsets of a topology. Observe that the closed sets are all infinite, hence no point inX is closed.


The open sets in X are the following. In addition to the whole space X andthe empty, one has Un = t0, 1, 2, . . . , nu. They form an increasing sequence, andone has

ď

iPI

Ui =

#

X if I is infinite

Un if I is finite and n = max I .

There are many ways of putting a sheaf of rings on X. Just take any sequenceof rings and homomorphisms like

B1 B2α2oo ¨ ¨ ¨

α3oo ¨ ¨ ¨ Bnαnoo Bn+1

αn+1oo ¨ ¨ ¨

αn+2oo (24.3)

and define a presheaf B on X by declaring the sections to be

Γ(U,B) =#

Bn if U = Un

limÐÝ

Bn if U = X ,

The restriction maps Γ(Un+i,B) Ñ Γ(Un,B) are the compositions αn ˝ ¨ ¨ ¨ ˝

αn+i, and in case U = X, the restriction Γ(U,B) Ñ Γ(Un,B) is the canonicalmap lim

ÐÝBn Ñ Bn. One easily verifies that B is a sheaf—this amounts to nothing

more than the definition of the projective limit—and so we have put a structureof ringed space on X. For a general sequences (24.3), this ringed space is farfrom being a scheme, the open sets Un equipped with the sheaves B|Un asstructure sheaves are not affine. However, for a juicy choice of the sequence,that we are about to describe, they will be and X is a scheme—hence a schemewithout closed points. What we need is that topologically Un = Spec Bn, andthat Bn+1 Ñ Bn induces an open immersion UnĎUn+1. Note that Un has theterminal topology; the open sets are the segments ti, i + 1, . . . , nu, n is the onlyclosed point and 1 the generic point.

Assume given a diagram of rings and ring homomorphisms where the rightsquare is cartesian (i.e. dual of a push-out diagram of schemes)

K B

L A C,

φ

(24.4)

where B is a local ring with maximal ideal mB, and A is a discrete valuation ringwith maximal ideal mA, residue field L and field of fractions K. By the definitionof a Cartesian diagram of rings it holds true that C = t (a, b) | φ(b) = a u, so wemay identify C with the inverse image φ´1(A). This implies that mBĎC. Lett P mA be a generator, and s P C a lift of t, i.e. an element such that φ(s) = t.

Lemma 24.16 One has B = Cs. Hence Spec B is the open subscheme D(s) of Spec C.The ring C is a local ring, and Spec C = Spec BYtmCu where mC denotes the maximalideal of C.

every finite partially ordered set is a spectrum 415

Proof: Let b P B be any element. We may write φ(b) = utr, where u isunit in A and r an integer. Hence b = vsr + w where v P C is a lift of u, andw P Ker φ = mBĎC. Hence b P Cs, and B = Cs. As mBĎC and C/mB is adiscrete valuation ring, it follows that φ´1(mA) is the only prime ideal in Ccontaining mB. o

To accomplish the construcion of the scheme X without a closed point, weshall recursively construct a sequence of diagrams like (24.4) above:

Kn Bn

Kn+1 An+1 Bn+1,

φn

π ψ

αn+1

φn+1

(24.5)

where the fields Kn all are of infinite transcendence degree over a base field k. Therings An+1 are all discrete valuation rings with Kn as fraction field and Kn+1 asresidue field. That done, using lemma 24.6 above, the sequence

B1 B2 ¨ ¨ ¨ ¨ ¨ ¨ Bn Bn+1 ¨ ¨ ¨α2 α3 αn αn+1 αn+2 (24.6)

gives a scheme structure on X with Un = Spec Bn.The point of departure for the construction of the Bn’s is a discrete valuation

ring B1 whose residue field K1 is of infinite transcendence degree over the basefield k, and φ1 is the residue map. Assuming φn : Bn Ñ Kn is constructed, thefollowing lemma shows that the sought-for diagrams (24.5) can be found:

Lemma 24.17 If K is a field of infinite transcendence degree over the base field k, thereexists a discrete valuation ring A contained in K whose residue field is of infinitetranscendence degree over k.

Proof: One may find a subfield LĎK and an element x P K with K = L(x)and x transcendental over L. Then A = L[x](x) is a discrete valuation ring as wewant. o

Note that the residue field and fraction field of A are in fact isomorphic.Exercise 24.4 Prove Proposition 24.5. Hint: Show that a quasi-compact topo-logical space possesses non-empty minimal closed subsets. Use that irreducibleclosed subsets of a scheme have a unique generic point. M

24.5 Every finite partially ordered set is a spectrum

A pertinent question is, at least for mathematicians working with scheme forsome time: what topological spaces can be underlying the spectrum of a ring?The answered was first given by Melvin Hochster, who proved that the threenatural properties of being sober (that is, every closed irreducible subset has


a unique generic point), of being quasi compact and of having a basis for thetopology consisting of quasi compact opens, suffice.

In particular, all finite T0-spaces are spectra, or equivalently, every finitepartially ordered set P is the spectrum of a ring. Pursuing the gluing techniquefrom the previous chapter, we shall give a simple and direct proof of this, whichalso serves as a nice illustration of some of the techniques we have developed.

We’ll be working with a fixed field K purely transcendental of infinitetranscendence degree over another field k, which we for simplicity assume isalgebraically closed, and we aim at proving:

Proposition 24.18 For any finite partially ordered set P, there is a ring Aso that Spec A » P as partially ordered set. One may choose A so that everyresidue field of A is isomorphic to K.

The natural way of attacking such a problem is by induction on the numberof elements in P. The idea is to split off the ‘cap’ P+ of P; this is the set ofelements of depth at most one (the depth of a point is the length of the longestascending chain emanating from the point). Then P = P+ Y P´ where P´denotes the set of non-maximal elements in P. The intersection P+ X P´ consistsof the minimal points in P+ which as well are the maximal ones in P´. Thefollowing lemma is trivial but useful:

Lemma 24.19 Given two partially ordered set P and Q and assume they have the samecap decomposition; that is, there are isomorphisms (as partially ordered sets) P+ » Q+

and P´ » Q´ that coincide on the intersection P+ X P´. Then P and Q are isomorphic.

Proof: The two bijections, f+ and f´, obviously match up to give a bijectionf from P to Q. We must verify that f (x) ď f (y) whenever x and y are twoelements from P with x ď y. If both either belong to P+ or to P´, it holds thatf (x) ď f (y). If x P P+ and y P P+, there is a z P P+ X Pi lying between them,and hence f (x) = f´(x) ď f´(z) = f+(z) ď f+(y) = f (y). o

For the induction to work P+ and P´ must of course have fewer elements thanP. This fails precisely when P´ is contained in P+, or in other words, whenpoints in P are either maximal or minimal (P is of dimension at most one, if youwant). So that case must be treated separately. For the moment we assume theproposition is valid for these.

Cap and pasteThe cap P+ of P is of dimension at most one, and we have assumed it isisomorphic to Spec A+ for some ring A+ with all residue fields isomorphic toK. The set P´ has fewer elements than P´ (there is always at least one maximalelement), and by induction P´ is isomorphic to Spec A´ for some ring Aáll whose residue fields are isomorphic to K. The minimal points of P+ are


the maximal ones of P´; and our challenge t is to glue Spec A+ and Spec A´together along these points, which is what Lemma 24.10 below will do for us.

Let the minimal prime ideals in A+ be p1, . . . , pr, and denote the correspond-ing points in Spec A+ by y1, . . . , yr. Furthermore let x1, . . . , xr be closed pointsof Spec A´ with m1, . . . ,mr the corresponding maximal ideals of A´. The fieldsK(xi) and K(yi) are isomorphic to K, we choose an isomorphism, identify themand call them Ki. Consider the Cartesian square

A´ś

Ki

C A+

φ

ψ

ι (24.7)

where φ is induced by the canonical maps A´ Ñ A´/mi = Ki and ι by the mapsA+ Ñ A+/piĎKi. The map ι is injective because A+ is reduced and the pi’s areall the minimal primes in A+.

Lemma 24.20 In the setting just described, the following hold true:

i) Spec A+ is a closed subset of Spec C;ii) For each i the maximal point xi in Spec A´ and the minimal point yi of

Spec A+ map to the same point in Spec C;iii) There is an element t P C such that the complement of Spec A+ satisfies

(Spec A+)c = Spec Ct and Ct = (A´)t;iv) All residue fields of C are isomorphic to K.

Proof: The first statement is obvious since ψ : C Ñ A+ is surjective (because φ

is).Considering the compositions of φ and ι with the projection from

ś

i Ki ontoone of the factors Kj, one infers that mi X C = ψ´1(pi), which is the algebraicversion of the second statement.

For the third, we begin by observing that m1 X ¨ ¨ ¨ Xmr = ψ´1(p1)X ¨ ¨ ¨ X

ψ´1(pr) holds since Ker φ = Ker ψ. Let q1, . . . , qs be the finitely many non-maximal prime ideals in A´ (corresponding to the points in Spec A´ztx1, . . . , xru).We contend that there is an element t P m1 X ¨ ¨ ¨ Xmr such that t R qj for eachj; and crucially, t lies in C. The open subset D(t) = Spec(A´)t of Spec A´ willthen be equal to Spec A´ztx1, . . . , xru. The element t being chosen to kill A+

and each of the Ki’s as a C-modules, one has (A+)t = (Ki)t = 0, and localizingthe exact sequence of C-modules

0 // C // A+ ‘ A´ //ś

i Ki // 0,

one infers that Ct = (A´)t. In other words Spec Cz Spec A+ equals the distin-guished open subset D(t).

To exhibit the element t, notice that from a potential inclusion relationm1 X ¨ ¨ ¨ XmrĎ q1 Y ¨ ¨ ¨ Y qs it would ensue by prime avoidance that one of the


qj’s be equal to one of the ideals mi, hence we find a t lying in each mi, but notin any qi, and t P C because m1 X ¨ ¨ ¨ XmrĎC.

The forth claim in the lemma, follows readily; the only critical points beingpoints on the brim, but as ψ is surjective, C/ψ´1pi = A+/pi whose fraction fieldis Ki = K. o

Proof of Proposition 24.8: In view of the trivial lemma 24.9, the above lemmaimplies the proposition; indeed, P+ » Spec A+ and P´ » Spec A´ moreoverthe isomorphisms extend over the intersection since Spec A+ X Spec A´ =

tx1, . . . , xru = ty1, . . . , yru. o

Example 24.21 One can not expect the resulting ring to be Noetherian whenone glues a closed point to an open. A simple example arises from the gluingdiagram

Q[t] Q

R Z(p),

ev

where ev is the evaluation map that sends f (t) to f (0). The ring R consists of thepolynomial assuming a value at zero whose denominator, when written in lowestterms, is without the prime p as factor. It is well known to be non-Noetherian(the ideals ai = (pít) form an ascending chain that is not stationary). K

Sets of dimension oneWe also attack the dimension one case by induction, and we continue with theglueing business, but this time we shall glue closed points to closed points. Soassume that P is one dimensional, fix one minimal point q and let Q be thesubset of P consisting of q and the maximal points dominating q.

If P = Q; that is, if q is the only minimal point in Q, one realizes P asthe spectrum of a semi-local ring A. Take as many algebraically independentelements αi P K as there are maximal elements in P, say r, and consider A =Ş

i K[t](t´αi)ĎK(t) where t is a variable. It is a semi-local integral domain withr maximal ideals and Spec A » P. Note that K(t) » K, so all residue fields areisomorphic to K.

If Q is strictly smaller than P, we may apply the induction hypothesise andPztqu is isomorphic with Spec B for some K-algebra B all whose residue fieldsare isomorphic with K. Each point xi in Qztqu corresponds to a unique point yi

in Pztqu, and the tactics are to glue these together pairwise. Successively doingthis, we arrive at adhering Spec A to Spec B in way to give the desired scheme.

To make induction work, the following lemma is perfect, and we shall applyit with C = Aˆ B and C/a » k(x1)‘ k(y2). (Remember that both residue fieldsk(xi) and k(yi) are isomorphic to K).

a most peculiar scheme 419

Lemma 24.22 Assume given a push-out diagram of rings

C C/a

D Kψ

φ ι

where C is a K-algebra, aĎC and ideal and ι the map induced from the K-algebrastructure of C. Then the ideal a is contained in D and is a maximal ideal there. Moreover,the morphism Spec φ : Spec C Ñ Spec D restricts to an isomorphism Spec CzV(a) »

Spec Dztau.

Proof: First of all, D is a K algebra in a natural way. Trivially, the ideal alies in D since K Ñ A/a is injective and the diagram is Cartesian. It must be amaximal ideal because D/a = K which is a field and D is a K-algebra. One hasa sequence of D-modules

0 D C‘ K C/a 0 (24.8)

Let q be an ideal in D different from a, and let t P a, but t R q. Localizing (24.8)in t gives

0 Dt Ct ‘ Kt (C/a)t 0

but bot K and C/a are killed by t, hence Dt » Ct. Letting t vary, we obtainan open affine cover tD(t)u of Spec Dztau such that the map Spec C Ñ Spec Drestrict to isomorphisms (Spec φ)´1(D(t)) = Spec Ct Ñ D(t), hence the laststatement in the lemma. o

Finally we comment that when coalescing two points, our glueing diagramis shaped like

C K‘ K

D K,

so obviously C is finite as a module over D generated lifts of the two non-trivialidempotents in K‘K. Hence if C is Noetherian, D will be as well after a theoremof Eakin and Nagata. In the one dimensional case, every partially ordered setwill therefore, by induction, be the spectrum of a Noetherian ring. In higherdimensions this is never true as it would violate the corollary of Krull’s Principalideal theorem that in between two primes in a Noetherian ring there is alwaysinfinitely many primes if any at all.

24.6 A most peculiar scheme

In almost every text book in algebraic geometry one meets the map π : A2C Ñ A2

C

that sends the point (x, y) to (x, xy). The map collapses the so-called exceptional


divisor, the line E = V(x), to the origin. Outside E it is an isomorphism, andthe image is just the subset D(x)Y t(0, 0)u. We shall refer to π is as the affineblow up of the origin, since it is the restriction of the blow up map from Section5.6 to one of the affine charts.

u

E=V(u)

v

x

y

In what follows, we shall work over an arbitrary algebraically closed groundfield k.

The behaviour of an irreducible curve C in A2k when pulled back along π is

simple. There are two different cases according to C passing by the centre ornot.

If C does not pass by the centre of blow up, its inverse image in A2 persistsbeing irreducible; any intersection C has with the line V(x) is pushed out toinfinity. For instance, the line y = c becomes the hyperbola x ¨ y1 = c. Indeed,if f is irreducible in k[x, x´1, y1] so if f = ab in k[x, y1], either a or b is a unitin k[x, x´1, y1]. In other words it either equals a power of x or a scalar, but asf R (x) it must be a scalar.

In the second case when C passes through the centre, so that f P (x, y),the polynomial f (x, y) is without constant term, and f (x, y) = f (x, xy1) =

xµ f1(x, y1) for some µ and with f1 R (x, y1).The main example will be the limit of a sequence of iterated birational maps

of the type:

A8 . . . . . . Ai Ai´1 . . . A1 A0, (24.9)

where all the Ai’s are isomorphic to A2k ; we let Ai = Spec Ai with Ai = k[x, yi].

The maps between the Ai’s are basically affine blow ups, but at each stage wetranslate along the exceptional line Ei+1 = V(x) by an amount of ai+1 beforecomposing with the blow up map. In other words, the map πi : Ai+1 Ñ Ai willbe given by the assignment (x, yi+1) ÞÑ (x, x(yi+1 + ai+1)). The scheme A is theinverse limit of the sequence, and will be the spectrum of the direct limit (in factthe union as it will turn out) of the rings Ai.

The map πi : Ai+1 Ñ Ai is induced by the inclusion k[x, yi] ãÑ k[x, yi+1]

where yi = x(yi+1 + ai+1). The Ai’s share the rational function field k(x, y) as


their common quotient field where they form an ascending chain of rings. Theirunion is the ring A = k[x, y, y1, y2, . . . ], and the projective limit appearing in(24.9) is just A8 = Spec A.

k[x, y] . . . k[x, yi] k[x, yi+1] . . . A k(x, y)

Note that k[x, x´1, yi] = k[x, x´1, yi+1] so the induced map (Ai)x Ñ (Ai+1)x

between localizations are not only isomorphisms, but equalities. In the samevein, Ax = k[x, x´1, y]. The open subset U8 = D(x)ĎA8 maps isomorphicallyto each Ui = D(x)ĎAi (and in fact, the map is the identity).

One easily verifies that the principal ideal (x)A is a maximal—indeed, killingx entails killing all yi—and if p8 P A8 denotes the corresponding closed point,it holds true that A8 = tp8u YD(x). The point p8 maps to the centre of blowup in each Ai and is the inverse image of each exceptional line Ei.

In general the ring A is not Noetherian, for instance if all the ai’s are zero, itholds true that

Ş

i(xi) = (x, y, y1, . . . ). The astonishing point, however, is thatfor a sufficiently generic choice of the points taiu, the ring A will be Noetherian.We shall prove

Proposition 24.23 If the power series τ =ř

i aixi is transcendental, the ringA is Noetherian. It is an an integral domain of dimension two, and all its localrings are regular, and A is even a ufd. Furthermore, it has a closed point ofheight one through which no curve passes.

Technically speaking, the closed point p8 is a Cartier divisor, as it is givenby one equation!

The hypothesis in the proposition enters the proof in the following way. Forany power series there is a corresponding ring homomorphism ι : A Ñ k[[x]]which is injective when τ is transcendental. To define the map, define for eachr P N the “tail” of τ P k[[x]] by:

τr =ÿ

iě1

ai+rxi = arx + ar+1x2 + . . . .

Then τ =ř

iăr aixi + xr´1τr, and it holds true that τr = x(τr+1 + ar+1), whichpermits us to define the map ι : A Ñ k[[x]] by x ÞÑ x and yr ÞÑ τr.

Lemma 24.24 If the power series τ =ř

i aixi is transcendental, the map ι is inJective.

Proof: If ι is not injective, for some r there is a polynomial F(x, yr) withF(x, τr) = 0. Putting G = xN F for a sufficiently large natural number N, we getG(x, τ´

ř

iăr aixi) = 0, which shows that τ is algebraic. o

Proof of of the proposition: We first recall Cohen’s criterion which says that aring is Noetherian if all prime ideas are finitely generated. So consider a prime


ideal pĎ A. There are two cases, either pĎ (x) or p Ę (x). For each i P N, definepi = pX Ai.

The case that p is not contained in (x) is the easy one. Pick an ielement f P p,but with f R (x). For some large i, we have f P Ai, but f R (x, yi) = (x)X Ai,and we may certainly assume that f is irreducible. Since V( f ) does not pass bythe centre of blow up, it persists being irreducibel (and prime) in Aj for all j ą i,hence f is irreducible and prime in A. So if p is a height one ideal, we readilyget p = ( f ). If the height is two, pX Ai = (x´ a, y´ b) and one easily obtainsp = (x´ a, y´ b).

For future reference, we treat the salient case when p Ď (x), in a separatelemma. o

Lemma 24.25 If for a prime ideal p in A it holds p Ă (x), then p = 0.

Proof: Assume p Ď (x). Then the ideal p0 = pX k[x, y] is strictly contained in(x, y) and is therefore principal generated by an irreducible polynomial f (x, y).To keep the geometric intuition, we let C denote the corresponding curve inA0 = A2. For each index i the polynomial f belongs to Ai and may be expressedas f (x, y) = fi(x, yi). Geometrically speaking, the polynomial fi defines theinverse image Ci of C in Ai. We contend that for each i there is a factorisationfi = xigi in Ai; of course, this entails that gi P p, and consequently gi mustbe without a constant term. Geometrically, these factorisations correspond tochopping a copy of the exceptional divisor off the inverse images of C at eachstage in the tower (24.9).

Proceeding by induction we find

fi+1(x, yi+1) = fi(x, x(yi+1 + ai+1)) = xigi(x, x(yi+1 + ai+1)).

Since gi does not have a constant term, gi(x, x(yi+1 + ai)) has x as a factor, andwe may write gi(x, x(yi+1 + ai)) = xgi+1(x, yi+1), and the claim is established.

So we have f PŞ

i(xi). Now, the crux is the inclusion ι : A ãÑ k[[x]]. Ofcourse in the power series ring k[[x]] it holds true that

Ş

(x)i = 0. It follows thatι( f ) = 0; consequently f = 0 and p = 0. o

A Noetherian domain that is not catenaryThe making of such an example needs two ingredients. The first is a constructionvery similar to the previous one but with a parameter z. That is, instead ofblowing up a sequence of points in A2, we blow up a sequence of lines in A3.The second ingredient is a pinching manoeuvre as in Section 24.5.

In algebraic terms we start out with an ascending chain

A0Ď . . . Ď AiĎ Ai+1Ď . . . Ď A


of rings where Ai = k[x, yi, z] and yi+1 = x(yi + ai), and we put A =Ť

i Ai =

k[x, z, y0, . . . ]. Geometrically we have a sequence

A8 . . . . . . Ai Ai´1 . . . A1 A0,

where this time Ai » A3 and each map is the translation yi ÞÑ yi + ai followedby the affine blow up of the line Li = V(x, yi). The exceptional divisors Ei are inthis case the planes V(x) = Spec k[z, yi] and the restriction of the map πi is justthe projecion onto Spec k[z] = Li´1

As before one checks that (x, z´ c)Ď A is a maximal (killing x kills all yi andsetting z = c transform A into k) and that the ideal (x) is prime with A/(x) =k[z]. The closed subset L8 = V(x) maps to each of the lines Li = V(x, yi)ĎAi,which are blown up. And the (x, z´ c)Ď A constitute the closed points of L(8).

Proposition 24.26 If the power series τ =ř

i aixi is transcendental, the ringA is Noetherian.

Before proceeding to the proof, let us finish with the application and from Aderive a ring which is not catenary. Observe that the maximal ideal m = (x, z)defines a closed point p although being of height two. Take any closed point qnot lying on L8. Its maximal ideal n is of height three since q is lying in openpart D(x) of A which is isomorphic to A3

k . Now, the idea is to coalesce p and qto one point r, just like we did in Section 24.5. To this end, consider the pushoutdiagram of rings

A k‘ k

B k

φ

ι

where φ is the evaluation at p and q and ι the canonical diagonal map. Thering B is the pushout ring defined by the diagram. Clearly A is generated overB by two elements, and citing the Eakin–Nagat theorem we deduce that B isNoetherian.

So why is B not catenary? From (0) to mp we have the staurated chain0 Ă (x) Ă (x, z) = m and from (0) to n we have saturated chains of length three,more over Spec A and Spec B differ merely in that the two pints p and q areidentified to say r, and the two chains survives intactly as chains from (0) to mr;one is of length two and the other has length three.

Proposition 24.27 The ring B is a Noetherian domain that is not catenary. Ithas a maximal ideal joined to zero by saturated chains of length two and three.

Proof of Proposition 24.15: We shall reduce to the previous case througha projection onto the plane z = 0; algebriacally this correponds to the idealq = pX k[x, y, y1, . . . ] and qi = pX k[x, yi]. There are two cases to consider:


i) pX k[x, y, y1, . . . ] ‰ 0;

ii) pX k[x, y, y1, . . . ] = 0;

In the first case, when q ‰ 0, it follows from the previous case that qi Ę (x, yi)

for i ąą 0 and hence that V(pi) eventually will be disjoint from Li. By the littlelemma 24.17 below it ensues that pi Ai+1 = pi (two primes one contained in theother which are equal in D(x), must be equal). A set f1, . . . , fr of generators forpi, will generate pj for j ě i as well and hence also p; indeed, any element in p

lies in pi for some i ąą 0. .In the second case, it must so that pi is of height one; indeed k[x, yi]Ď k[x, yi, z]/pi

show that dim k[x, yi, z]/pi = 2—and it is therefore generated by an irreduciblef P k[x, yi]. Now by the same argument as above, f stays irreducible (and henceprime) in Aj for j ě i and we deduce that p = ( f ). o

An easy little lemmaOur situation may be summarized by the diagram where X = Spec B andY = Spec A are two Noetherian affine schemes, and π is a dominant mapcorresponding to an inclusion AĎ B:

X Y

E F

π

π

Moreover, F = V(x) is a divisor and E = V(x) is the inverse image of F; finallywe make the a crucial assumption that π induces an isomorphism XzE » YzF.The following little lemma is heuristically concincing, but needs a proof, whichis a nice recapitulation of primary decomposition:

Lemma 24.28 In the staging just described, if Z = V(p) is disjoint from F, then pB isa prime ideal.

Proof: Since X is Noetherian, there is an irredundant primary decompositionpB = qX q1 X ¨ ¨ ¨ X qr, where q is such that V(q)X D(x) = π´1Z, and q is aprime ideal because π is an isomorphisn between XzE and YzF. Let pi =

?qi.

In XzE = D(x), the qi’s disappear from the decomposition. Hence for each iit holds that V(qi)Ď E, which is impossible since V(pB) does not meet E. Indeed,cover XzV(pB) by distinguished open subsets V( fi) with 1 ď i ď s, which meansthat ( f1, . . . , fs) = pB. Then pBĎ p1Y ¨ ¨ ¨ Y pr and prime avoidance gives pBĎ pi

for at least one i, which contradicts the decomposition being irredundant. o

Appendix A

Some results from Commutative Algebra

A.1 Discrete valuation rings

Let K be a field, and let Kˆ denote the set of non-zero elements of K.

Definition A.1 A discrete valuation on K is a map v : Kˆ Ñ Z such that

o v(xy) = v(x) + v(y)

o v(x + y) ě min(v(x), v(y)).

where we must assume that x + y ‰ 0.

It is common to formally set v(0) = 8 to get around this exception, so thata valuation is a map v : K Ñ ZY t8u. We usually assume that v is normalized,i.e., v is surjective.

Note that the first condition implies that v(1) = 0. Furthermore, we musthave v(x´1) = ´v(x).

The valuation ring of v is the subring of K given by

A = tx P Kˆ|v(x) ě 0u Y t0u

This is a local ring with maximal ideal m Ă A generated by all elements x withv(x) ą 0. The group of units in A is given by the subgroup

Aˆ = tx P K|v(x) = 0u.

An integral domain A is called a discrete valuation ring if it is the valuationring of a valuation v : Kˆ Ñ Z, where K is its field of fractions. This means thatfor any x P K, either x P A or x´1 P A.Example A.2 Let K = k(x) be the field of rational functions in one variable.Let f P k[x] be an irreducible polynomial. Then any element y P K can bewritten as y = f dg/h where d P Z; and g, h are coprime to f . We can define a

unique factorization domains 426

valuation v f : Kˆ Ñ Z by setting v(y) = d. In this case, the valuation ring is thelocalization of k[x] at f :

A = k[x]( f )

K

Example A.3 Let K = Q be the field of rational numbers, and let p be a primenumber. Any y P Q can be expressed as y = pda/b where d P Z and a, b arecoprime to p. We can define a valuation vp : Qˆ Ñ Z by setting v(y) = d. Inthis case, the valuation ring is the localization of Z at (p):

A = Z(p) =!m

nP Q |gcd(p, n) = 1

)

K

Here is a general characterization of such rings:

Proposition A.4 Let A be a noetherian local integral domain with maximalideal m. The following conditions are equivalent:

o A is normal and dimension 1

o A is normal and m is a principal ideal

o A is a principal ideal domain of dimension 1 (and hence discrete valuationring)

o A is a regular ring of dimension 1.

Corollary A.5 An integral domain A is a discrete valuation ring if and only if thereis an element t P A such that every non-zero a P A may be written uniquely as a = utd,where d P Z and u is a unit.

The element r is said to be a uniformizing parameter for A. The integer d is calledthe order of a and we write ord(a) = d.

A.2 Unique factorization domains

Lemma A.6 Let A be a noetherian domain. Then A is a UFD if and only if everyheight 1 prime ideal is principal

Proof: Suppose that A is a UFD. Let p be a height 1 prime ideal. Take x P p

non-zero and let x = x1 ¨ ¨ ¨ xn be a factorization into irreducible elements. Sincep is prime, we must have, say, x1 P p. However, also (x1) is prime (since A isUFD), so since p has height 1, we must have p = (x1).

Conversely, suppose that every height 1 prime is principal. Since A is noethe-rian, every non-zero non-unit x has a factorization into irreducible elements. It

hartog’s extension theorem 427

suffices to prove that an irreducible element is prime. Let (x) Ă p be a minimalprime over (x). Then p has height 1 (localize at p and use minimality to seewhy).

o

A.3 Hartog’s extension theorem

Proposition A.7 Let A be a noetherian normal integral domain of dimensioně 1 with fraction field K. Then

č

pPSpec A, ht(p)=1

Ap = A

A.4 Projective modules

A.5 Dimension theory

Appendix B

More on sheaf cohomology

In Chapter 13, we introduced the Cech cohomology of sheaves, which is wellsuited for computations, and in an fact is most efficient road (if not the only)to find the explicit necessary results on cohomology. There is however anotherstandard way of introducing cohomology which works in greater generality. Itgoes by the so-called derived functors, in our case the right derived functors (thereis also the notion of left derived functors). This goes back to

Grothendieck’srevolutionaryTôhoku-paper from1957, and this 65 yearsold approach seems tobe old fashioned today;there is a modern andextremely generaldefinition in the theoryof model categories.

The idea is to approximate an object A (in any abelian category) by ‘coho-mologically trivial objects’. Such an approximation, or a acyclic resolution asit is called, is an exact complex (C ‚, d‚) with an isomorphism A Ñ Ker d0; itdisplays as

0 A C 0 C 1 C 2 . . . (B.1)

and, the key point, the C i are ‘cohomologically trivial’ (we’ll come back withwhat that means in our concrete situation, typically the C i will be so-called‘injective’ objects ).

Recall that a complexis exact if the kernel ofeach map is equals theimage of the precedingone; that isIm di = Ker di+1.

Then one applies the functor F to C ‚ and thus obtains thecomplex F(C ‚), which displayed appears as

F(C 0) F(C 1) F(C 2) . . .

The value of the (right) derived functor (or the i-cohomology group) of F atA will be the homology of that complex; that is, for each i P N0 one hasRiF(A) = Hi(F(C ‚)).

There is a serious issue brought on by the choices involved in this process.The homology Hi(F(C ‚)) must be well-defined so it must, in some sense, beindependent of the choice of the complex (B.1), and the precise condition is it beunique up to a unique isomorphism. Uniqueness of the isomorphism is requiredto have the necessary functorial properties, one wants equalities between inducedmaps.

We shall not dive into the deep see of abelian categories and homologicalalgebra, but merely concentrate on our present interest, the global section functor.And we shall circumvent the unicity issues by using so-called flabby sheaves as

flabby sheaves 429

resolving objects; with those there is a completely canonical resolution of anyabelian sheaf, which also depends functorially on F .

Part of the story is also to show that the two definitions of cohomologycoincide in most situations. In the case of general (separated) schemes thishinges on a theorem of Henri Cartan with a longish proof, which we refrainfrom giving. We contend ourselves with a proof for Noetherian seperatedschemes; then things are considerably much easier.

B.1 Flabby sheaves

Let X be a topological space and F a sheaf X. One calls F as Flabby sheaves (butteeller slappe knipper)

flabby if therestriction map

F (X)Ñ F (U)

is surjective for every open subset U Ă X. Flabby sheaves are quite differentfrom the coherent sheaves one most often work with in algebraic geometry, theytend to be rather large and ‘formless’. Here are two prototypical examples:Example B.1 Godement sheaves. Back in Chapter 1, we constructed the Godementsheaves A. They were constructed by choosing an arbitrary family of abeliangroups Ax, one for each point x P X, whose group of sections over an open U is

A(U) =ź

xPU

Ax

and whose restriction maps are induced appropriate projections. These sheavesare obviously flabby. Indeed, the restriction map A(X) =

ś

xPX Ax Ñś

xPU Ax =

A(U) is just the projection that keeps the components of (ax) with indices x P Uand throws the others away.

In particular the Godement sheaf Π(F ) associated to an abelian sheaf Fbelongs to the class of flabby sheaves; just let the family of abelian groups bethe family of stalks Fx. K

Example B.2 If X = Spec A is affine and M is a divisible A-module (that is,all multiplication maps x ÞÑ f x with f ‰ 0 are surjective), then ĂM is flabby.Indeed, the localization maps M Ñ M f are surjective. In particular, this appliesto injective modules over an integral domain. K

So to the words ‘cohomologically trivial’. Heuristically, the origin of coho-mology of sheaves is that taking global section does not preserve surjections,and the next lemma may be view as an indication that flabby sheaves are‘cohomologically trivial’:

Lemma B.3 Given an exact sequence

0 F G H 0

flabby sheaves 430

of sheaves F , G and H on the topological space X. If F is flabby, the correspondingsequence of global sections

0 F (U) G(U) H(U) 0

is exact for every open set U Ď X.

Proof: By restricting F , G and H to U, it suffices to prove the statement forU = X. The global section functor is left exact, so we only need to check thatthe sequence is exact on the right. Let σ P H(X). Consider the family Σ of pairs(U, s) of open subsets U of X and sections s P G(U) that maps to σ|U . The setΣ has a partial order for which (U, s) ď (U1, s1) if UĎU1 and s = s1|U , and it isquite clear that under this ordering every ascending chain in Σ is bounded. SoZorn’s lemma ensures there is a maximal pair (U0, s0).

Aiming for a contradiction, assume that U0 is not the entire space X and picka point x P XzU0. Let U1 be an open neighbourhood of x small enough that σ|U1

lifts to a section s1 in G(U1). On the intersection V = U0 XU1 both sections s0|V

and s1|V maps to σ|V , and hence their difference s0|V ´ s1|V belongs to F (V).Now F is flabby, so the difference is the restriction of a section t P F (X). Thens1 + t|U1 maps to σ|U1 and coincides with s0 on V. Hence the two can be gluedtogether to a section of G over U0 YU1 that maps to σ|U0YU1 , contradicting themaximality of (U0, s0). o

Lemma B.4 Suppose we are given an exact sequence of sheaves

0 // F // G // H // 0.

If F and G are flabby, then so is H.

Proof: Let U Ă X be a subset of X. Then each section h P H(U) is representedby a section g P G(U) by the previous lemma. Since G is flabby, g can beextended to a section g1 of G(X). Then g1 maps to an element h1 P H(X)

extending h; that is, h1|U = h. o

Lemma B.5 Suppose we are given an exact complex of flabby sheaves

0 F 0 F 1 ¨ ¨ ¨ F i F i+1 . . .d0 d1 di(B.2)

Then for each open set U Ă X, the complex

0 F 0(U) F1(U) ¨ ¨ ¨ F i(U) F i+1(U) . . .d0(U) d1(U) di(U)

(B.3)is exact.

Proof: One chops the complex (B.3) up into short exact sequences

0 Im di F i+1 Im di+1 0.

the godement resolution 431

Bearing the two preceding lemmas in mind, we see by induction that each Im di

is flabby (the base of the induction follows as Im d0 = F0 which is flabby byassumption) and that each sequence

0 Im di(U) F i+1(U) Im di+1(U) 0

is exact. o

B.2 The Godement resolution

Given a sheaf F on a topological space X, we are about to construct a resolutionof F in terms of flabby sheaves which we shall use to define the cohomologyof F . There are no choices involved, so the construction is canonical, andmoreover it has the virtue of being functorial (in every conceivable way) sowe get unambiguously defined cohomology groups, and all their functorialproperties come almost for free.

To explain how this works, recall the Godement sheaf Π(F ) with sectionsover U being

Π(F )(U) =ź

xPU

Fx

and restriction maps the appropriate projection, and the canonical inclusionκF : F Ñ Π(F) which over an open set U sends a section s P F (U) to theelement (sx)xPU . Defining C 0F = Π(F ) and Z 1F as the cokernel of κ, we geta canonical exact sequence

0 F C 0F Z 1F 0.

Remember that Π is a functor AbShX Ñ AbShX which is compatible with κ;that is, it holds true that Π(α) ˝ κF = κG ˝Π(α) for each map α : F Ñ G. ThusΠ(α) passes to the quotient, and we have commutative diagrams

0 F C 0(F ) Z 1F 0

0 G C 0(G) Z 1G 0.

α C 0α Z 1α

This makes Z 1 a functor.The Godement functor Π(F ) is even an exact functor. This hinges on the

fundamental quality that being exact is a local property of sequences of sheaves;so if the sequence

0 F G H 0

is exact, the sequence of sections over an open U

0ś

xPU Fxś

xPU Gś

xPU H 0

sheaf cohomology 432

is exact for all U; indeed, it is obtain by taking the product (which preservesexactness) of the stalk-wise sequences (which are exact). The snake lemma thenshows that also Z1 is an exact functor.

We now iterate this construction and recursively put C i+1F = C 0Z iF andZ i+1F = Z 1Z iF . These sheaves all fit into short exact sequences, one for eachi, shaped like

0 Z 1F C iF Z i+1 0.

Proceeding to assemble the Godement resolution we splice these sequencestogether to a complex C ‚F . The sheaves in this complex will of course be theC iF ’s, and the differentials di : C i Ñ C i+1 will be the compositions C i Ñ Z i Ñ

C i+1.

Proposition B.6 Given a topological space X and an abelian sheaf F on X.

i) The Godement complex C ‚F is a flabby resolution of F .ii) The complex C ‚ depends functorially on F , and the functor

C ‚ : ShAbX Ñ CpxShAbX is an exact functor.

Proof: Most has already been done. By construction the sheaves C iF are flabbyand C ‚ is exact in positive degrees. For i = 0 it holds, also by construction, thatKer d0 » F . This takes care of i).

Claim ii) is an immediate consequence of C i and Z i being exact functors. o

B.3 Sheaf cohomology

We are now ready for defining the cohomology of an abelian sheaf F . Theprocedure is: first form the Godement resolution C ‚F of F , then take globalsection of C ‚F , which yields a complex of abelian groups C ‚F (X), and finally,take the homology of that complex:

Definition B.7 Let F be an abelian sheaf on the topological space X. We definethe i-th cohomology group Hi(X,F ) by the formula

Hi(X,F ) = Hi(C ‚F (X)).

For each map α : F Ñ G we define Hi(X, α) : Hi(X,F ) Ñ Hi(X,G) by theformula

Hi(X, α) = Hi(C ‚α(X)).

The notation Hi(X, α) is exceptonally cumbersome and one usually abbrevi-ats it to αi

˚ or sometimes even to α˚ with the index i being tacitly understood.The cohomology is a functor in that (α ˝ β)˚ = α˚ ˝ β˚ whenever α and β arecomposable maps between abelian sheaf and of course id˚ = id.


Recall that any short exact sequence of complexes of groups induces a longexact sequence in homology. And for any functor to have the right to bear thetitle ‘a cohomology theory’ an absolute requirement is similarly to induce longexact sequences from short ones:

Proposition B.8 (Long exact sequence) With each short exact sequence

0 F G H 0α β

of abelian sheaf on the topological space X and each non-negative integer iis associated a connecting map δ : Hi(X,F ) Ñ Hi+1(X,H) so that the longsequence

. . . Hi(X,G) Hi(X,H) Hi+1(X,F ) Hi+1(X,G) . . .β˚ δ α˚

is exact. Moreover, the connecting map δ depends functorially on the sequence.

Again, including the dependence on the sequence and on i in in the notationδ would make it unnecessarily cluttered; but of course, when needed anyappropriate decoration is possible. That δ depends functorially on the sequencemeans that for any map between two exact sequence, that is a set up like

0 F G H 0

0 F 1 G 1 H1 0

α β γ

with squares commuting, it holds true that αi+1˚ ˝ δ = δ ˝ γi

˚; or for loversof diagrams, that for all i the middle red square in the following diagramcommutes:

. . . Hi(X,G) Hi(X;H) Hi+1(X,F ) Hi+1(X,G) . . .

. . . Hi(X,G 1) Hi(X,H1) Hi+1(X,F 1) Hi+1(X,G 1) . . .

βi˚

δ

γi˚ αi+1

˚ βi+1˚

δ

The other squares commute as well simply because the cohomology H‚(X,F )

is functorial in F .Proof: Proposition B.4 tells us that the sequence

0 C ‚F C ‚G C ‚H 0

formed from (B.6) is an exact sequence of complexes. In each degree there is anexact sequence of sheaves which is exact and consists of flabby sheaves, and byLemma B.1 it follows that it persists being exact after global sections are taken.But that means precisely that the complex

0 C ‚F (X) C ‚G(X) C ‚H(X) 0


of abelian groups is exact, and taking homology yields a long exact sequence ofhomology groups (see Chapter 13).

The second statement about functoriality follows from the correspondingproperty of complexes of abelian groups since both C ‚ and Γ(X,´) are functors.

o

Proposition B.9 If F is flabby all higher cohomology of F vanish; i.e.Hi(X,F ) = 0 for i ě 1.

Example B.10 Flabby resolutions furnish good tools for establishing generalstatements, but in concrete situations they are usually rather difficult to study inan explicit manner. There are however a few exceptions, and one comes here:

Let X = Spec A be a reduced and irreducible affine scheme; that is, the ringA is an integral domain. The field of fractions K of A induces the sheaf rK onX, and since K is divisible, this sheaf is flabby. One effortlessly checks thatalso the quotient K/A is divisible, hence ĆK/A is flabby, and we have the flabbyresolution

0 OX rK ĆK/A 0.

It follows using acyclicity of flabby sheaves and the long exact sequence thatHi(X,Ox) = 0 for i ą 1, and H1(X,OX) = 0 since the global section of the maprK Ñ ĆK/A is just the surjection K Ñ K/A. K

Example B.11 Let X be an integral scheme, In Chapter 15 we defined thegroup of CaDiv as Γ(X, K ˚/O˚X where KX is the constant sheaf with value thefunction field K(X) of X, and Cartier divisor class group as the cokernel of themap K Ñ Γ(X, K ˚/O˚X) induced from the exact sequence

1 O˚X K ˚X K ˚

X /O˚X 1

We saw that K ˚X is a flabby and hence it follows that CaCl(X) » H1(X,O˚X). K

Exercise B.1 Let X be a topological space and let ι : Z Ñ X be the inclusion of asubset. Show that for a sheaf F on Z,

Hi(X, i˚F ) = Hi(Z,F ) (B.4)

for all i. M

Acyclic sheavesSince the Godement resolution often is difficult to handle and the involvedsheaves are both rather enormous and structureless, one looks for other andmore workable resolutions. The following proposition, where resolutions byso-called acyclic sheaves are used, gives this flexibility.

Definition B.12 A sheaf F on the topological space X is called Acyclic (asyklisk)acyclic ifHi(X,F ) = 0 for all i ą 0.


Given a resolution C ‚

0 F C 0 C 1 C 2 . . .

of F (which by definition means that the sequence is exact), the resultingsequence C ‚(X)

C 0(X) C 1(X) C 2(X) . . .

may fail to be exact, but is still a complex of abelian groups, so it makes senseto ask about its cohomology, and the point is that if the Ci’s are acyclic, we getback the cohomology of F :

Lemma B.13 If the sheaves C i in (B.3) are acyclic, then there is a natural isomorphism

Hi(X,F ) » Hi(C ‚(X))

Proof: Define K´1 = F , and Ki = Ker(C i+1 Ñ C i+2) for i ě 0. By exactness ofthe complex C ‚, we have for each i ě 0 an exact sequence

0 Ñ Ki´1 Ñ C i Ñ Ki Ñ 0

Taking the long exact sequence, gives

0 H0(Ki´1) H0(C i) H0(Ki) H1(Ki´1) H1(C i) = 0 (B.5)

where the right-most group it zero because C i is acyclic. Also, the same sequenceshows that Hp(Ki) = Hp+1(Ki´1) for every p ě 1. The maps in these sequencesfit into the diagram

0

H0(Ki)

H0(C i´1) H0(C i) H0(C i+1)

H0(Ki´1) H0(Ki+1)

0

From this, we see that

Im(

H0(C i)Ñ H0(Ki))= Im

(H0(C i)Ñ H0(C i+1)

)and that

H0(Ki´1) = Ker(

H0(C i)Ñ H0(C i+1)).

cech vs sheaf cohomology 436

In particular,

H0(F ) = Ker(

H0(C 0)Ñ H0(C 1))= H0(C ‚(X)),

and the theorem holds in degree i = 0. By the same token, we have

H0(Ki) = Ker(

H0(C i+1)Ñ H0(C i+2)).

From (B.5), and the isomorphisms Hp(Ki) » Hp+1(Ki´1) we therefore get

Hi+1(C ‚(X)) = Ker(

H0(C i+1)Ñ H0(C i+2))

/Im (H0(C i)Ñ H0(C i+1))

= H0(Ki)/Im (H0(C i)Ñ H0(Ki))

= H1(Ki´1)

= H2(Ki´2)

= ¨ ¨ ¨

= Hi+1(K´1)

= Hi+1(F ).

o

B.4 Cech vs sheaf cohomology

A ever returning phenomenon is the diversity of homology and cohomologytheories. The seems to be that there one categorical pattern and then various par-ticual vesrions adapted to different purpose, but for most spaces they coincide.So also in the present notes. We have two definitions of sheaf cohomology onegiven by the Godement resolution and then Cech cohomology. In this sectionwe shall show that they coincide for quasi-coherent sheaves on Noetheriansepareted schemes. The Noetherian hypothesis is not necessary, but disposingof it requires a rather long proved result of Henri Cartan, which we merely willcite.

Note the important point that in Leray’s theorem, only uses a fixed cover -this is indispensable when it comes to concrete calculations.

The tactics of the proof of Leray’s theorem are first to exhibit resolution ofthe sheaf in question by making a complex of sheaves out of the Cech resolutionassociated with an affine cover, and then verify the salient point that those Cechsheaves will be acyclic onces we know the affine schemes are cohomologicallytrivial.

The Cech resolutionWe start the program by introducing the sheafy version of the Cech complex.The setting is a scheme X with a quasi-coherent sheaf F . We are further givenan finite open affine cover U = tUiuiPI of X, and if α = (i0, . . . , iq) is a sequence


of indices from I we use the notation Uα = Ui0...iq = Ui0 X ¨ ¨ ¨ XUiq . These are allaffine since X is separated. Moreover, we let ια : Uα Ñ X be the open inclusionof Uα into X.

The covering U = tUiuiPI induces a covering UV = tUi XVuiPI of each opensubset V in X, and with it is associated a Cech complex C ‚(UV ,F |V) as inSection 13.2 on page 252. Furthermore there are for each open subset V1ĎVobvious restriction maps C p(UV ,F |V) Ñ C p(UV1 ,F |V) (they are simple casesof the refinment maps described in (13.3)), and these make each C p(UV ,F |V)a sheaf; which we shall denote by C p(U ,F ). The sections over an open V aregiven as

C p(U ,F )(V) =ź

(i0,...,ip)PIp+1

F (V XUi0 X ¨ ¨ ¨ XUip)

and with a few moments of reflection, one convinces oneself that this meansthat

C p(U ,F ) =ź

α=(i0,...,ip)PIp+1

ια˚F |Ui0X¨¨¨XUip.

The restrictions of the sheaves C p(U ,F ) are compatible with the coboundarymaps of the Cech complexes, and hence we obtain a complex C‚(U ,F ) of sheaves.The sheaf version of the formula given in Chapter 13 for the coboundary mapreads

(dσ)i0...ip =

p+1ÿ

j=0

(´1)jσi0 ...ij ...ip|VXUi0X...Uip

where σ is a section in C p(U ,F )(V).

Lemma B.14 This gives a resolution

0 F C0(U ,F ) C0(U ,F ) C0(U ,F ) ..d0 d1 d2(B.6)

Moreover, Γ(X, C‚(U ,F )) = C ‚(U ,F ).

Proof: The second statement and that B.6 is a complex, are central featuresfrom the the definition of C‚(U ,F ).

So the main content is that — contrary to the ordinary Cech complex — thesheafy version of the Cech complex is exact. Since this is a sequence of sheaves,we may check exactness on on stalks.

The proof consists of writing down a homotopy operator on the complexC‚(U , F)x of stalks at a point x P X. For a general complex C ‚ with differentiald, a homotopy operator is a map k : C ‚ Ñ C ‚ of degree ´1 (that is, a bunch ofmaps kp : C p Ñ C p´1, one for each p ą 0 so that kd + dk = idC ‚). Having sucha homotopy operator forces the complex to be exact in positive degrees; indeed,if dσ = 0, one has σ = dhσ + kdσ = dkσ, and so σ is a coboundary.


We are about to define a map kp : C p(U ,F )x Ñ C p´1(U ,F )x: An element inthe stalk C p(U ,F )x is induced from a section (σα) over an open neighbourhoodV of x, and we can assume that some r P I it holds that VĎUr (just shrink V ifneeded). Then V XUi0 ...ip´1 ĎV XUri0...ip´1 , and there is a restriction map

ρ : C p(U ,F )(V XUri0...ip´1)Ñ C p(U ,F )(V XUi0...ip´1)

which allows us to define

(kpσ)i0...ip´1 = ρ(σri0...ip´1).

Now the crux is that k is a homotopy operator on the complex C‚(U ,F )x ofstalks; that is,

dk + kd = id.

Establishing this is just a matter of writing down the definitions: on the onehand we have

(dkσ)i0...ip =

pÿ

j=0

(´1)j(kσ)i0...ij ...ip=

=

pÿ

j=0

(´1)jσri0...ij ...ip,

and on the other hand

(kdσ)i0...ip = dσri0 ...ip =

= σi0 ...ip +

pÿ

j=0

(´1)j+1σri0 ...ij ...ip),

and adding the two yields the formula. o

Theorem B.15 (Leray) Assume that X is a topological space with a sheaf F onX and let U = tUiuiPI be an open covering of X. If all sheaf cohomology groupsHp(Ui0 X ¨ ¨ ¨ XUiq ,F ) = 0 for all p ą 0 and all sequences (ij) of indices, thenthe Cech cohomology and the sheaf cohomology of F coincide; more precisely,H(U ,F ) = Hp(X,F ).

There two comments to make: firstly, the conclusion is that actually Cech coho-mology of just one covering gives the sheaf cohomology, a property importantfor the computations. Secondly, we underline that it is the sheaf cohomologyHp(Ui0 X . . . Uiq ,F ) that is supposed to vanish as opposed to the Cech cohomol-ogy. As mentioned in the introduction, there is a related result:

Theorem B.16 (Cartan) Let X be a topological space and F an abelian sheafon X. If there is a set of open subsets B, forming a basis for the topology andbeing closed under finite intersections, and is such that Hp(U, F) = 0 for allU P B and all p ą 0, then Cech and sheaf cohomology of F coincide.


Proof: The sheaves C p(U ,F ) will be acyclic and we can activate Lemma B.9on page 435. Indeed, in view of Exercise B.1, this ensues from the expression in(B.4) for the Cech complex. o

The affine case

Theorem B.17 Assume that X = Spec A is Noetherian affine scheme and F isa quasi-coherent sheaf on X. Then Hi(X,F ) = 0 for i ą 0.

The condition that F be quasi-coherent is essential. For instance, as weobserved in Example 4 in good cases one has Pic X » H1(X,O˚X), and rathermany affine schemes have a non trivial divisor class group. Examples can beSpec A for any Dedekind ring that is not a ufd (e.g. any affine elliptic curve).

As mentioned above, the result holds true without the Noetherian hypothesis(see EGA III 1.3.1??):

Corollary B.18 Let X be a Noetherian scheme and F a quasi-coherent OX-module.

i) If X is separated, sheaf- and Cech cohomology on X agree: it holds thatHi(X,F ) » H(X,F ) for all i ą 0;

ii) If U = tUiuiPI is an open affine covering so that any finite intersectionUi1 X ¨ ¨ ¨ XUir of members of U is affine, then Hi(X,F ) = Hi(X,F ) =

H1(U,F ).

If there is a covering Ui of affines closed under finite intersections, the resultstill is true (and the proof still holds water.).Example B.19 Glue the spectrum X = Spec A of a dvr A to it self at the genericpoint η. Then X is covered by two open affine subsets Ui = Spec A whoseintersection is the open affine tηu = Spec K. Sheaf- and Cech cohomologycoincide, and to compute Hi(X,OX) we have the sequence Cech complex.

0 H0(X,OX) Aˆ A K H1(X,OX) 0α

where α(a, b) = a´ b. Thus H0(X,OX) = A, and H1(X,OX) = K/A. This is arather large module. For instance, in case A = Zp for a prime p, it equals thegroup Zp8 of roots of unity a power of p. K

Proof of Theorem B.13: There are three parts, in the first we establish thetheorem for the special case of the structure sheaf F = OX of an integralscheme, subsequently for coherent sheaves and finally reduce it to that case.

To begin with we assume X integral and F = OX = A. If K is the fractionfield of A, the sheaf K = K is constant and therefore flabby. One easily see thatthe qoutebt K/A is divisible that sheaf is flabby as well, and we have the flabbyresolution

0 OX K K/OX 0


of OX. The globale sections of K Ñ K/OX is surjective so H1(X,OX) = 0, andthe long exact sequence yields Hi(X,OX) = Hi´1(X,K/OX) = 0.

Secondly we consider any coherent sheaf F and write F = M with M afinitely generated A-module (A is Noetherian). A result from commutativealgebra (xxxx) tells us there is a descending sequence tMju of submodules of Msuch that each subquotient Mj´1/Mj = A/pj with pj’s being pime ideals. Hence

0 Fj Fj´1 OX| 0

where Fi = Mi and Xi = V(pi). Induction on i and the first point above yieldsthat Hi(X,Fi) = 0 for all i and j, and this establishes the theorem for coherentmodules.

Finally we treat the case that F is quasi-coherent, and to reduce the proof tothe previous case, we write M as the union

Ť

j Mj = M of its finitely generatedsubmodules Mj.

Quite generally, if F is the union of a bunch of subsheaves tFiu, one readilyverifies that the Godement resolution Π(F ) is the union of the Π(Fi)’s Π(Fi)/Fi

(the sections of Π‚(F) over opens are just products of stalks, and forming stalksis an exact operation). Hence the Godement resolution Π‚(F) has sub complexesΠ(Fi) such that each Πj(F) =

Ť

i Πj(Fj).In our case, the subsheaves Fi = Mj are coherent and each a Π‚(Fi) is exact

by the second point above, and so we may finish the proof by the following littleobservation:

Lemma B.20 If (C‚, d) is a complex with subcomplexes (C‚j , dj) and each C i =Ť

j Cij

is exact for i ą 0, then C ‚ is exact for i ą 0.

Proof: Indeed, if x is a cocycle in C ‚, that is dx = 0. For some index j theelement x belongs to C‚j and persists being cocycle, so because C ‚j is exact, it iscoboubdary dy = djy = x. o

o

Proposition B.21 If F is flasque, then so are the sheaves C p(U ,F ) for p ą 0.Hence (??) is an acyclic resolution for F and

Hp(X,F ) = Hp(C‚(U ,F ))

Proof: If F is flasque, then so is each restriction to each Ui0X¨¨¨XUp , and productsof flasque sheaves are flasque, so

ś

i0ă¨¨¨ăipi˚F |Ui0X¨¨¨XUp

is flasque. o

godement vs. cech 441

B.5 Godement vs. Cech

It remains to see why these two definitions are equivalent. So let U = tUiu be acovering for F . We will assume that this is Leray in the sense that Hi(UI ,F ) = 0for all multi-indices I and i ą 0. We claim that there is a natural isomorphism

Hi(X,F ) » Hi(U ,F ),

where we, in order to avoid confusion, let Hi(U ,F ) denote the Cech cohomologygroup. The statement is clearly true for i = 0, since both coincide with Γ(X,F ).

Lemma B.22 Let 0 Ñ F Ñ G Ñ HÑ 0 be an exact sequence. If U is Leray, there isa long exact sequence

0 H0(U ,F ) H0(U ,G) H0(U ,H)

H1(U ,F ) H1(U ,G) H1(U ,H) ÝÑ ¨ ¨ ¨

Proof: Since U is Leray, we have H1(UI ,F ) = 0 for all multi-indexes I (in fact,this is the only property we need from the covering U ). Hence the followingsequences are exact:

0 Ñ F (UI)Ñ G(UI)Ñ H(UI)Ñ 0

Then applying the Cech complex, we get an exact sequence of complexes

0 Ñ C‚(U ,F )Ñ C‚(U ,G)Ñ C‚(U ,H)Ñ 0

Now the claim follows from Proposition 13.1. o

Hence we get our desired theorem:

Theorem B.23 (Leray) Suppose U is a cover of X and Hq(U1 X ¨ ¨ ¨ X

Up, F ) = 0 for all p, q ą 0 and all U1, . . . , Up P U . Then there is a natu-ral isomorphism between cohomology and Cech cohomology:

Hp(X,F ) » Hp(U ,F )

Proof: We use induction on p. For p = 0, the claim is clear. Note that we havethe exact sequence

0 Ñ F Ñ Π(F )Ñ Z 1 Ñ 0

and H1(X,F ) = Coker(Γ(X, Π(F ))Ñ Γ(X, Z 1F )), and

Hp(X,F ) = Hp´1(X, Z 1F )

for p ě 2. On the other hand, we also have the corresponding result for Cechcohomology:

godement vs. cech 442

0 H0(U ,F ) H0(U , Π(F )) H0(U , Z 1)

H1(U ,F ) H1(U , Π(F )) = 0

where H1(U ,F ) = 0 by Lemma ??, since Π(F ) is acyclic. Hence also

H1(X,F ) = Coker(Γ(X, Π(F ))Ñ Γ(X, Z 1F )) = H1(X,F )

Hence the theorem also holds for p = 1.We continue by induction on p. Since also Hi(U , Π(F)) = 0 for all i ą 0,

same long exact sequence of Cech cohomology also shows that Hp(U ,F ) =

Hp´1(U , Z 1). Moreover, the cover U is also Leray with respect to Z 1: Hi(UI , Z 1) =

Hi+1(UI ,F ) = 0). Hence replacing F with Z 1, we get the desired conclusion.o

Appendix C

Solutions

Exercise 1.1 One can for instance take the constant presheaf with F (U) = Z

for every non-empty U: this does not satisfy the gluing axiom. For an examplewhich fails Locality: write X = tp, qu and define F by F (X) = Z3, F (tpu) = Z,F (tqu = Z and F (H) = 0. Also define the restriction maps ρp : F (X) Ñ

F (tpu) and ρq : F (X)Ñ F (tqu) by the first and second projection map Z3 Ñ Z

respectively. This is easily seen to be a presheaf, but it is not a sheaf since thetwo elements (0, 0, 0), (0, 0, 1) P F (X) both restrict to the same element 0 in tpuand tqu.

Exercise 1.2 Let U Ă X be a connected open set. If the derivative D f iszero, the function f is locally constant, hence constant (since U is connected).Therefore A (U) = C.

If U has connected components tUiuiPI , we can define a map A (U)Ñś

i C

by sending ( f : U Ñ C) to the tuple ( f (xi))iPI where xi P Ui is any point. Thisis clearly an injective and surjective map which commutes with the restrictionmappings.

Exercise 1.3 (A Riemann surface)The presheaf F is not a sheaf, because you can’t glue; e.g. the function

f (x) = x is bounded on any open ball Br(0) = tx P Rn|x ă ru, but therestrictions f |Bn do not give you a bounded continuous function on all ofŤ

ně0 Bn = Rn.In this example, any continuous function is locally bounded, so the saturation

of Cb(X, R)+ in C(X, R) is simply all of C(X, R).

Exercise 1.9 Consider the ring maps vn : Z[[x]] Ñ Z/pn given by sendingx to p. These maps are consistent with the maps Z/pn+1 Ñ Z/pn, so by theuniversal property of the inverse limit we get a map

Z[[x]]Ñ limÐ

Z/nkZ = Zp.

The map is clearly surjective, and the kernel is indeed

X8k=1(x´ p, xk) = (x´ p).

444

Hence we get the desired isomorphism Z[[x]]/(x´ p) » Zp.

Exercise 1.10 a) Let G = limÝÑiPI Gi. Each Gj admits a map to Gi for each i ě j,

and these are consistent with the directed maps of the directed system. Hencewe get for each j, an induced map vj : Gi Ñ G. The maps vj are compatible withthe directed maps Gj Ñ Gj‘ for j‘ ě j. So by the universal property of lim

ÝÑ, we

get an induced map limÝÑjPJ Gj Ñ G.

b) We only need to construct an inverse to the map in a). Since J is cofinal,for each i, we have a map Gi Ñ lim

ÝÑjPJ Gj which is compatible with the directedmaps Gi Ñ Gi‘. Hence by the universal property, we get an induced maplimÝÑiPI Gi Ñ lim

ÝÑjPJ Gj.

Exercise 1.14 Let Gi = t0, 1, . . . , iu and define for each i ď j fij : Gi Ñ Gj to bethe inclusion. The direct limit exists in sets, but it is not a finite set.

Exercise 1.16 Take any non-closed subset Z of your favourite ringed space,and define a Godement sheaf A with the property that Ax ‰ 0 if and only ifx P Z.

Exercise 1.20 Let φ : F Ñ G be a map of sheaves. If s P Ker φ(V), then clearlysx P Ker φx. We can therefore define Φ : (Ker φ)x Ñ Ker φx by sending (s, V)

to sx. Φ is clearly a group homomorphism. Φ is injective: If (s, V) maps tozero, then (s|W , W) = (0, W) for some W Ă V and hence the element is zero inthe stalk (Ker φ)x. Φ is surjective: any element sx P Ker φx Ă Fx is induced bysome section (s, V) for some V Ă X. φ(s) is an element so that φ(s)x = 0, soby shrinking V we may assume that φ(s) = 0 on V, and hence s P Ker φV . Inparticular, sx is induced by (s, V).

Exercise 1.23 To prove Theorem 1.28, it will be convenient to introduce somenotation. Let us define an f -map Λ : G Ñ F to be a collection of homomorphismsΛV : G(V)Ñ F ( f´1(V)) indexed by open subsets V Ď Y such that

G(V) F ( f´1V)

G(V1) F ( f´1V1)

ΛV

ρG ρF

ΛV1

commutes for all inclusions V1 Ď V of open sets in Y. Bearing this definition inmind, we can now prove the following lemma, which implies Theorem 1.28:

Lemma C.1 Let f : X Ñ Y be a continuous map. Let F be an abelian sheaf on X andlet G be an abelian sheaf on Y. There are canonical bijections between the following foursets:

i) The set of f -maps Λ : G Ñ F ;ii) The set of sheaf maps G Ñ f˚F ;

iii) The set of sheaf maps f´1G Ñ F ;iv) The set of presheaf maps f´1

p G Ñ F .

445

Proof: The bijection between iii) and iv) follows by the adjoint property ofsheafification (as in 1.8) because F is a sheaf, so it suffices to consider the sets ini), ii) and iv).

Let us first consider i) and ii). If we are given a map of sheaves φ : G Ñ f˚F ,we have a map φV : G(V) Ñ f˚F (V) = F ( f´1V) for each open set V Ď Y.By the definition of a sheaf map, this commutes with the various restrictionmappings to smaller opens V1 Ď V, so we get a well-defined f -map Λ : G Ñ F .Conversely, it is clear that any f -map Λ appears from a map of sheaves in thismanner, so we have established the desired bijection.

For the bijection between the sets i) and iv), suppose we are given a map ofpresheaves f´1

p G Ñ F , so that we have a map

limÝÑ

WĚ f (U)

G(W)Ñ F (U).

Applying this to U = f´1(V), and compositing with the map G(V) into thedirect limit, we get a map G(V)Ñ F ( f´1V). Again, this is compatible with therestriction maps to smaller open sets V1 Ď V, so we get a well-defined f -mapΛ : G Ñ F . Conversely, it is clear that any f -map Λ arises in this manner, i.e.,each ΛV factors as

G(V) F ( f´1V)

limÝÑWĚV G(W)

ΛV

φ f´1V

for some map of presheaves φ : f´1p G Ñ F : Just define φU for U Ď X by

composing ΛW with the restriction map to get a map G(W) Ñ F ( f´1W) Ñ

F (U) for W Ě f (U) – the fact that Λ is an f -map means that we get an inducedmap in the direct limit. Over U = f´1V, this φ makes the above diagramcommute. This completes the proof of the lemma. o

Exercise 2.1 f P?añ f n P a for some n ą 0 ñ f n P p for all primes p Ą añ

f P p for all p Ą añ f PŞ

pĄa p.We follow the hint. Let ι : A Ñ A f denote the localization map. If f R

?a

then aA f is a proper ideal (1 =ř

ai fń implies that f n P a for some n). Hencethere is a maximal ideal m Ą aA f . The preimage p = ι´1(m) is then a maximalideal containing a, but not f . Hence f R

Ş

pĚa p.

Exercise 2.2 f nilpotent ñ f n = 0 for some n ą 0 ñ D( f ) = D( f n) =

D(0) = H. Conversely, D( f ) = H ñ V( f ) = X ñ f P p for all p P Spec A ñ

f PŞ

p p =?

0 ñ f is nilpotent.

Exercise 2.4 Assume first thatŤ

i D( fi) = Spec AzV(a). Then

V(a) = (ď

i

D( fi))c =

č

i

V( fi) = V(( fi|i P I)

),

446

and consequently a and ( fi|i P I) have the same radical. On the other hand, ifthey share radicals, the same equalities hold true (but read in a different order):

ď

i

D( fi) = V(( fi|i P I))c = V(a)c = Spec AzV(a).

Exercise 2.5 Let U and V be two non-empty open subsets of Z. Then bothU X Z and V X Z are nonempty, and being open in Z, they have a nonemptyintersection since Z is assumed to be irreducible. For the second statementassume that Z is irreducible and that f (Z) = Z1 Y Z2 with Z1 and Z2 closedsets. Then Z = f´1(Z1)Y f´1(Z2), and it follows that either Z = f´1(Z1) orZ = f´1(Z2). In the former case f (Z) = Z1, and in the latter f (Z) = Z2.

Exercise 2.6 Let tZiu be an ascending chain of irreducible subsets containingZ. We contend that the union W =

Ť

i Zi is irreducible. Indeed, if U and V areopen subsets of W, there must be an index ν so that both U X Zν and V X Zν

are non-empty. Both are open in Zν and Zν being irreducible, their intersectionis non-empty. Hence by Zorn’s lemma, there is a maximal irreducible setcontaining Z. To second task, any x P X is contained in an irreducible set;indeed, the closure x is irreducible.

Exercise 2.8 Assume first that Spec A is disconnected; say it is the disjointunion Spec A = U1 YU2 with Ui a proper open set. Then each Ui is closedas well, and hence it is shaped like Ui = V(ai) for some radical ideal ai inA. Since U1 XU2 = V(a1)XV(a2) = H, it holds that a1 + a2 = A, and sinceU1 YU2 = Spec A it holds that

?a1 X a2 =

?a1 X

?a2 = a1 X a2 = 0. Then the

Chinese Remainder theorem yields that A » A/a1 ˆ A/a2.If e is an idempotent, 1´ e is also idempotent, so when e is distinct from 0

and 1, the pair 1´ e and e form a pair of non-trivial orthogonal idempotent withsum equal to unity, and they determine a non-trivial representation of A as adirect product. Hence Spec A is disconnected according to Example 10.

Exercise 2.9 Assume to begin with that X is Noetherian and let Σ be a familyof closed sets without a minimal element. One then easily constructs a strictlydescending chain that is not stationary by recursion. Assume a chain

Xr Ă Xr´1 Ă ¨ ¨ ¨ Ă X1

of length r has been found; to extend it just append any subset in Σ strictlycontained in Xr, which does exist since Σ by assumption has no minimalmember.

Next, assume that every non-empty family of closed subsets has a minimalmember and let an open covering tUiu of an open subset U of X be given.Introduce the family Σ consisting of the closed sets that are finite intersections ofcomplements of members of the covering; i.e. the sets of the shape Uc

i1 X ¨ ¨ ¨ XUcir .

It has a minimal element Z. If Uj is any member of the covering, it follows thatZXUc

j = Z, hence UjĎZc, and by consequence U = Zc.

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Finally, suppose that every open U in X is quasi-compact and let tXiu be adescending chain of closed subsets. The open set U = Xz

Ş

i Xi is quasi-compactby assumption, and it is covered by the ascending collection tXc

i u, hence it iscovered by finitely many of them. The collection tXc

i u being ascending, we caninfer that Xc

r = U for some r; that is,Ş

i Xi = Xr and consequently it holds thatXi = Xr for i ě r.

Exercise 2.11

a)

I = (x2, y2)X (x2, z)X (y2, z)

b)

I = (x2, y2x)X (y, y2x)

= (x)X (x2, y2)X (y)X (x, y2)

= (x)X (y)X (x, y2)

Another primary decomposition is given by

I = (x)X (y)X (x´ y, y2)

c)

I = (x3, xy4)X (y, xy4)

= (x)X (x3, y4)X (y)

= (x)X (y)X (x3, y4)

d)

I = (x2 + (x2 ´ 1)2 ´ 1, y´ x2)

= (x2, y´ x2)X (x´ 1, y´ x2)X (x + 1, y´ x2)

= (x2, y)X (x´ 1, y´ 1)X (x + 1, y´ 1)

Exercise 2.12 It is trivial that?aĎ

?a+ n. To prove the reverse inclusion, let x

be an element in A so that xn =ř

1ďiďr aiyi with ai P a and yi P n. Then each yi

is nilpotent, and they being finite in number, there is an m P N so that ymi = 0.

Appealing to the binomial theorem, one may express a power xN as a sum ofterms, each having as factor a monomial yν1

1 . . . yνrr of degree N; so choosing

N ě rm each term will vanish.To the second question, observe that if n is a locally nilpotent ideal in a ring

A, then n is contained in every prime ideal of A.

Exercise 2.13 (Direct products of rings)

448

a) Elements in Ai are of shape x = aei with a P A and those in ai of shapey = bei with b P a; hence xy = (aei)(bei) = abei, since e2

i = ei, and xy P ai asab P a. Given x P a; it follows from 1 =

ř

i ei that x =ř

ixei, and it holds

that xei P ai. Finally, the sum is direct, since ai X aj = 0 when i ‰ j, due tothe orthogonality relations eiej = 0.

b) Assume that p is a prime ideal in A; it is proper, so at least one of theei’s does not belong to p, say it is ei0 . The ideal pi0 must be prime since ifaei0 ¨ bei0 = abei0 P pi0 , it follows that either a or b belongs to p as ei0 does notbelong there. If i ‰ i0, the orthogonality relation eiei0 = 0 entails that ei P p,and consequently pi = Ai.

c) The sets Ui = t pi | p P Ai u are both open and closed, being equal bothto D(ei) and to V(ri) where ri = t x | xei = 0 u. Each projection A Ñ Ai

induces a homeomorphism between Ui and Spec Ai.

Exercise 2.14 We are given φ : A Ñ B, a = Ker φ locally nilpotent. e2 = eidempotent in B. Let x P A such that φ(x) = a and define y = 1´ x.

a) xy = x(1´ x) = x´ x2 maps to e´ e2 = 0. Hence xy P a.

b) Let f =ř

iąn (2ni )xiy2ní and g =

ř

iďn (2ni )xiy2ní. We have

1 = (x + (1´ x))2n = (x + y)2n = f + g

by the binomial theorem. f g = 0 since every monomial in the expandedproduct is divisible by (xy)n.

c) f 2 = f (1´ g) = f ´ f g = f . So φ( f ) =ř

iąn (2ni )φ(x)iφ(1´ φ(x))2ní) =

ř

iąn (2ni )e

i(1´ e)2ní =ř

2nąiąn (2ni )e(1´ e) + e = e.

d) Suppose X = Spec A = U1 YU2 with U1 XU2 = H and Ui closed and open.Suppose U1 = V(I) and U2 = V(J). We get that H = U1 XU2 = V(I + J)and X = U1 YU2 = V(I X J). Hence I + J = (1) and I X J =

?0.

Pick z P I, w P J such that z + w = 1. Then z = z(z + w) + z2 + zw. Thusz2 ´ z P I X J Ă

?0. Hence z2 = z in A/

?0. Now apply part c) to A/

?0:

We find x P A idempotent mapping to z.Now define φ : A Ñ B ˆ C by a ÞÑ (ax, a(1 ´ x)), where B = Ax andC = A(1´ x) (these are naturally subrings of A). φ is surjective: givenb = a1x P Ax and c = a2(1´ x), we have (b, c) = φ(a1 + a2). φ is injective:φ(a) = (0, 0) ñ ax = 0 = a(1´ x) = a´ ax = a, so a = 0. Done.

Exercise 2.15 X =š

iPI Spec Ai is not quasi-compact, hence not homeomorphicto the spectrum of any ring (the cover Ui = Spec Ai has no finite subcover).

Exercise 2.16 The implication ñ: Given a Q-algebra A there is a canonicalstructure map φ : Q Ñ A. This induces maps Z ãÑ Q Ñ A and hence mapsSpec A Ñ Spec Q Ñ Spec Z, which is the desired factorization.

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Assume given a factorization

Spec A Spec Z

Spec Q

canonical

We apply the global section functor Γ and get the diagram

A Z

Q

canonicalφ

which gives us a morphism φ : Q Ñ A showing that A is a Q-algebra.

Exercise 2.17 Let φ be the morphism Spec Z[i] Ñ Spec Z be induced by theinclusion Z Ă Z[i], and note that Z[i] = Spec Z[x]/(x2 + 1).

a) The preimage of φ over (p) P Spec Z consists of prime ideals q Ă Z[i] suchthat qXZ = (p); that is, prime ideals in Z[i]/(p) = Fp[i], or in other words,prime ideals in Fp[x]/(x2 + 1). We also have

dimFp

(Fp[x]/(x2 + 1)

)= dimFp(Fp + Fpx) = 2.

b) The ring A = Fp[x]/(x2 + 1) is a field if and only if x2 + 1 does not havea root in Fp: Assume that a P Fp is a root of x2 + 1 in Fp; that is, x2 + 1 =

(x´ a)(x´ b) for some b P Fp. Hence (x´ a) is not a unit in A, and A isnot a field.Assume that A is not a field, and let a P Az0 be so that (a) Ă A is a properideal. Then A/(a)A is a Fp-vector space of dimension one, and there is anisomorphism A/(a) » Fp. Thus we get a surjective ring homomorphismφ : Fp[x]/(x2 + 1) Ñ Fp. Note that 0 = φ(x2 + 1) = φ(x)2 + 1, so a = φ(x)is a root of x2 + 1 in Fp.

c) The ideal (p)Z[i] is prime if and only if A = Z[i]/(p) » Fp[x]/(x2 + 1) isan integral domain. If A is not a field, then there is an a P A which generatesa proper ideal, so we conclude as in b).

Exercise 2.18

a) The prime ideal of A are all maximal (CA Proposition 10.55) since A is offinite type over k. Then the Chinese Remainder Theorem gives a surjectionφ : A Ñ A/m1 ˆ ¨ ¨ ¨ ˆ A/mr where the mi’s are the maximal ideals in A.Hence r ď dim A. In case r = dim A, the map φ is an isomorphism andeach A/mi is of dimension one over k.

b) The fibre over a point p in Spec A equals Spec BbA K(A/p) and the ringBbA K(A/p) is a vector pace of dimension n over K(A/p) since A is free ofrank over B. It follows from a) that the fibre has at most n points.

450

Exercise 2.19

a) By Lemma 2.8 on page 52 prime ideal p is maximal if and only if V(p) is aclosed point.

b) Some possibilities include: i) A = Q, ii) A = C[x](x) (or any dvr) and iii)A = C[x, y](x).

c) The point is that because mB is a proper ideal and elements in A not in mA

are units, it always holds that φ´1(mB)ĎmA. Hence φ(mA)ĎmB if and onlyif φ(mA) = mB; ot in other words if and only if φ´1(mB) = mA.

d) The inclusion C[x](x) Ñ C(x) is not local: C(x) is a field and has (0) asits only (maximal) ideal, but this is not maximal in C[x](x). Geometricallythis corresponds to mapping the point Spec C(x) to the open point η P

Spec C[x](x).

Exercise 2.20 Assume first that Spec A is a singleton and let N =?

0 denote thenilradical of A. Clearly A is a local ring (all rings have at least one maximal ideal)with maximal say m. We have N =

Ş

pPSpec A p = m. Hence AzN = Azm = A˚,and every element in m is nilpotent.

Next, assume that A is local and that all non-units are nilpotent. Considerthe quotient map π : A Ñ A/N where N =

?0. It is surjective, and maps units

in A to units in A/N. By assumption AzN = A˚, so each element in A/N iseither zero or a unit. Hence A/N is a field, and consequently N must be themaximal ideal. Every prime ideal p is contained in a maximal ideal; that is,contained in N, and on the other hand the nilradical N is contained in everyprime ideal, so we conclude that p = N. Hence there is only one prime ideal.

Exercise 2.21 Since f is invariant g7 ˝ f 7 = f 7; hence f 7 : B Ñ A takes values inAG, and defining h amounts to letting h7 be the ensuing map B Ñ AG.

Exercise 2.22

a) For each a P A consider the polynomial P(t) =ś

γPG(t´ aγ); which ev-idently is monic, and since G has an identity element, it holds true thatP(a) = 0. It is moreover clear that P is invariant under G; indeed,

P(t)g =ź

γPG

(t´ aγ)g =ź

γ

(t´ aγg) = P(t),

since γg runs through G when γ does. It follows that P(t) is invariant; thatis, P(t) P AG[t]; and so P(a) = 0 is an integral dependence relation for a.Citing Going–Up, we conclude that π is closed and surjective.

b) Consider a prime ideal p in AG. Let q and r be two prime ideals in the fibreof π over p; that is, they satisfy qX AG = tX AG = p. Assume that r is notin the orbit of q. This means that r Ę

Ť

γPG qγ, since by prime avoidance itwould be conained in one of the qγ, and hence equal to qγ by Lying–Over(CA Prop??, ??). So let a P r but r R qγ for all γ; wich menas that a R qγ for

451

any γ. Considerś

γ aγ: It is invarinat under G and lies in r but not q, whichis absurd since rX AG = qX AG

Exercise 3.2 Starting with ( f , f 7) : X Ñ Y and (g, g7) : Y Ñ Z, we defineh = (g ˝ f ) : X Ñ Z by g ˝ f : X Ñ Z on the level of topological spaces, andh7 : OZ Ñ h˚OX by

OZ(W)Ñ OY(g´1(W))Ñ OX( f´1(g´1(W)),

that is, h7 = f 7g´1W ˝ g7W . The induced map OZ,h(x) Ñ OX,x coincides with

f 7x ˝ g7f (x), so being a composition of morphisms of local rings, it is a morphismof local rings.

Exercise 3.9 (Three-point-schemes) Let X be a space with three points andorganize the possible topologies according to the number of closed sets. Thediscrete topology, which has all eight subsets closed, are realized as the spectrumof products of three fields k1 ˆ k2 ˆ k3. It there are seven, the topology is alsodiscrete; indeed, either all points are closed or all doubletons are, and in bothcases the topology will be discrete.

[

There is merely one topology on a three space X having six closed sets: X musthave two closed points and two closed doubletons; and the union of the twoclosed points is one of the closed doubletons. The point not lying in a doubletonis both open and closed and is a connected component of X. One may e.g.realize X as Spec(V ˆ k) where V is a dvr and k a field.

There are two topologies on X having five closed sets: one with two andone with just one closed point, but with two closed doubletons. The latter isirreducible, but has no generic point, so ot is excluded from being a scheme.The former can be realized as the spectrum of any one dimensional semilocalintegral domain with two maximal idea, le.g. Z6.

[[ [

Having four closed subsets forces X to have exactly one closed point andone closed doubleton. They can be organised in two ways, either the closeddoubleton contains the closed point or not. In the latter case X is irreduciblebut has no generic point (and the closed set has two!), so it is not underlyinga scheme. The former is realized as the spectrum of some non-noetherianvaluation rings (for an explicit example, see CA Example ?? or Section 24.5 onpage 415 below. )

452

[ [ [ [

No three-point-space with three closed subsets underlies a scheme. If it has aone closed point, there are two generic points, and if it has a closed doubleton,that doubleton has two generic points. The trivial topology, has three genericpoints, and is of course not a scheme.

Exercise 3.10

a) If X is affine, say X = Spec A, an irreducible Z is of the form V(p) for aprime ideal p which is the unique generic point of Z. In general, if UĎXis an open and affine subset meeting Z, the set U X Z has a generic pointz in U; that is, the closure of tzu in U equals ZXU. Now U X Z is densein Z since Z is irreducible, so that the closure of tzu in X must be equal toZ. If z1 and z2 are two generic points of Z, both must lie in U X Z since itscomplement is a proper closed set, hence they coincide by the affine case.

b) Any scheme having a closed irreducible subset with more than one pointis not Hausdorff. Indeed, if Z = ¯tzu and y P Z is different from z, anyopen neighbourhood of y contains z. In a Hausdorff and sober topologythe irreducible components are the singletons, and if additionally the spaceis quasi-compact, they must be finite in number. It follows that the spacediscrete (and finite); hence the spectrum of an Artinian ring.

c) Given different points x and y in X we are to exhibit an open set containingone of them but not the other. If x R ¯tyu, the open set ¯tyuc contains x but noty. If x P ¯tyu, it holds that y R ¯txu since otherwise we would have ¯txu = ¯tyuand x = y by uniqueness of generic points; hence ¯txuc is an open subset asdesired.

d) In a Quasi-compact set every descending chain tZiu of closed sets has anon-empty intersection, and by Zorn’s lemmas we deduce that there areminimal non-empty closed sets. Such a minimal closed set Z has a uniquegeneric point z, and being minimal Z reduces to tzu.

Exercise 4.5 Starting with a morphism ( f , f 7) : Spec K Ñ X, we let x be theimage of f (Spec K consists of a single point x0, so this is a well-defined pointof X.) The map between stalks is just f 7x : OX,x Ñ OSpec K,x0 = K, which gives amap between residue fields k(x) = OX,x/mx Ñ OSpec K,0 = K. As always withnon-zero maps of fields, this has to be an injection.

Conversely, suppose we are given x P X and k(x) Ñ K. We can define thecorresponding map of topological spaces f : Spec K Ñ X, which takes Spec K tox P X. We also construct a map of structure sheaves f # : OX Ñ f˚OSpec K in thefollowing way: for opens U Ď X not containing x, the map f #

U is the zero map(which it has to be, as f´1(U) = H), while for opens U with x P U, we need

453

maps OX(U)Ñ K. These maps are constructed using the given map k(x)Ñ Kvia the compositions

OX(U) OX,x k(x) K,

and thus we obtain the desired morphism of schemes ( f , f #) : Spec K Ñ X. It isclear that these two constructions are mutually inverse.

Exercise 5.1 A1k minus the origin is naturally identified with the open set

D(x) Ă A1k , and D(x) = Spec k[x, x´1] is affine.

Exercise 5.2 Let X = Spec k[x, y, z, w]/(xw´ yz) and consider the open setU = X´V(x, y) = D(x)YD(y). We have

OX(D(x)) = (k[x, y, z, w]/(xw´ yz))x = k[x, x´1, y, z, w]/(xw´ yz) » k[x, x´1, y, z]

and similarly, OX(D(y)) » k[x, y, y´1, w]. We also have D(x)X D(y) = D(xy),and

OX(D(xy)) = k[x, x´1, y, y´1, z]

The sheaf sequence gives

0 Ñ OX(U)αÝÑ k[x, x´1, y, z]ˆ k[x, y, y´1, w]

βÝÑ k[x, x´1, y, y´1, z]

where β(p(x˘1, y, z), q(x, y˘1, w)) = p(x˘1, y, z)´ q(x, y˘1, x´1yz)). We see thatOX(U), the kernel of β, is identified with elements p in k[x˘1, y˘1, z] which aresimoultaneously polynomials in x˘1, y, z as well as x, y˘1, x´1yz. Hint: view monomials

as elements in Z3.We leave it to

the reader to check that p must be a polynomial in x, y, z, x´1yz, i.e.,

OX(U) » k[x, y, z, x´1yz] » k[x, y, z, w]/(xw´ yz)

Thus shows that OX(X)Ñ OX(U) is an isomorphism, so U is not affine, by theargument of Section 5.2.

Exercise 5.3 Morphisms f : P1k Ñ Spec A are in one-to-one correspondence

with ring maps A Ñ Γ(P1,O) = k. However, each ring map A Ñ k mustcorrespond to the ‘constant map’ Spec k Ñ Spec A.

Exercise 5.8 Answer: Γ(X,OX) = Z[x, y] and X Ñ Spec Γ(X,OX) is exactlythe blow-up morphism p.

Exercise 5.11 With reference to the diagram on page 111 that gives the gluingrecipe for the blow-up, we put u = ts´1; the relation xt = ys defining theblow-up then becomes y = ux. The blow-up will be glued together of thetwo affine planes Spec k[x, u] and Spec k[y, u´1] along the common open subsetSpec k[x, y, u, u´1] = Spec k[x, u, u´1] = Spec k[y, u, u´1]. So in terms of the Ln-terminology x plays the role of s and y that of t; and the glue derives from therelation y = ux and becomes s ÞÑ u´1t. In other words, the result of the gluingwill be L´1. The exceptional divisor E is given by respectively x = 0 and y = 0

454

in the two A2’s; that is respectively by s = 0 and t = 0, which are the equationsfor the zero section.

Exercise 5.14 Using homogeneous coordinates (x : u) on P1k , points (x, y) in

X1 map to (x : 1) and points in X2 to (1 : u). Therefore the image of D(x) isD+(x), and of D(u) it is D+(u).

Assume first that the characteristic of k is not two. The fibre over a point(x : 1) are the points (x, y) where y is a solution of the quadratic equationy2 = (x2g ´ 1)x. There are precisely two, save when the right side vanishes, andthis occurs at (0 : 1) and the 2g points (ξ : 1) with ξ a 2g-th root of unity. Thefibre over (1 : u) in X2 are points (u,v) with v2 = u(1´ u2g). We have alreadyaccounted for all of them except over (1 : u) which reduces to the one point(0, 0). Hence we find all together 2g + 2 “ramification points”, as they are called.

If the characteristic of k equals two, numbers have at most one square root,so in that case all fibres have just one point (but there is of course a multiplicityaround).

Exercise 5.15 Let g(n) = 4n4 ´ 4n3 + 12n2 + 20 and note that g(n) is a perfectsquare if and only if n4 ´ n3 + 3n2 + 5 is (i.e. there is a solution to n4 ´ n3 +

3n2 + 5 = y2). It is easy to show that

(2n2 ´ n + 2)2 ă g(n) ă (2n2 ´ n + 5)2

so the only way that g(n) can be a perfect square is if it equals (2n2 ´ n + 3)2

or (2n2 ´ n + 4)2. Trying both of these possibilities gives that n = 2 is the onlysolution. In that case 24 ´ 23 + 3 ¨ 22 + 5 = 25. Hence X1(Z) = t(2,´5), (2, 5)u.To determine X(Z), we need only find X2(Z)´ X1(Z). But this corresponds toZ-points where u = 0, i.e., (u, v) = (0,˘1). We conclude that X(Z) consists ofthe four points (2,´5), (2, 5) and (0˘ 1).

Exercise 6.3 By Noether normalization, there are finite morphisms X Ñ Am,and Y Ñ An. Then XˆY Ñ Am+n is also finite, which gives the claim.

Exercise 6.6 I) A is not integrally closed because y P Frac(A)zA is integralover A (it satisfies T2 ´ y2 = 0) II) The morphism

f : A2k Ñ A4

k : (x, y) ÞÑ (u = x, v = xy, w = y2, z = y3)

has as image X, the surface X Ă A4k defined by the three equations

u2w = v2, u3z = v3, w3 = z2

III) If we put O = (0, 0) P A2k and O1 = (0, 0, 0, 0) P X Ă A4

k , the morphism frestricts to an isomorphism f0 : A2

kztOu”Ñ XztO1u. Its inverse

f´10 : XztO1u ”Ñ A2

kztOu : (u, v, w, z) ÞÑ (x, y)

is given by:x = uy = v/u if u ‰ 0 or y = z/w if w ‰ 0

455

Exercise 6.7 (Noether normalization)(i) This follows immediately from the statement of the Noether Normaliza-

tion lemma in Commutative Algebra.(ii) Write A = B[z]/(z2 ´ xy), where

B = C[x, y]/(x2y´ xy3 + x2y2 ´ 1)

If we perform a change of variables u = ´x, v = x + y, then the relation herebecomes

x2y´ xy3 + x2y2 ´ 1 = u2(u + v) + u(u + v)3 + u2(u + v)2 ´ 1

= 2u4 + (lower order terms in u)

If we divide by 2 we see that u is integral over C[v]. Hence a Noether normal-ization is given by

Spec A Ñ Spec C[x + y]

Exercise 7.6 (Basic properties)The verifications are entirely functorial only relying on the universal property

and so are valid in any category (where involved products exist).

a) It is totally tautological that the bottom square in diagram

Z

X X

S S

ψψ

=

φX φX

=

(C.1)

is Cartesian; indeed, the upper part shows that any map ψ : Z Ñ X is itsproper lifting to X.

b) The order of X and Y is just apparent and a typographical phenomenon;X and Y enter the formulation of the universal property in a completesymmetric way; hence XˆS Y and XˆS Y are identical, just denoted in twodifferent ways.

c) There are three natural maps from XˆS (YˆS Z) to respectively X, Y andZ; the first, pX, is just the projection onto X, the two next, pY and pZ, arethe projection onto YˆS Z followed respectively by the projections onto Yand Z. For (XS ˆS Y)ˆS Z there are corresponding maps p1X, p1Y and p1Z.We contend that giving an S-map ψ : Z Ñ X ˆS (YˆS Z) is the same asgiving three S-maps ψX,ψY and ψZ from Z to X, Y and Z respectively.Indeed, given ψ, one just composes with the maps pX, pY and pZ; and ithe triple ψX, ψY and ψZ is given, the map

(ψX, (ψy, ψZ)

)is as desired. The

analoguos statement clearly holds for the product (XS ˆS Y)ˆS Z. Hence

456

we obtain maps(

pX, (pY, pZ))

and ((p1X, p1Y), p1Z) between the products andone easily verifies they are mutually inverse maps.

d) On the spot one obtains a map X ˆS Y Ñ X ˆS T, which enters in thecommutative diagram

XˆS Y Y

Z XˆS T T

X S

and using that the lower and the large square both are Cartesian, one inferseasily that the upper square is Cartesian.

Exercise 7.14 (Flat base change)Let tUiu be an affine covering of X. Letting Ui,B = Ui ˆSpec A Spec B and

Uij,B = Uij ˆSpec A Spec B, then tUi,Bu is an open affine covering of XB and oneverifies that Uij,B = Ui,B XUj,B Moreover, the sequences (4.2) on page 97 forX and XB give rise to the commutative diagram below, where β1 and β2 areisomorphisms (check all the details!):

0 Γ(X,OX)bA BÀ

i Γ(Ui,OUi)bA BÀ

i,j Γ(Uij,OUij)bA B

0 Γ(XB,OXB)À

i Γ(Ui,B,OUi,B)À

i,j Γ(Uij,B,OUij,B)

β β1 β2

It follows that we have a map β as desired, and if B is A-flat, the upper sequenceis exact and the Five-lemma shows that β is an isomorphism.

Exercise 7.16 (Answer:)The fibre product equals the empty scheme Spec 0 = H. Geometrically,

this means that Spec(Z/2) Ñ Spec Z and Spec(Z/3) Ñ Spec Z have distinctimages.

Exercise 8.2 That the ι is locally closed immersion follows from Proposition 8.2above and the fact that being a locally closed immersion is preserved underpullback . To see that the diagram is Cartesian, two S-maps ψ : Z Ñ XˆS X andφ : Z Ñ XˆS X so that f ˝ φ = f ˝ ψ are also T-maps, hence the pair induces amap Z Ñ XˆT X.

Exercise 8.3 (Pullback of diagonals)This is in a formality valid in any category with fibre products. Let πX : XˆS

T Ñ X be the projection. Two T-morphisms f and g from a T-scheme intoXT = XˆS T are equal if and only if πX ˝ f = πX ˝ g (this follows from unicitypart of the universal property of products), so the map ρ : XT ˆT XT Ñ XˆS Xthat sends a pair ( f , g) of T-morphisms to the pair (πX ˝ f , πX ˝ g), is injective.

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If (πX ˝ f , πX ˝ g) lies in the image of the diagonal ∆X/S, it holds that πX ˝ f =

πX ˝ g hence f = g, and the pair factors trhrough ∆XT/T. That is, the followingdiagram is Cartesian as required.

XT XT ˆT XT

X XˆS X.

∆XT /T

∆X/S

Exercise 8.4

a) This is entirely a formality: if g, g1 : Z Ñ X are two morphisms such thatΓ f ˝ g1 = Γ f ˝ g one has g1 = πY ˝ Γ f ˝ g1 = πY ˝ Γ f ˝ g = g.

b) It is trivial to verify that the diagram is Cartresian, just think about it.

Exercise 8.9 That the diagram is Cartesian is just the statement that twoT-morphisms f and g are equal independently of wether one regards them asT-morphisms or S-morphisms. All diagonals are separable and pullbacks ofseparable maps are separable according to iii) of Proposition 8.5. If X separable,the diagonal ∆X/S is a closed immersion, hence ι is a closed immersion.

Exercise 8.12 Consider the morphism h : X Ñ YˆS Y induced by the diagram

X

YˆS Y Y

Y S

g

f

We claim that Z = h´1(∆(Y)). Ď. Let x P Z, so that f (x) = g(x) = y. This givesπ1(∆(Y)) = y = f (x) = π1(h(x)) and π2(∆(Y)) = π2(h(x)). Since π1 and π2

are monic, we conclude that ∆(y) = h(x), so that x P h´1 ˚ (∆(Y)).Ě. Let x P h´1(∆(Y)). h(x) = ∆(y) for some y P Y. Hence f (x) =

π1(h(x)) = π1(∆(y)) = y and similarly g(x) = y, so x P Z.Now, since Y is separated, ∆(Y) is closed in Y ˆS Y, so h is continuous,

Z = h´1(∆(Y)) is closed in X.

Exercise 8.13 cacac

Exercise 8.16

a) The salient point in the solution of this exercise, is the diagram

X XˆZ Y Y

X Z

Γ f η

g

f˝g

(C.2)

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where the square is Cartesian and Γ f is the graph of f . It follows that η isa separated, being the pullback of the separated map f ˝ g, and all graphsbeing separated f = η ˝ Γ f is separated.

b) For the example just take Y to be the disjoint union Y = Y1 Y Y2 andf1 : Y1 Ñ Z being separable while f2 : Y2 Ñ X is not.

Exercise 8.16 Hint: Reduce to the affine case, showing that OS Ñ f˚OX issurjective.

Exercise 8.17 There is a general fact that if f : X Ñ Y and g : Y Ñ Z aremorphisms of schemes, and P is a property of morphisms preserved undercomposition and base change, then under the hypothesis that g ˝ f has propertyP and the diagonal of g has property P, we can conclude f has property P.In our case, we can take P to be the property of separatedness, and then forZ = Spec Z, g ˝ f must be the unique morphism X Ñ Spec Z (which is separatedby assumption), and the diagonal of g is separated because it is a locally closedimmersion (this is true of the diagonal of any morphism), so we conclude f isseparated.

Exercise 9.1 One easily checks that q is an ideal. Every x P (Rp)0 is shaped likex = f g´1 with deg f = deg g. When x R q it holds that f R p, and f is invertiblein Rp; it follows that f´1g P (Rp)0, hence q is maximal.

Exercise 9.2 One way is trivial, so let us prove that xy P p implies thatx P p or y P p when x and y are homogeneous. Assume that neither x nor ybelongs to p. By subtracting all homogeneous component belonging to p, wemay assume that no component of x or of y lies in p. But if xn and ym are thecomponents of highest degree of x and y, the product xnym is a component ofxy and consequently lies in p because p is homogeneous. Contradiction.

Exercise 9.6 This follows from the ‘homogenization’ procedure: Since X isintegral, each open affine U = D+( f ) is dense and have the same function field.Since U is the spectrum of an integral domain, we have k(U) = K((R f )0).

Exercise 9.8 The following ring works in all three cases. Let t and xi for i P N0

be variables and let a be the homogeneous ideal in k[t, x0, x1, . . . ] generated byall products xixj and txi. Then Proj R/a is just one point; indeed, D+(t) = Spec ksince inverting t kills all the xi’s, and D+(xi) = H since inverting a nilpotentkills everything.

Exercise 9.9 Let R = A[x0, . . . , xn]. There is a commutative diagram of

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localizations

(Rxi)0 Rxi

(Rxixj)0 Rxixj

(Rxj)0 Rxj

which shows that the morphisms Spec(Rxi)Ñ Spec(Rxi)0 glue to a morphismfrom

Ťni=0 D(xi) = An+1

A ´V(x0, . . . , xn) to PnA =

Ťni=0 D+(xi).

Exercise 9.12 Consider the map of graded A-algebras φ : A[u, v] Ñ R givenby u ÞÑ xt and v ÞÑ yt. It is clearly surjective, and it will suffice to show thatKer φ = a where a = (xv´ yu). The inclusion aĎ Ker φ is clear. Conversely, wecan write, modulo a, any element p of k[x, u, y, v] as

p =ÿ

ai,j,kxiujvk +ÿ

bi1,j1,k1xi1yj1vk1 .

If now p P Ker φ, we have

0 = φ(p) =ÿ

ai,j,kxi+jyktj+k +ÿ

bi1,j1,k1xj1yi1+j1 tk1 .

The monomials in x, y and t being linearly independent over k, the coefficientsai,j,k, bi1,j1,k1 must vanish except possibly when the same monomials appear ineach sum; i.e. when i + j = i1, k = j1 + k1,i = i1 + j1 and j + k = k1, and in whichcase we must have ai,j,k = ´bi1,j1,k. These conditions imply that j = ´j1 whichmust then both be 0, and so i = i1, k = k1. It follows that p = 0 mod a and soKer φ Ď a.

Exercise 9.13 The map α sends the irrelevant ideal R+ = (x, y, z) into the ideal(x, y, wp) whose radical equals (x, y, w) = A+. Hence G(α) = Proj A = P2

k , andwe get a morphism π : P2

k Ñ P(1, 1, p).If k is of characteristic different from p, the best way of thinking about the

map π is by introducing an action of the cyclic group µp of roots of unity. Anelement η P µp acts on P2

k by (x : y : w) ÞÑ (x : y : ηw) which on on the ringlevel is expressed as w ÞÑ ηw. Then clearly k[x, y, wp] is the ring of invariants.

To identify the fibres, we examine π over each of the distinguished opensets. We start with D+(x) which equals Spec (Rx)0 and maps into Spec (Ax)0.On the level of rings this is the map k[yx´1, zx´p] Ñ k[yx´1, wx´1] that sendsz ÞÑ wp; i.e. zx´p ÞÑ (wx´1)p. Simplifying the notation, it presents itself ask[u, v]Ñ k[u, w] with v mapping to wp. On the geometric level it sends (u, v) to(u, vp). The fibre over m = (u´ a, v´ b) will be Spec k[u, w]/(u´ a, wp ´ b). Ifthe characteristic is different from p, and b ‰ 0, the fibre has p distinct pointscorresponding to the p distinct pth-roots of b. If b = 0 or k has characteristic p,the fibre is isomorphic to the non-reduced scheme Spec k[t]/tp.

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Next, let us see what happens in D+(w), which is more interesting. Afterthe coordinates being simplified, the map will be

k[t0, . . . , tn]Ñ k[u, v] ti ÞÑ upívi

If fibre m = (u´ a, v´ b) belongs equalsŞ

ηPµp(u´ ηa, v´ ηb). If (a, b) ‰ (0, 0),

the points (ηa, ηb) are different and fibre has exactly p-point. In case (0, 0) thefibre is reduced to one point, but the scheme-theoretical is highly non-reducedfibres being equal to Spec k[u, v]/(upívi|i = 1, . . . , p).

Exercise 10.7 One can for instance take F = G equal to the constant sheaf onZX on a space X with two connected components.

Exercise 10.12

a) Let Y = A1 = Spec k[x], X = Spec k and f : X Ñ Y the inclusion of a k-pointy. Then f˚OX is a skyscraper sheaf at y, which is certainly not isomorphicto OY. f to be the inclusion of a point x P X; then i˚OX

b) k Ă K induces f : Spec K Ñ Spec k; F = f´1OY is a sheaf satisfyingF (Spec K) = k, where as OX(Spec K) = K.

Exercise 10.13

a) A morphism of sheaves is an isomorphism if and only if it is an isomorphismon stalks. The stalk of ( f´1OY)Y, f (x) and the induced map is the stalk mapOY, f (x) Ñ OX,x.

b) See https://mathoverflow.net/questions/286828/when-is-the-inverse-image-of-the-structure-sheaf-the-structure-sheaf

Exercise 10.14 From its definition, it is straightforward to check that applyingf ˚ commutes with taking tensor products of sheaves. On the level of presheaves,we have

f ˚(G bH) = f´1(G bOY H)b f´1OYOX

= ( f´1G b f´1OYf´1H)b f´1OY

OX

= ( f´1G b f´1OYOX)bOX b( f´1H)b f´1OY

OX)

= f ˚G bOX f ˚H.

and sheafifying, we get an isomorphism of the corresponding sheaves.However, the pushforward f˚ rarely commutes with taking tensor products

of sheaves (we will see several examples of this later). There is however, at least,a map f˚(F )b f˚(G)Ñ f˚(F b G) for OX-modules F ,G: If U Ď Y is an open,and s P f˚(F ), t P f˚(G), then sb t is an element of F b G over f´1(U), andhence sb t defines a section of f˚(F b G) over U.

Exercise 10.15 For every open sets U Ď X, V Ď Y such that f (U) Ď V, wehave a map F (V) Ñ G(U). Note that both terms here are OY(V)-modules(in view of the map OY(V) Ñ OX(U), and the map is a homomorphism ofOY(V)-modules.

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Suppose φ : G Ñ f˚F is an OY-module homomorphism. Then by the adjointproperty of f˚ and f´1 (in the categories ShX and ShY), we get a map f´1G Ñ F ,which is f´1OY-linear. Now F is an OX-module, so we get an OX-linear mapf´1G b f´1OY

OX Ñ G by the universal property of the tensor product. Hencewe obtain a map of OX-modules f ˚G Ñ F .

Conversely, let φ : f ˚G Ñ F be OX-linear. Again by properties of the tensorproduct, there is a map f´1G Ñ f ˚G which is f´1OY-linear. Consequently thereis a f´1OY-linear map f´1G Ñ F . This induces a OY-linear map G Ñ f˚F bythe earlier adjointness property of f˚ and f´1.

Exercise 10.21

HomX( f ˚G,F ) = HomX( f ˚ rN, ĂM)

= HomX((N bA B)„, ĂM)

= HomB(N bA B, M)

= HomA(N, HomA(B, M))

= HomA(N, MA)

= HomY( rN, ĂMB)

= HomY(G, f˚F )

Exercise 10.26 Let ι : U Ñ X denote the inclusion. Let α : OrU Ñ F |U be a map

with G as image. Consider the composition where η is the canonical map from(10.1) on page 193:

θ : OrX ι˚ι˚Or

X ι˚ι˚Fι˚α

The sheaf ι˚ι˚F is quasi-coherent after Theorem 10.28, and hence the image of θ

is coherent, being the image of OrX. And, in fact, it is the wanted sheaf: the map

θ restricts to α on U (and images to images).Consider the restriction to U of the limit lim

ÝÑH of all coherent subsheaves

of F . We contend it equals the limit limÝÑ

M of all coherent subsheaves of F |U ,indeed every term in the latter limit occurs as the restriction of term from theformer.

Exercise 10.32

Proof: The only difference between X and Xred is the structure sheaf, sodefine g on the level of topological spaces by f . On the level of sheaves wefind that (over any open U Ď X) the map f #(U) : OX(U)Ñ f˚OY(U) takes allnilpotents to zero, as Y is reduced. By the universal property of quotients theremust exist a unique morphism of rings g#(U) : OXred(U)Ñ f˚OX(U) such thatf #(U) = g#(U) ˝ r#(U). This gives the required morphism (g, g#) of schemes. o

Exercise 10.33 The condition (i) is clearly necessary. If there is a sequenceY Ñ Z Ñ X, then there is a sequence of sheaves OX Ñ OX/I Ñ f˚(OY), whichmeans that the map OX Ñ f˚(OY) factors through OX/I , and so also (ii) holds.

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Conversely, we define the map g on topological spaces by the inclusion (i).To define it on sheaves, we use the map OX Ñ f˚(OY). This annihilates I , so wethus get a map OX/I Ñ f˚(OY) = g˚(OY). This gives us the map g : Y Ñ Zfactoring f .

Exercise 10.35 Pick a non-empty open affine U1 = Spec A (where A is anintegral domain) and represent F |U 1 as F |U1 = ĂM where M is finitely generatedA- module. By a general principle in commutative algebra there is an f P Aso that M f » Ar

f for some non-negative integer r (it might be zero!!). LetU = Spec A f and ι : U Ñ X the inclusion. Now, it holds that H1 = ι˚F |U = ι˚Or

Uis quasi-coherent by Theorem 10.28 on page 210, and by the adjoint property,as in (10.1) on page 10.1, there are canonical maps α : F Ñ ι˚˚F = H1 andβ : Or

X Ñ ι˚˚OrX = H1. Let H be Im α + Im β. It is quasi-coherent being a

subscheme of the quasi-coherent sheaf H1 and being the quotient of a coherentsheaf it is coherent. Moreover, both α and β are isomorphisms when restricted toU, so their respective kernels and cokernels are supported in the proper closedsubset X Ă U.

Exercise 11.1 We may assume that X = Spec(A), where A is Noetherian,and F = ĂM, where M is a finitely generated A-module. Let x1, . . . , xn begenerators for M as an A-module. We have Ex = Mp, for the prime ideal p Ă Acorresponding to x P X. By assumption, Mp » Ar

p is free, so let m1, . . . , mr be abasis of Mp as an Ap-module. We can write, in Mp:

xi =ÿ

cijmj

Clearing denominators, we see that some multiple dixi (with di P A´ p) is alinear combination of the elements mi with coefficients in A. Let s = d1 ¨ ¨ ¨ dr,and consider the open subset D(s) Ď X. Now, s is invertible in As, so there is asurjective map Ar

s Ñ Ms. This is also injective, since any relation between themi in Ms must survive in Mp (since s R p). Hence Ms » Ar

s. It then follows thatF|D(s) »

ĂMs » OrX|D(s), is free on the open neighbourhood D(s) of x.

Exercise 11.2 Let A =ś8

i=0 Z. We may regard M = Z as an A-module, byembedding it as the 0-th component in

ś8i=0 Z. Thus M is projective, since

Z‘ś8

i=1 Z = A. However, M is not free, since A (and hence any free module)is uncountable. In this example Spec A is an infinite disjoint union of Spec Z’s;F = M restricts to OSpec Z on one of these components and 0 on the others.

Exercise 12.2 Let x1, . . . , xr be degree one generators of R. Let α : R Ñ R1 =Γ˚(OX), be the map above. It is clear that the map is injective: If r P R is anelement so that r/1 = 0 over every (R f )0, then r = 0.

To show integrality, let s P R1 be a homogeneous element of non-negativedegree. By quasi-compactness, we can find an n ą 0, so that α(xn

i )s P α(R) forevery i. Rm is generated by monomials in xi of degree m, so α(Rm)s Ă α(R)for m large (e.g., m ě kn). Let Rěkn be the ideal of R generated by elements ofdegree ě kn. We have that α(Rěkn)s Ă α(Rěkn). Moreover, since R is noetherian,

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Rěrn is finitely generated, so applying the Cayley–Hamilton theorem, we getthat s satisfies an integral equation over R. Hence R1 is integral over R.

Exercise 12.4 Let X = Proj R, Y = Proj R1 and Z = Proj S. Let f P R be ahomogeneous element. Define

Z f =ď

gPR1Spec S f deg gbgdeg f

We claim that there is a natural isomorphism of graded rings

S f 1bg1 Ñ (R f )0 bA (R1g)0

rb r1

( f 1 b g1)s ÞÑrf 1sb

r1

g1s

where f 1 = f deg g and g1 = gdeg f . Indeed, the inverse is given by the map(R f )0 bA (R1g)0 Ñ S f 1bg1 defined by

rf r b

r1

gt ÞÑrt deg g b r1r deg f

( f 1 b g1)rt

Hence we see thatZ f =

ď

gPR1Spec((R f )0 bA (R1g)0)

On the overlaps, we have

Spec((R f )0 bA (R1g)0)X Spec((R f )0bA)R1h)0) = Spec(

S f deg g+deg hb(gh)deg f

)= Spec

((R f )0 bA R1gh

)From this is is clear that

Z f = D+( f )ˆR Y

Moreover, for any other f 1 P R we have Z f 1 = Z f f 1 = X f f 1 ˆR Y. Hence

Z =ď

fPR

Z f = XˆR Y.

Exercise 14.5 (Hints:)

a) Consider D(x) and compute the reduction of R(x).

b) X is covered by U = Spec k[ yx , w

x ]/((wx )

2) and V = Spec k[ xy , z

y ]/((zy )

2). Us-

ing the Cech complex, compute that

dim H0(X,OX) = n + 1.

Exercise 15.3 Using the exact sequence ?? we find that Cl(P) = ZD. Asexplained in xxx, P is isomorphic to the projective cone over the rational normalcurve C Ă Pd of degree d. Let D Ă P correspond to a line ` » P1 passing

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through the vertex. Clearly dD is Cartier, since OP(dD) » OPd(1)|P is invertible.On the other hand we have OPn(1) restricts to OP1(1) on the line `. If D isCartier, then OX(D)|` » OP1(a) for some a P Z. However, this a would have tosatisfy da = 1, which is a contradiction.

Exercise 15.4 (The Picard group of the affine line with two origins) The samereasoning as for P1

k can be applied to the affine line X with two origins. Recallthat X is obtained by gluing U0 = Spec k[s] and U1 = Spec k[t] along U01 =

Spec k[s, s´1] = Spec k[t, t´1] using the identification t = s. So much of thesame argument applies: Given an invertible sheaf L on X, the restriction of itto each U0, U1, U01 must be trivial and over U01 we obtain and automorphismψ : φ1 ˝ φ´1

0 : OU01 Ñ OU01 . Again this is induced by a map k[s, s´1] Ñ k[s, s´1]

which must be of the form p(s, s´1) = sn for some n P Z. As for P1k , the sheaves

we obtain from sn are non-isomorphic (e.g., since they have non-isomorphicΓ(X, L)). So we have Pic(X) = Z.

Exercise 16.1 i) This is a consequence of the Hilbert syzygy theoremii) Any locally free sheaf on An is trivial (the Quillen–Suslin theorem).iii) The glueing condition on Xzto1, o2u (o1, o2 are the two origins) is an

automorphism of the trivial line bundle on A2ztou, hence extends to an auto-morphism on A2 by Hartog’s Lemma. This implies that the initial vector bundleis trivial.

iv) The ideal sheaf Ix of a closed point x P X is not a quotient of a locallyfree sheaf (e.g., since it is not globally generated).

Exercise 17.5

a) Pick an elementř

i aib bi that belongs to I, which means thatř

i aibi = 0.Then

ř

i aib 1(1b bi ´ bib 1) =ř

i aib bi ´ (ř

i aibib 1) =ř

i aib bi.

b) This is trivial: bb 1´ 1b b is a member of I.

c) Since the two B-module-structures coincide, one has

adb+ bda = ab 1(bb 1´1b b)+ 1b b(ab 1´1b a) = abb 1´1b ab = d(ab).

d) Let D : B Ñ M be an A-derivation and define an A-linear map α1 : BbA B ÑM by ab b ÞÑ bD(a).If we give BbA B the B-module structure from the second factor, α1 will beB-linear, and it vanishes on I2; indeed, we have

(ab 1´ 1b a)(bb 1´ 1b b) = abb 1´ ab b´ bb a + 1b ab

and applying α1 to this we obtain

D(ab)´ bD(b)´ aD(b) + abD(1) = 0.

By a) we infer that α1(I2) = 0. The map α1 thus passes to the quotient andgives a map α : I/I2 Ñ M, which satisfies α(d(b)) = α(bb 1´ 1b b) =

1 ¨D(b)´ bD(1) = D(b), and we are through.

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Exercise 17.7 We begin by checking that the choice of representative for bs´1

does not matter. So assume that bs´1 = at´1; that is, uas = ubt for some u P S.Leibniz’ rule gives

asD(u) + uaD(s) + usD(a) = btD(u) + ubD(t) + utD(b).

After having multiplied through by u, we may cancel the terms uasD(u) andubtD(u), and we find the equality

u2(aD(s) + sD(a)) = u2(bD(t) + bD(t)).

Multiplied through by st it becomes

u2(staD(s) + ts2D(a)) = u2(stbD(t) + st2D(b)),

and after a slight reorganizing this gives

u2s2(tD(a)´ aD(t)) = u2t2(sD(b)´ bD(s)).

To see that the map defined by (17.10) abide by Leibniz’ rule, we computeand find

s2t2d(at´1 ¨ bs´1) = D(ab)st´ abD(st) =

= staD(b) + stbD(a)´ absD(t)´ abtD(s) =

= sa(tD(b)´ bD(t)) + tb(sD(a)´ aD(s)),

from which the desired equality follows upon division by s2t2.

Exercise 17.8 The exercise is trivial if k is of characteristic two or three, soassume this is not the case. To see η is non-zero, consider ΩA/k[x]. On one hand,it equals A/(2y)Ady = k[x, y]/(y, y2 ´ x3)dy = k[x]/(x3)dy, and on the otherhand, it lives in the sequence

0 k[x]dx ΩA/k ΩA/k[x] 0

k[x]/(x3)dy

Our torsion element η = 3ydx ´ 2xdy maps to 2xdy in ΩA/k[x] = k[x]/x3dy,which is non-zero, and so must be non-trivial.

In fact, η generated the torsion part of ΩA/k: The torsion part maps injectivelyinto ΩA/k[x] since k[x]dx is torsion free, so the torsion is bounded by k[x]/x3dy.If say ξ is maps to dy, we have dy = ξ´ p(x)dx, with yξ = 0. Then ydy = p(x)dxhence (x3 ´ ypx)dx = 0 impossible since x3 ´ yp(x) can not be zero in A.

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Exercise 17.10 The Kähler differentials live in the diagram with exact rowsand columns

0 0

0 A K H 0

0 A Adx‘ Ady ΩA/k 0

A A

»

α

β

»

where α(a) = ( fxdx + fydy)a and β(adx + bdy) = a fy ´ b fx. The lower left‘hook’ in the diagram is the Koszul complex on the partial derivatives fx and fy

tensorized by A. It follows that H = TorR1 (T, A) where T = R/( fx, fy): indeed,

since f does not have multiple components, the singularities are isolated. Hencefx and fy form a regular sequence, and T is resolved by the Koszul complexbuilt on them. On the other hand, this Tor-module can be computed as thekernel of the ‘multiplication-by- f -map’ in the sequence

R/( fx, fy) R/( fx, fy) R/( f , fx, fy) 0,f

and this kernel is the image in R/( fx, Ry) of the transporter t x | f x P ( fx, fy) u.Finally, since T = R/( fx, fy) is Artinian, multiplication by f in T is injective

if and only if it is an isomorphism; that is, if and only if f is a unit mod ( fx, fy),which is equivalent to ( f , fx, fy) = 1.

Exercise 17.13

a) By induction on the number of generators, it suffices to consider the caseL = K(x). If x is transcendent the dimension ΩL/k increases by one asdoes the transcendence degree, so we may assume that x is algebraic. ThenL = K[t]/(P(t)), let I = (P(t)). There is a diagram

I/I2 ΩK[t]/kbK[t] L ΩL/k 0

ΩK/kbK L

δ

φψ

where the upper row is the conormal sequence and φ is induced by theinclusion KĎK[t]; i.e. it sends dK f b 1 to dK[t] f b 1. It holds that ΩK[t]/k =

ΩK/kdt, hence dimL ΩK[t]/kbK[t] L = dimL ΩK/kbK L + 1. Now, I/I2 is aone-dimensional vector space over L since I is principal. It follows that

dimL ΩL/k ě dimL ΩK[t]/kbK[t] L´ 1 = dimL ΩK/kbK L.

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b) Assume then that dimK ΩK/k = trdeg K/k = r, and let x1, . . . , xr be ele-ments in K so that dx1, . . . , dxr is a basis for ΩK/k over K. It follows thatΩK/k(x1,...,xr) = 0, and hence K is separable and algebraic (e.g. by a)) overk(x1, . . . , xr). But since trdeg k(x1, . . . , xr) = r, the elements x1, . . . , xr aretranscendent over k.

c) Finally, assume that k is perfect. Now δ[P] is the class of dPb 1. Onecomputes dP(x) = P1(x)dx +

ř

i daixi. If P1(x) ‰ 0, it follows that dP ‰ 0. IfP1(x) = 0, the element x is inseparable over K, but is not inseparabel over k,which is perfect, hence not all the coefficients of P(t) lie in k: it follows thatat least one dai ‰ 0, and hence dP ‰ 0.

Exercise 17.17 Since M is finitely generated one may lift a basis for Mbk k andobtain a a sequence

0 E Ar M F 0φ

where φb k is an isomorphism and r dimk MbA k. Nakayama’s lemma yieldsthat φ is sutjecyive and hence F = 0. Tenzored with K sequence C becomes

0 EbA K Kr MbA K 0.

Now dimK MbA K ě r, so EbA K = 0. Since A being an integral domain, istorsion free, it follows that E = 0.

Exercise 17.18 We may assume that X = Spec A. Let K be the fractionfield of A. By Exercise c) it holds that K is separably generated over k (whichis perfect by hypo). Hence dim ΩK/k = trdeg K/k = dim X and in view ofΩK/k = ΩA/kbA K, it holds that ΩX/kbOX K is free of rank dim X. A basisextends to a basis for ΩX/k|U for some open dense subset U. For x P U. we havedim X = dimk(x) ΩX/kb k(x) = trdegk K ď dimk(x) mx/m2

x = dimK(x) ΩK/k. Bydefinition X is smooth at each x P U, and it follows by Theorem 17.27 that x isregular.

Exercise 17.19 (Hint:)Computing the local ring explicitly, shows that it is a domain, and every

local ring with dimension 1 is a DVR hence regular.

Exercise 18.1 If b1, . . . , br is a basis for BbA K(A) ober K(A) one may beadjusting the denominators of the bi’s assume that they belong to B. Considerthe A-linear map φ : Ar Ñ B that sends the i-th standard basis vector to bi. Sinceφb idK(A) is an isomorphism, the cokernel is not of global support, hence thereis an open U over which φ is surjective and we assume that φ is surjective.Choose generators for the kernel of φ and concider the ensuing exact sequence

An Ar B 0ψ φ

The ψbA K(A) = 0 image φ(x) for any x P An vanishes in ArbA K(A) = K(A)r,hence is zero because A is an integral domain.

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Exercise 18.2 By property i) there is basis B for the topology of X consistingof open affine subsets all having P . Let VĎX be an open affine subset. Itcan be covered by members of the basis B each of which in its turn can becovered by finitely many distinguished open subsets of V; hence there is a finitecovering tD( fi)u of V with each D( fi) being contained in some U from B. Now,D( fi)XU = D( fi|U), so if D( fi)ĎU, it holds that D( fi) = D( fi|U), and by i) thedistinguished open D( fi) has P , and finally, since the D( fi)’s cover V, propertyii) gives that U has P .

As to the latter, use induction on the number r of fi’s. Write a1 f1 + ¨ ¨ ¨+

ar fr = 1, and let g = a2 f2 + ¨ ¨ ¨+ ar fr. Then each D( fig) with i ě 2 is dis-tinguished in D( fi) and hence has P by i); on the other hand, they are alsodistinguished in D(g) and cover D(g). Hence D(g) has P by induction, and Ubeing the union of D( f1) and D(g) has P by r = 2 case.

Exercise 18.3 We appeal to the previous exercise and shall verify that f´1(U)

being affine, is a distinguished property. So we have to see that the two re-quirements are fulfilled. Number one is the easiest: if f´1(U) = Spec B ands P Γ(U,OX), it holds true that f´1(D(s)) = B( f 7(s)).

The second requirements is more demanding. Let U = Spec A and lettD(s1), D(s2)u be a finite cover of U by two distinguished opens so that f´1(D(si)) =

Spec Bi. A crucial observation is that f´1(D(s1))X f´1(D(s2)) = f´1(D(s1s2)) =

Spec(B2)s1 . Now consider the “gluing sequence” associated to the coveringt f´1(D(s1)), f´1(D(s2))u and which computes the space B = Γ( f´1(U),OX):

0 B B1 ˆ B2 B12.

It is a sequence of A-modules, and the right hand map sends (b1, b2)Ñ ι1b1 ´

ι2b2 where ιi : Bi Ñ B12 are the localization maps. When being localized in s1,the sequence takes the form

0 Bs1 (B1)s1 ˆ (B2)s1 (B2)s1

and becomes split exact, hence it follows that Bs1 = (B1)s1 = B1. It also holdstrue that B = Γ( f´1(U),OX), hence there is a map X Ñ Spec B, inducing openembedding on Spec Bi. Hence it is an isomorphism.

Exercise 18.6 If g : T Ñ Y is a morphism, the inverse images g´1(Ui) forman open cover of T. Each restriction fT| f´1

T g´1Uibeing the pullback of f along

the g´1(Ui)Ñ Ui Ñ S, is a closed map since f | f´1(Ui)is supposed to be proper.

Hence fT is closed since being closed is local on the target.

Exercise 18.8 If both A[x] = aA[x] and A[x´1] = aA[x´1] hold true, there arerelations

1 = a0 + a1x + ¨ ¨ ¨+ arxr

1 = b0 + b1x´1 + ¨ ¨ ¨+ bsx´s

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where ar ‰ 0 and as ‰ 0. We may assume s ď r and that r is minimal.Multiplying the second relation by arb´1

s xr and subtracting from the first, weobtain a relation of degree less than r; contradiction. As to the second claim,consider the set Σ of local subrings of K dominating A. It is non-empty since Aitself belongs to it and the union of an ascending chain of rings dominating Aclearly dominates A. So there is a maximal one, say R in Σ. If R is not a valuationring there is an x P K with neither x nor x´1 lying in R, by Chevalley’s lemma,either mRR[x] or mRR[x´1] is a proper ideal, say mrR[x]. Then mRR[x]X R = mR

and by localizing we obtain a local domain strictly larger than R that dominatesA.

Exercise 18.9

a) Let u = ab´1 be an element in the intersectionŞ

Ap not lying in A. Leta = t x | xa P (b) u. Then clearly (b)Ď a. Moreover a R (b) since u R A,so that a is not a zero-divisor of A/(b), and hence there is a prime idealp associated to (b) not containing a. It is straightforward that aĎ p. Now,that u P Ap means that u = cd´1 with d R p; that is, ab´1 = cd´1. This givesad = bc, and so d P a contradicting that d R p.

b) If a principal ideal (a)Ď A is prime, the maximal ideal of A(a) will begenerated by a, and hence A(a) being Noetherian is a dvr. So assume that(a) is not a prime ideal and let p be associated to (a); because A Noetherianp = (a : b) for some b R (a). We contend that pAp is generated by ab´1. Itholds true that ba´1pĎ A so that also ba´1mĎ Ap. If ba´1m were containedin m, ba´1pĎ A would be contained in p and the element ba´1 would beintegral over A; hence it would belong to A which it does not. We deducethat ba´1m = Ap; or in other words, m = (ab´1). Thus the maximal ideal ofAp is principal, and since Ap also is Noetherian, it is a dvr.

c) Combine a) and b).

d) Translate c) into geometry.

Exercise 18.10 It holds that AĎ B, and they have a common fraction field K . Ifp is a height one prime ideal in A, the local ring Ap is dvr because A is normal.Let qĎ B be a prime ideal such that qX A = p, and let V be valuation ring inK with BĎV and mV X B = q. Then, since AĎ BĎV and mV X A = p, it holdsthat ApĎV; but dvr’s are maximal rings in their fraction fields so that Ap = V.It follows by Hartog’s therorem that AĎ BĎ

Ş

Ap = A; where the intersectionextends over all primes of height one in A.

Exercise 18.13 Let K be an algebraically closed field of sufficiently heightranscendence degree that it contains all the Ki’s, and choose an embeddingKiĎK for each i. These embeddings give rise to sections σi of the projections

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from A1K ˆk Spec Ki onto A1

K. Perform now the base change

A X

A1Spec K Spec k,

where the resulting A is the disjoint union A =Ť

i A1 ˆk Spec Ki. Choseinfinitely many (for instance countably many, not to run into cardinality issues)closed points xi in A1

K and consider Z = tσi(xi)u in A. It is closed, but maps tothe non-closed set txiu in A1

K, so f is not universally closed.

Exercise 18.14

a) If t(Ui, Zi)uiPI be an ascending chain in Σ and let U =Ť

i Ui and Z =Ť

i Zi.We contend that Z is discrete and closed in U. Indeed, let x P Z; then forsome ν, it holds that x P Zν, and as Zν is discrete, there is an open V in Uν

with V X Zν = txu. But then V X Z = V XUν X Z = V X Zν = txu. Thisshows that Z is discrete. Similarly, if x P UzZ for it will, for some index ν,hold that x P Uν, and Zν being closed in Uν, there is an open neighbourhoodV of x in Uνnot meeting Zν. But now ZXUν = Zν, so V does not meet Zeither. Hence Z is closed in U, and (U, Z) belongs to Σ. By Zorn’s lemmathere is a maximal pair in Σ.

b) For a scheme X, let (U, Z) be a maximal pair as in previous point. If U isa proper subset, pick a point x P X not in U and let V be an open affinecontaining x. Then U XV is a non-empty proper open subset of the affineV, and its complement has a closed point y. The set ZY tyu is then discreteand closed in V YU, contradicting the maximality of (U, Z), and we mayconclude that U = X. Finally, if every such Z is finite, X would be coveredby finitely many Ui’s and hence quasi-compact.

c) Let Z be as in b). The image f (Z) is thus a union of closed points whichis closed because f is universally closed, and as every closed subset of anaffine scheme is quasi-compact, f (Z) must be finite. Hence infinitely manyof the members of Z map to a closed point x in Y, and we let Z1 be the unionof those. From the following commutative diagram and Proposition 18.4above

Z1 X

Spec k(x) Y

ι

f |Z1 f

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we deduce that the restriction f |Z1 : Z1 Ñ Spec k(x) is universally closed,which is absurd in view of the previous exercise (Exercise 18.14).

d) Follows directly from the definition of a quasi-compact map.

Bibliography

Documents

Introduction to Schemes