1 Schemes - Les-Mathematiques.net 1 Schemes It is the purpose of this first chapter to give the necessary introduction to schemes following the functorial approach of [DG]. This approach

1 Schemes

It is the purpose of this first chapter to give the necessary introduction to schemes following the functorial approach of [DG]. This approach appears to be most suitable when dealing with group schemes later on. After trying to motivate the definitions in 1.1, we discuss affine schemes in 1.2-1.6. What is done there is fundamental for the understanding of everything to follow.

As far as arbitrary schemes are concerned, it is most of the time enough to know that they are functors with some properties so that all affine schemes are functors and so that over an algebraically closed field any variety gives rise to a scheme in a canonical way. Sometimes, e.g., when dealing with quotients, it is useful to know more. So we give the appropriate definitions in 1.7-1.9 and mention the comparison with other approaches to schemes and with varieties in 1.1 1. The elementary discussion of a base change in 1.10 is again necessary for many parts later on.

There is also a discussion of closed subfunctors and of closures (1.12-1.14). Finally, we describe the functor of morphisms between two functors (1.15) and prove some of its properties. Again, these results are used only in few places.

A ring or an associative algebra will always be assumed to have a 1, and homomorphisms are assumed to respect this 1. Let k be a fixed commutative ring. Notations of linear algebra (like Hom, 0) without special reference to a ground ring always refer to structures as k-modules. A k-algebra is always

3

4 Representations of Algebraic Groups

assumed to be commutative and associative. (For noncommutative algebras we shall use the terminology: algebras over k.)

1.1 Before giving the definitions, I want to point out how functors arise naturally in algebraic geometry. Assume for the moment that k is an algebraically closed field.

Consider a Zariski closed subset X of some k" and denote by I the ideal of all polynomials f E k[T,, T,, . . . , T,,] with f(X) = 0. Instead of looking at the zeroes of I only over k, we can look also at the zeroes over any k-algebra A, i.e., at %(A) = {x E A" ( f ( x ) = 0 for all f E I } . The map A H %(A) from {k-algebras} to {sets} is a functor: Any homomorphism cp: A -+ A' of k-algebras induces a map cp": A" -+ (A')",

x E A" and f E k[T,,. . . , T,,]. Therefore cp" maps %(A) to %(A'). Denote its restriction by %(cp): %(A) -+ %(A'). For another homomorphism cp': A' -+ A" of k-algebras, one has obviously %(cp')%(cp) = %(cp' o cp), proving that % is indeed a functor.

A regular map from X to a Zariski closed subset Y of some k" is given by rn polynomials f 1 , f 2 , . . ., f, E k[T, , T,, . . ., T,] as f : X -+ Y, x H ( f , ( x ) , f 2 ( x ) , . . . , fm(x)) . The fi define for each k-algebra A a map f(A): A" + A", x H ( f , ( x ) , . . . , f m ( x ) ) . The comorphism f ( k ) * : k [ T , , . . ., T,] -+ k[Tl,. . . , T,] maps the ideal defining Y into the ideal I defining X . This implies that any f(A) maps %(A) into Y(A). The family of all f(A) defines a morphism f :%-+% of functors, i.e., a natural transformation. The more general discussion in 1.3 (cf. 1.3(2)) shows that the map f H f is bijective (from {regular maps X --+ Y } to {natural transformations % -+ Y}).

Taking this for granted, we have embedded the category of all affine algebraic varieties over k into the category of all functors from {k-algebras} to {sets} as a full subcategory. This embedding can be extended to the category of all algebraic varieties, see 1.11.

One advantage of working with functors instead of varieties (i.e., of working with % instead of X ) will be that it gives a natural way to work with "varieties" over other fields and over rings. Furthermore, we get new objects over k (algebraically closed) in a natural way. Instead of working with I , we might have taken any ideal I' c k[T,, . . . , T,,] defining X , i.e., with X = {x E k" I f ( x ) = O for all f~ 1') or, equivalently by Hilbert's Nullstellensatz, with f i = I . Replacing I by I' in the definition of X, we get a functor, say X’, with %'(A) = %(A) for each field extension A 2 k (or even each integral domain), but with %'(A) # %(A) for some A if I # 1'. Such functors arise in a

(a,, a, , . . * 7 a,) H (cp(a,), cp(a,), ' . * 9 with f(cp"X)) = cp(f(x)) for all

Schemes 5

natural way even when we deal with varieties, and they play an important role in representation theory.

Before giving the proper definitions, let us describe the functor % without using the embedding of X into k”. For each k-algebra A, we have a bijection Homk-,,,(k[Tl, T,,. . . , T,], A ) --f A“, sending any a to (a(T1),a(T2), . . . , a(T,)). The inverse image of %(A) consists of those a with 0 = f(a(T1), . . . , a(T,,)) = a ( f ) for all f E I, hence can be identified with Horn,.,,,(k[ Tl, T,, . . . , T,,]/Z, A) . As k[Tl, T,, . . . , T,,]/Z is the algebra k[X] of regular functions on X, we have thus a bijection .%-(A) N Homk.,,,(k[X], A ) . If cp: A + A’ is a homomorphism of k-algebras, then %(q) corresponds to the map Homk~,l,(k[X], A ) +

Hom,,,,(k[X], A’) with u H cp 0 a. A morphism f: X + Y is given by its comorphism f*: k [ Y ] + k[X]. Then the morphism p: 3 + g is given by / ( A ) : Hom,_,,,(k[X], A ) --f Hom,_,,,(k[ Y ] , A) , ct H ct 0 f* for any k-algebra A .

1.2 (&-functors) Let us assume k to be arbitrary again. In the definitions to follow, we shall be rather careless about the foundations of mathematics. Instead of working with ‘‘all’’ k-algebras, we should (as in [DG]) take only those in some universe. We leave the appropriate modifications to the interested reader.

A k-functor is a functor from the category of k-algebras to the category of sets.

Let X be a k-functor. A subfunctor of X is a k-functor Y with Y ( A ) c X(A) and Y(cp) = X ( V ) / ~ ( ~ ) for all k-algebras A, A’ and all cp E Homk-,&i, A’) .

Obviously, a map Y that associates to each k-algebra A a subset Y ( A ) c X(A) is a subfunctor if and only if X(cp)Y(A) c Y(A’) for each homomorphism cp: A + A’ of k-algebras.

For any family (Qor of subfunctors of X, we define their intersection niEr 5 through (flier y ) ( A ) = niEI q ( A ) for each k-algebra A. This is again a subfunctor. The obvious definition of a union is not the useful one, so we shall not denote it by UiEl Y;:.

For any two k-functors X,X’, we denote by Mor(X,X’) the set of all morphisms (i.e., natural transformations) from X to X . For any f E Mor(X,X’) and any subfunctor Y’ of X’, we define the inverse image f - ‘ ( Y ) of Y under f through f - ’ ( Y ) ( A ) = f ( A ) - ’ ( Y ’ ( A ) ) for each k- algebra A. Clearly f - ’ ( Y ’ ) is a subfunctor of X. (The obvious definition of an image of a subfunctor is not the useful one.) Obviously, f- commutes with intersections.

For two k-functors X,, X,, the direct product XI x X, is defined through (X, x X,)(A) = X,(A) x X,(A) for all A. The projections pi:X, x X, 4 Xi


are morphisms and ( X , x X,,p,,p,) has the usual universal property of a direct product.

For three k-functors X , , X,, S and two morphisms f,: X , + S, f, : X 2 + S , the jibre product X , x s X , (relative to f,, f2) is defined through

The projections from X , x s X , to X , and X , are morphisms, and X , x s X , , together with these projections, has the usual universal property of a fibre product. Of course, we may also regard X , x s X 2 as the inverse image of the diagonal subfunctor D, c S x S (with Ds(A) = {(s, s) I s E S ( A ) } for all A ) under the (obvious) morphism ( f 1 , f 2 ) : X , x X , -+ S x S. (On the other hand, inverse images and intersections can also be regarded as special cases of fibre products.)

1.3 (Affine Schemes) For any n E N, the functor A" with A"(A) = A" for all A and A"(q) = q":(a, , . . . ,an) H ( q ( a l ) , . . . , q(a,)) for all cp: A + A' is called the afJine n-space over k. (We also sometimes use the notation A; when it may be doubtful which k we consider.) Note that A' is the functor with Ao(A) = (0) for all A. Hence there is for each k-functor X exactly one morphism from X to Ao (i-e., Ao is a final object in the category of k-functors), and we can regard any direct product as a fibre product over A'.

For any k-algebra R, we can define a k-functor Sp,R through (Sp,R)(A) =

HOmk-alg(R, A ) for all A and (Sp,R)(q): HOmk.alg(R, A ) + HOmk-alg(R, A'), a H q 0 a for all homomorphisms q: A + A'. We call Sp,R the spectrum of R. Any k-functor isomorphic to some Sp,R is called an afJine scheme over k. (Note that the Sp,R generalize the functors 9" considered in 1.1.) For example, the affine n-space A" is isomorphic to Sp,k[T,, . . . , T,] (and will usually be identified with it), where k[T,, ..., T,] is the polynomial ring over k in n vari- ables T,, ...,T,.

We can recover R from Sp,R. This follows from:

Yoneda's Lemma: f H f(R)(id,) is a bijection

For any k-algebra R and any k-functor X , the map

Indeed, take any k-algebra A and any a E Homk-alg(R, A ) = (Sp,R)(A). As f is a natural transformation, we have X ( a ) 0 f (R) = f(A) 0 (Sp,R)(a). Let us ab-

Schemes 7

breviate xs = f(R)(id,). As (SpkR)(a)(idR) = a 0 id, = a, we get

(1) f(A)(a) = X(a) (xs ) .

This shows that f is uniquely determined by x f and indicates how to con- struct an inverse map. For each x E X ( R ) and any k-algebra A, let f , (A): ( S p k R ) + X ( A ) be the map Z H X(r)(x). Then one may check that f , ~ Mor(SpkR, X ) and that x H f, is inverse to f H x f .

An immediate consequence of Yoneda’s lemma is

(2) MOr(SpkR, SpkR’) 1 HOmk-a,g(R’, R )

for any k-algebras R,R‘ . We denote this bijection by f ~ f * and call f * the comorphism corresponding to f. As Homk-,,,(k[T1], R ) r R under a H a( T,) we get especially

(3) Mor(SpkR, A ’ ) 3 R.

For any k-functor X , we denote M o r ( X , A ’ ) by k [ X ] . This set has a natural structure as a k-algebra and (3) is an isomorphism k [ S p k R ] 7 R of k-algebras. (For f i , f 2 E k [ X ] , define f, + f2 through (f, + f , ) ( A ) ( x ) = f , ( A ) ( x ) + f , ( A ) ( x ) for all A and all x E X ( A ) . Similarly, f1f2 and af, for a E k are defined.) We shall usually write f ( x ) = f ( A ) ( x ) for x E X ( A ) and f E k [ X ] . Note that for X = SpkR and f E R N k [ X ] we have f ( x ) = x ( f )

The universal property of the tensor product implies immediately that a direct product X, x X , of affine schemes over k is again an affine scheme over k with k [ X , x X , ] N k [ X , ] 0 k [ X , ] . More generally, a fibre product X , x s X , with X , , X , , S affine schemes is an affine scheme with

for X E (SPkR)(A) = HOmk-alg(R, A ) .

1.4 (Closed Subfunctors of Affine Schemes) Let X be an affine scheme over k . For any subset I c k [ X ] , we define a subfunctor V ( I ) of X through

(1) V ( I ) ( A ) = {x E X ( A ) I f ( x ) = 0 for all f E I }

N { a E Homk-,,,(k[X],A)Ia(I) = o} for all A. (One can check immediately that this is indeed a subfunctor, i.e., that X((p )V(I ) (A) c V ( I ) ( A ’ ) for any homomorphism q: A + A’.)

Of course, V(1) depends only on the ideal generated by I in k [ X ] . We claim:

The map I H V ( I ) f rom {ideals in k [ X ] } to {subfunctors of X } is injective. (2)


More precisely, we claim for two ideals I , I ’ of k [ X ] :

(3) I c I ’ 0 V ( I ) 3 V(Z’).

Of course, the direction “+-” is trivial. On the other hand, consider the canonical map a : k [ X ] + k [ X ] / I ’ , which we regard as an element of X ( k [ X ] / I ’ ) . As a(Z’) = 0, it belongs to V(Z’) (k[X] /Z’) . If V(Z’) c V ( I ) , then a E V ( Z ) ( k [ X ] / I ’ ) and a(I ) = 0, hence I c I ’ .

We call a subfunctor Y of X closed if it is of the form Y = V ( I ) for some ideal I c k [ X ] . Obviously, any closed subfunctor is again an affine scheme over k as

(4) V ( I ) = SPk(kCXI/I). For any family ( I j ) j E of ideals in k [ X ] , one checks easily

Thus the intersection of closed subfunctors is closed again. For each subfunctor Y of X , there is a smallest closed subfunctor F of X

with Y ( A ) c P ( A ) for all A. (Take the intersection of all closed subfunctors with the last property.) This subfunctor 7 is called the closure of Y. We really do not have to assume here that Y is a subfunctor: Any map Y will do that associates to each A a subset Y ( A ) c X ( A ) . We can, for example, fix an A and consider a subset M c X ( A ) . Then the closure M of M is the smallest closed subfunctor of X with M c M ( A ) .

Let I , , I , be ideals in k [ X ] . Because of (3), the closure of the subfunctor A H V ( I , ) ( A ) u V(I , ) (A) is equal to V ( I , n I , ) . If A is an integral domain, then one checks easily that V ( I , ) ( A ) u V(Z,)(A) = V(I1 n I,)(A). For arbitrary A, this equality can be false. Still, we dejne the union as V ( I , ) u V ( I , ) =

VUl n 121.

Let f: X ’ --t X be a morphism of affine schemes over k. One easily checks for any ideal I of k [ X ] that

(6) f - ’ V ( I ) = V ( k [ X ’ ] f * ( I ) ) .

Thus the inverse image of a closed subfunctor is again a closed subfunctor. For any ideal I ’ c k [ X ’ ] , the closure of the subfunctor A H f ( A ) ( V ( I ‘ ) ( A ) ) is V(( f*) - ’ I ’ ) . This functor is also denoted as f(V(Z’)), but we do not want to define f ( V(I’)) here.

For two affine schemes X , , X , over k and ideals I , c k [ X , ] , I , c k [ X , ] , one checks easily

(7) V(I1) x V(I2) N V(I1 0 k [ X , l + k [ X i l @

Schemes 9

If S is another affine scheme and if morphisms X, -, S , X, --+ S are fixed, then one gets

(8) v(zl) xS v ( z 2 ) 21 V(zl @k[S] k[x21 + k[xll @k[S] I 2 ) *

(Use, e.g., that V ( I , ) xs V(Z2) = p i ’ V(Il) n p i ’ V(I,) together with (5 ) , (6), where pi: XI xs X, -, Xi for i = 1,2 are the canonical projections.)

1.5 (Open Subfunctors of Affine Schemes) Let X be an affine scheme over k . A subfunctor Y of X is called open if there is a subset I c k[X] with

Y = D ( I ) where we set for all k-algebras A :

(1) D ( I ) ( A ) = {x E X(A) I 1 A f ( x ) = A ) i e r

= { a E HOmk-,,,(k[X],A)I Aa( l ) = A }

Note that (1) defines for each ideal I a subfunctor: For each cp E Homk.alg(A, A ’ ) and each x E D(I) (A) , one has x f E r A’f(X(cp)x) = E r B r A’cp(f(x)) = A ’ q ( x , , , Af(x)) = A’cp(A) = A’. Obviously:

(2) If A is a j e l d , then D(I ) (A) = U f B r ( x E X(A) (f(x) # 0).

Of course, the right hand side in (2) would be the obvious choice for something open. But it does not define a subfunctor, as homomorphisms between k- algebras are not injective in general. Therefore, we have to take (1) as the appropriate generalization to rings.

For I of the form I = {f} for some f E k[X], one writes X, =

D ( f ) = D({f}) and gets

(3) Xf(A) = {. Homk-alg(k[X1, A ) 1 @ ( f ) E A x 1, hence

(4) xf Spk(k[XIf)

where k[X], = k[X][f-’] is the localization of k[X] at f. So the open subfunctors of the form X, are again affine schemes. For arbitrary I , however, D ( I ) may be no longer an affine scheme.

Obviously, D ( I ) depends only on the ideal of k[X] generated by I . As any proper ideal in any ring is contained in a maximal ideal, we have for any A

D ( I ) ( A ) = { a E Homk-,,,(k[X], A ) I a ( ] ) $ m for any M E Max(A)}

= { a E Homk-,,,(k[X], A ) I a, E D ( I ) ( A / ~ ) for any rn E Max(A)}

where Max(A) is the set of all maximal ideals of A and a, is the composed map k[X] 4 A % A/rn. This shows that D ( I ) is uniquely determined by its


values over fields and especially that D ( I ) = D ( J I ) for any ideal I c k [ X ] . Denote for each prime ideal P c k [ X ] the quotient field of k [ X ] / P by QP and the canonical homomorphism k [ X ] + k [ X ] / P + QP by a p . Then

up 4 D ( I ) ( Q p ) 0 a p ( I ) = 0 0 P 3 I .

As &is the intersection of all prime ideals P I> I of k [ X ] , we get for any two ideals I , I ‘ of k [ X ]

( 5 ) D ( I ) c D(I’ ) o 8 c f i Thus I H D ( I ) is a bijection (ideals I of k[X] with I = J I } + {open subfunctors of X } .

(6 )

and gets especially for any f, f ’ E k [ X ]

(7)

For two ideals I , I ’ in k [ X ] , one checks easily

D ( I ) n D(I’ ) = D(I n 1’ ) = D ( I . 1 ’ )

X, n X,. = X,,,.

For any ideal I in k [ X ] one has

(8) If A is a field, then X ( A ) is the disjoint union of D ( I ) ( A ) and V(I ) (A) ,

For arbitrary A, the union may be smaller. Also, the next statement may be false for arbitrary A : Consider a family ( I j ) j e J of ideals in k [ X ] . Then obviously

(9) If A is a field, then ujEJ D ( I j ) ( A ) = D ( c j E J Ij)(A).

For any morphism f: X ’ + X of affine schemes over k, one has

(10) f - ’ D ( I ) = D ( k [ X ’ ] f * ( I ) )

for any ideal I c k [ X ] , as one may check easily. We get especially for any f ’ E “ 1 (1 1) f - ’ ( x , . ) = X&.

For any fibre product X , xs X , of affine schemes over k (with respect to suitable morphisms) and any ideals I, c k[X,], I, c k [ X , ] , one has

(12) DV,) xs W Z ) = D(I1 Ok(S1IZ).

(Argue as for 1.4(8).)

1.6 (Affine Varieties and Affine Schemes) An affine scheme X is called algebraic if k [ X ] is isomorphic to a k-algebra of the form k[T, , . . . , % ] / I for some n E N

Schemes 11

and a finitely generated ideal I in the polynomial ring k[Tl,. . . ,TI . It is called reduced if k [ X ] does not contain any nilpotent element other than 0.

Assume until the end of Section 1.6 that k is an algebraically closed field. Any affine variety X over k defines as in 1.1 a k-functor 3 which we may identify with S p , [ X ] . One gets in this way exactly all reduced algebraic affine schemes over k . For two affine varieties X , X ‘ , one has M o r ( X , X ‘ ) = Hom,-,,,(k[X’], k [ X ] ) ‘v Mor(X, 3’). So we have indeed embedded the category of affine varieties as a full subcategory into the category of affine schemes.

When doing this, one has to be aware of several points. Any closed subset Y of an affine variety X is itself an affine variety. The functor CiY is the closed subfunctor V ( I ) c 3, where I = { f E k [ X ] I f( Y ) = O}. In this way one gets an embedding {closed subsets of X } + {closed subfunctors of X}. On the level of ideals (cf. 1.4(2)), it corresponds to the inclusion {ideals I of k [ X ] with I = f i } c {ideals of k [ X ] } . The embedding is certainly compatible with inclusions (i.e., Y c Y’ c> Y c CiY ), but in general not with intersections: It may happen that Y n Y’ is strictly larger than the functor corresponding to Y n Y’. Take for example in X = k 2 (where k [ X ] = k[Tl, T,]) the line Y = {(a,O) 1 a E k } and the parabola Y’ = {(a, a’) I a E k } . Then Y n Y’ = {(O,O)]. The ideals I,I of Y, Y’ are I = (T,) and I‘ = ( T t - T,), hence I + I’ =

(Tf, T,) # (TI, T2) and Y n CiY = V ( I ) n V(I’) = V(I + 1’) differs from the subfunctor V(Tl, T2) corresponding to Y n Y’.

So, when regarding affine varieties as (special) affine schemes, we have to be careful, whether intersections are taken as varieties or as schemes. The same is true for inverse images and (more generally) for fibre products.

Similar problems do not arise with open subsets. To any open Y c X we can associate the ideal I = { f ~ k [ X ] I f ( X - Y ) = 0) and then the open subfunctor D ( I ) , which we denote by Y. Because of 1.5(5), the map Y H Y is a bijection from {open subsets of X > to {open subfunctors of z } that is compatible with intersections. It follows from 1.5(10), (12) that this bijection is also compatible with inverse images and fibre products. (In case Y is affine, the notation C?l is compatible with the earlier one.)

1.7 (Open Subfunctors) (Let k again be arbitrary.) Let X be a k-functor. A subfunctor Y c X is called open if for any affine

scheme X ‘ over k and any morphism f: X ’ + X there is an ideal I c k[X ] with f - ‘ (Y) = D ( I ) .

Note that this definition is compatible with the one at the beginning of 1.5 because of 1.5(10). From 1.5(6) one gets

(1) I f Y, Y’ are open subfunctors of X , then so is Y n Y’.


Let f: X’ + X be a morphism of k-functors. Then one has, obviously,

(2)

some morphisms. Then one gets (using Y, xs Y, = p;’ (Y , ) n p i 1 ( Y,))

(3) If Y, c X , and Y, c X, are open subfunctors, then Y, xs Y2 is an open subfunctor of X , xs X , .

If Y is an open subfunctor of X , then f - I ( Y ) is an open subfunctor of X’.

Let X , , X, , S be k-functors and suppose X , xs X , is defined with respect to

Let Y, Y’ be open subfunctors of X . Then

(4) (Of course ‘‘3” is trivial. In order to show e , suppose Y # Y . Then there is some k-algebra A with Y ( A ) # Y’(A) . Assume that there is x E Y ( A ) with x # Y’(A) . Via Y ( A ) N Mor(SpkA, Y ) c Mor(SpkA,X), regard x as a morphism SpkA + X . Then idA E x- ’ (A) (A) , # x - ’ ( Y ) ( A ) , hence x - ’ ( Y ) # x-’(Y‘). Now the result follows from the discussion preceding 1.5(5).)

A family ( y j ) j E of open subfunctors of X is called an open covering of X , if X ( A ) = uje ? ( A ) for each k-algebra A which is a jield.

If X is affine and if 5 = D ( I j ) for some ideal Ij c k [ X ] , then formula 1.5(9) implies that the D ( I j ) form an open covering of X if and only if k [ X ] = cj, I j . A comparison with the case of an affine variety shows that this is the appropriate generalization of the notion of an open covering. Note that especially

( 5 ) Let X be afine and f i , f 2 , ...,f, E k [ X ] . Then the X f i form an open covering of X if and only if k [ X ] = ci= , k [ X l f i .

k-algebras. Then

(6 ) If A’ is a faithfully flat A-module via cp, then

Y = Y o Y ( A ) = Y ’ ( A ) for each k-algebra A that is a jield.

Let Y c X be an open subfunctor, and let cp: A + A‘ be a homomorphism of

Y ( A ) = X ( q ) - ’ Y ( A ’ ) .

We have to prove only ‘‘3”. Suppose at first that X is affine. Then Y = D(Z) for some ideal Z c k [ X ] . Consider some a E X ( A ) = Hom,,,,,(k[X],A) with cp 0 a = X(cp)(.) E Y(A’), i.e., with A’ = A’cp(a(I)). The isomorphism A OA A’ % A’, a 0 a’ H cp(a)a’ induces an isomorphism Aa(I) 0, A’ r A’cp(a(1)). Therefore A’ = A’cp(a(1)) together with the flatness of A‘ implies (A /Aa(I ) ) 0, A’ = 0, hence Aa(I) = A by the faithful flatness.

For arbitrary X , we regard x E X ( A ) as a morphism x: SpkA + X with x(A)(id,) = x , hence with X(cp)x = x(A’)Sp,(cp)(id,). So, if x E X(cp)-’ Y(A’), then id , E Spk(cp)-’(x-’(Y)(A‘)), hence id , E x- ’ (Y) (A) , as x- ’ (Y) is an open subfunctor of the affine scheme Spk(A). Now x = x(A)(id,) E Y ( A ) as desired.

Schemes 13

Of course, ( 6 ) implies that we can restrict to algebraically closed fields in (4). Also, a family ( I.;)je of open subfunctors of X is an open covering of X if and only if X ( A ) = U j , ?(A) for all k-algebras A that are algebraically closed fields.

1.8 (Local Functors) As the notion of an affine scheme generalizes the notion of an affine variety, we want to define the notion of a scheme generalizing the notion of a variety. Certainly a scheme should (by analogy) be a k-functor admitting an open covering by affine schemes. This is, however, not enough.

Consider two k-functors X , Y and an open covering ( I.;)j , of Y. If X , Y correspond to geometric objects, then a morphism f: Y + X ought to be determined by its restrictions jirJ to all I.;. Furthermore, given for each j a morphism fj: I.; -+ X such that filrJ r,, = y J , for all j, j’ E J , then there ought to be a (unique) morphism f: Y -+ X with fir, = fi for all j. In other words, the sequence

I y,

ought to be exact where a( f ) = (fir,)jcJ and p ( ( f j ) j , J ) resp. ~ ( ( f j ) ~ , ~ ) has

For arbitrary X , Y, (I.;), the sequence (1) will not be exact. So we define a k- functor X to be local if the sequence (1) is exact for all k-functors Y and all open coverings ( I.;)jc J . (One can express this as saying that the functor Mor(?, X ) is a sheaf in a suitable sense.)

RJ;. = R, the Sp,(Rf,) form an open covering of the affine scheme SpkR. In this case the sequence (1) takes (because of Yoneda’s lemma) the form

(j,j’)-component fil YJnY,. resp. f j ’ , Y , n Y , . .

For any k-algebra R and any fl,. . . ,f, E R with

where the maps have components of the form X(a) with a one of the canonical maps R + Rji or Rj, + R f i f j . Now one can prove (cf. [DG], I, 51, 4.13).

Proposition: fl, . . . , f , E R with

A k-functor X is local if and only if for any k-algebra R and any RJ;. = R the sequence (2) is exact.

(Note that in [DG] the second property is taken as the definition of “local”.) For R and fl , . . . , f, as in (2) the sequence

(3)


(induced by the natural maps R --+ Rf i and R f i + R f i f j ) is exact. (This is really the description of the structural sheaf on Spec R, e.g., in [Ha], II,2.2.) For an affine scheme X over k the exactness property of Hom,,,,(k[X],?) = X(?) shows that the exactness of (3) implies the exactness of (2). Thus we get

(4) Any ajine scheme over k is a local k-functor.

Consider k-algebras A , , A , , . . , , A , and the projections p j : nl= , Ai + A j . If we apply (2) to R = nl=, A i and the f i = (0,. . . ,0,1,0,. . . ,O), then we get

( 5 ) If X is a local functor, then X ( n l = , Ai) r n;=, X(Ai) for all k- algebras A , , A , , . . . , A , .

(The bijection maps any x to ( X ( p i ) x ) l s i s , , . )

1.9 (Schemes) A k-functor is called a scheme (over k) if it is local and if it admits an open covering by affine schemes.

Obviously, 1.8(4) implies

(1) Any ajine scheme over k is a scheme over k.

The category of schemes over k (a full subcategory of {k-functors}) is closed under important operations:

( 2 ) subfunctor of X, then X ‘ is local (resp. a scheme).

If X is a local k-functor (resp. a scheme over k) and i f X‘ is an open

In the situation of 1.8(1), the injectivity of tl for X implies its injectivity for X ’ . In order to show the exactness for X ’ , one has to show then for any f~ M o r ( Y , X ) such that each f i Y j factors through X ’ , that also f factors through X ’ . The assumption implies r j c f - ’ ( X ’ ) for each j ’ , hence by the definition of an open covering that f - ’ ( X ’ ) ( A ) = Y ( A ) for each k-algebra A that is a field. Then 1.7(4) implies Y c f - ’ ( X ’ ) and f factors through X ’ . In order to get the affine covering of X ’ in case X is a scheme, one can restrict to the case where X is affine, hence X ’ = D ( I ) for some ideal. Then the ( X f ) f E I form an open affine covering.

Let X , , X , , S be k-functors and form X , x s X , with respect to suitable morphisms. Then:

(3) If X , , X,, S are local (resp. schemes), then so is X , xs X,,

The proof may be left to the reader. The most important non-affine schemes are the projective spaces and, more

generally, the Grassmann schemes gr,, for each r , n E N. For any k-algebra A, one sets gren(A) equal to the set of direct summands of the A-module A ‘ + ,

Schemes 15

having rank r. (Then P = 3’,,” is the projective space of dimension n.) In [DG], 1.1.3.9/13, there is a proof that all gr,” are schemes.

1.10 (Base Change) Let k‘ be a k-algebra. Any k’-algebra A is in a natural way also a k-algebra, just by combining the structural homomorphisms k -, k‘ and k’ A. We can therefore associate to each k-functor X a k’-functor Xk! by Xk.(A) = X(A) for any k’-algebra A. For any morphism f :X -, X’ of k- functors, we get a morphism fk‘: Xk, -+ Xi, of k‘-functors simply byfk.(A) = f (A) for any k’-algebra A . In this way we get a functor X H &,, f H fk,

from {k-functors} to {k’-functors}, which we call base change from k to k‘. For any subfunctor Y of a k-functor X, the k’-functor G. is a subfunctor of

xkf. Furthermore, the base change commutes with taking inverse images under morphisms, with taking intersections of subfunctors, and with forming fibre products.

The universal property of the tensor product implies that (SpkR),. = Spk.(R 0 k’) for any k-algebra R. In other words, if X is an affine scheme over k, then XkT is an affine scheme over k‘ with k’[Xk.] N k[X] 0 k‘. For any ideal I of k[X], one gets then V(& = V ( I 0 k’) and D(I)kr = D(I 0 k’). (we really ought to replace 1 0 k‘ in these formulas by its canonical image in k[X] 0 k‘, but for once we shall indulge in some abuse of notation.)

For any k’-algebras A, R one has

(Spk,R)(A) = Homk,-,,g(R,A) C HOmk-alg(R, A ) = (SPkR)k‘(A) .

Thus we have embedded Spkf R as a subfunctor into (SpkR),,. For any ideal I of R, denote the corresponding subfunctors as in 1.4/5 by V ( I ) , D ( I ) c SpkR and l$ ( I ) , Dkf(I) c Spk,R. Then one sees immediately Dkf(I) = (Sp,.R) n D(I)k, and

Using the last results, it is easy to show for any open subfunctor Y of a k- functor X that G. is open in Xk,. If X is a local k-functor, then obviously Xk. is a local k’-functor. Now it is easy to show that Xk! is a scheme over k‘ if X is one over k.

Let k, be a subring of k. We say that a k-functor X is dejinedouer k, if there is a fixed k,-functor X, with X = (X1)k.

h,(I) = (Spk. R) V(I)k,.

1.11 (“Schemes”) In text books on algebraic geometry (like that by Hartshorne to which I shall usually refer in such matters) another notion of scheme is introduced that I shall denote by “schemes” in case a distinction is useful. A “scheme” is a topological space together with a sheaf of k-algebras and an open covering by “affine schemes”. The “affine schemes” are the prime spectra Spec(R) of the k-algebras R endowed with the Zariski topology and a sheaf having sections R, on each Spec(R,) c Spec(R) for all f E R. To each


such “scheme” X one can associate a k-functor 95 via %(A) = Mor(Spec A, X) for all A.

On the other hand, one can associate in a functorial way to each k-functor X a topological space (XI together with a sheaf such that ISp,R( = Spec(R) for each k-algebra R. It turns out that 1x1 is a “scheme” if and only if X is a scheme and that X H 1x1 and X’ H %’ are quasi-inverse equivalences of categories. (This is the content of the comparison theorem [DG], I, §1,4.4.)

One property of this construction is that the open subfunctors of any k- functor X correspond bijectively to the open subsets of 1x1, cf. [DG], I, 41, 4.12. More precisely, if Y is an open subfunctor of X, then I Y I can be identified with an open subset of 1x1 and the k-algebra of sections in I Y I of the structural sheaf of 1x1 is isomorphic to Mor(Y,A’), ibid. 4. 14/15.

Suppose that k is an algebraically closed field. Consider a scheme X over k that has an open covering by algebraic affine schemes. We can define on X(k) a topology such that the open subsets are the Y(k) for open subfunctors Y c X. The map Y H Y(k) turns out to be injective ([DG], I, §3,6.8). We can define a sheaf ox(,) on X(k) through ox(,)( Y(k)) = Mor( Y, A’), Then X H (X(k), Ox,,,) is a faithful functor and its image contains all varieties over kin the usual sense.

There are some fundamental notions of algebraic geometry (like smooth- ness and dimension) that we shall have to consider only in a few places. The necessary definitions and the main properties from the point of view of k- functors are contained in [DG]. I do not want to repeat what is done there in order to keep the length of this book down. Any reader who is familiar with these notions in the context of “schemes” (e.g., from [Ha]) can use the correspondence of X and 1x1 as above to translate. For example, a scheme Xis smooth if and only if 1x1 is so.

1.12 (Closed Subfunctors) A subfunctor Y of a k-functor X is called closed if and only if for each affine functor X’ and any morphism f: X’ --f X of k- functors the subfunctor f -‘(Y) of X’ is closed in the old sense (as in 1.4). Because of 1.4(6), this is compatible with the old definition in case X is affine.

(1) If ( y i ) i e , is a family of closed subfunctors of a k-functor X, then nie, yi is closed in X. (2) Let fi X X‘ be a morphism of k-functors. If Y’ c X‘ is a closed subfunctor, then f -‘(Y’) c X is closed. (3) Let X , , X,, S be k-functors with fixed morphisms X, -+ S and X, -+ S . If Y, c X, and Y, c X2 are closed subfunctors, then Y, xs Y2 c X, xs X, is closed.

The following statements are clear from 1.4(5) or by definition:

Schemes 17

Because of (l), we can define the closure of any subfunctor Y of X as the intersection of all closed subfunctors containing Y. In order to get some deeper results we need

(4) Let X be an afJine scheme and ( X j ) j , J an open covering of X . If Y, Y' are local subfunctors of X with Y n Xj = Y' n X j for all j E J , then Y = Y'.

If X j = D(l j ) for some ideal 'j c k [ X ] , then cjsJlj = k [ X ] , cf. 1.7. We can choose a finite subset J, c J and & E lj for all j E Jo such that k [ X ] = cis J o k [ X ] f , . Then the D(4.) c 5. with j E J , form also an open covering of X (refining the original one). We have also Y n D(&) = Y ' n D(&) for all j E J,, so we may as well assume that J = {I, 2,. . . , r> and X j = X f j for some f j E k [ X ] with k [ X ] =

Consider now x E X ( A ) = Hom,-,,,(k[X], A ) for some k-algebra A ; set f; = x(h) E A and x i E X ( A f ; ) corresponding to the composed homomorphism k [ X ] A A 2 A J i . Now Xi=, k [ X ] L = k [ X ] implies A =

k [ X ] f j .

A f i , so the local property of Y and Y' yields

x E Y ( A ) 0 xi E Y ( A f ; ) for all i

o x i E ( Y n Xi)(AfJ for all i

o x i E (Y' n X i ) ( A f i ) for all i

o x E Y'(A).

In the affine case, any closed subfunctor is again an affine scheme, cf. 1.4(4), hence local, so we can apply (4) to it.

(5) Let subfunctor with Y 3 Xj for all j , then Y = X .

be an open covering of some k-functor X . If Y c X is a closed

Indeed, consider x E X ( R ) for some k-algebra R and let f: Sp ,R + X be the morphism with f (R)( idR) = x, cf. 1.3. We can apply (4) to the closed, hence local, subfunctors f - I ( Y ) and f - ' ( X ) = Sp,(R) of Sp,(R) and the open covering (f -’(xj))j,J. We get f - ' ( Y ) = f -’(X), hence idR E f - ' ( Y ) ( R ) and x E Y(R) .

( 6 ) Any closed subfunctor Y of a local functor X (resp. a scheme X ) is again local (resp. a scheme).

Indeed, consider a morphism f: X ' + X and an open covering ( X j ) j , , of X ' such that each f ix ; factors through Y, i.e., with X j c f - '( Y ) . As f - I ( Y ) is closed, ( 5 ) yields f - ' ( Y ) = X ' , hence f factors through Y. This together with the local property of X implies easily that Y is local. If X is a scheme and if


(Xj)js is an open covering by affine schemes, then ( Y n Xj)js is an open covering by closed subschemes of affine schemes, hence by affine schemes.

The proof of the following statement is left as an exercise:

(7) For any closed subfunctor Y of a k-functor X and any k-algebra k' the subfunctor &, of X,. is closed.

1.13 Lemma: Let X be a local functor and Y c X a local subfunctor of X. Let (Xj)je be an open covering of X. Then Y is closed in X if and only if each Y n Xj is closed in Xj.

Proof: One direction being obvious, let us suppose that each Y n Xj is closed in Xj. For any morphism J X' -+ X with X' affine also, f - ' (Y) z X'x,Y is local, the f - ' ( X j ) are an open covering of X ' , and each f - '( Y) n f -'(Xj) = f - '( Y n Xj) is closed in f - ’(4.). So we may as well assume that X is affine.

As in the proof of 1.12(4), we can assume J = { 1,2,. , . , r } and 4. = X,, for somefi E k[X]. Let I resp. Ij be the kernel of the restriction map k[X] --* k[ Y] resp. k[X] -+ k [ X j n Y]. Then F = V(I ) . As Y n X j is closed, we have Y n Xj = V(Zj),j. The Y n Xj form an open covering of Y. So the restriction induces an injective map k[Y] -+ ns= k[Y n X,], hence I = ns=l 4. We have for all i , j

hence ( I j ) , 8 f , = ( I J J i f j . So for any a E Ii, there is some n with (fih)"a E Ijfor all j , hence with f :a E 4 for all j and thus f :a E I = Zj . This implies ZSi = (Z i ) s i for all i , hence r n X i = Y n X , . Now apply 1.12(4) to the local subfunctors Y and F of r and get Y = F.

1.14 (Closures and Direct Products) Let us assume in this section that k is noetherian (in order to simplify the following definition). A scheme X over k is called algebraic if it admits a finite open covering by affine subschemes which are algebraic in the sense of 1.6. (One can check that this yields the old definition in the affine case.)

Let Y,Z be schemes and X a subscheme of Y such that X and Y are algebraic and such that Z is flat. (This means that Z admits open and affine covering (Z i ) i such that each k[Zi] is a flat k-module.) Then we have in Y x Z

(1)

This follows, e.g., by applying [DG], I, $2, 4.14 to Y' = Y x Z and the pro- jection Y x Z + Y.

x x z = 5? x 2.

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1.15 (Functors of Morphisms) For any k-functors X , Y , we can define a k- functor A o + ( X , Y ) through

(1) &oz(X, Y ) ( A ) = M o r ( X , , Y,)

for any k-algebra A. For any homomorphism cp: A -+A' of k-algebras A G ~ ( X , Y)(cp) maps any morphism f : X A + Y, to the morphism f A , : X A , 'v

( X A ) A f -+ ( YA)As = Y,. using the structure of A' as an A-algebra via cp. The construction of A o 2 ( X , Y ) is clearly functorial: To each morphism

X ' -+ X resp. Y' -+ Y of k-functors there corresponds an obvious morphism A o . c ( X , Y ) -+ A u t ( X ' , Y ) resp. A G 2 ( X , Y ' ) -+ A o h ( X , Y ) . If Y' is a subfunctor of Y, then we shall always regard &02(X, Y') as a subfunctor of &Oh(X, Y ) .

Consider an open covering ( X j ) j , of X and a closed subfunctor Y' of Y. Let pj : A G ~ ( X , Y ) -+ &&(Xj, Y ) be the obvious restriction map. We claim

Of course, one inclusion (’’c") is trivial. Consider on the other hand f E A o Q ( X , Y ) ( A ) = M o r ( X , , YA) for some k-algebra A with p j ( A ) f E

M o r ( X j A , Y a ) for all j E J, i.e., with X j A c f - ' (YA) for all j. Now the ( X j A ) j E J are an open covering of X, and f-’( Y a ) is a closed subfunctor of X,, so 1.12(5) yields f - ' ( Y > ) = X,, hence f E A G ~ ( X , Y)(A) .

If X is an affine scheme, i.e., if there is a k-algebra R with SpkR = X , then A u h ( X , Y ) can also be described as follows: One has for any k-algebra A

Auh(SpkR, Y ) ( A ) Mor((SpkR)A, YA)

2: Mor(Sp,(R 0 A) , Y ) N Y ( R 0 A ) .

Any Y E Y ( R 0 A ) defines by Yoneda's Lemma (1.3) a morphism f,: Spk(R 0 A ) -+ Y mapping any y E Homk-AI,(R 0 A, B) = Spk(R 0 A ) ( B ) to Y(y) (y) . Using the identification as above, we get also a morphism f k : SpkA -+ A&h(SpkR, Y ) . For any k-algebra B, we can regard f ; ( B ) as the map Homk-AI,(A,B) -+ Y ( R 0 B ) with p H Y(idR 0 p)(y). So f b ( B ) is the

with fJR 0 B). We claim: If R is free as a k-module and if Y' c Y is a closed subfunctor,

then fF'Ao@pkR, Y ' ) is closed in SpkA. To start with, we know that f ; ' ( Y ' ) is closed in Spk(R 0 A) , so there is an ideal I' c R 0 A with f;’( Y ' ) = V(1') . The description of f; as above implies (for any k-algebra B )

COmpOSitiOn Of p H id^ 0 p, Homk-A],(A, B ) + Homk-~],(R 0 A , R 0 B )

f;'&oh(SpkR, Y' ) (B) = { p E HOmk.Alg(A, B ) I I' c ker(idR 0 p)} = ( p E Homk-~],(A, B ) 11' c R 0 ker(p))

20 Representatiolrs of Algebraic Groups

using the freeness of R for the last equality. This freeness implies also R 0 n li = n R 0 li for any family (li)i of ideals in A. If we take as the li all ideals with R 0 li 3 I’, then I = n li is the smallest idea of A with I ' c R Q I . Then

I' c R Q ker(p) o I t ker(p) o p E V(I)(B) ,

SO f;'. ,doh(Sp,R, Y ' ) = V ( I ) is closed. As we can apply this to all A and all y , this implies that Jtio.t(Sp,R, Y ' ) is

closed in .,dob(SpkR, Y ) . Using ( 2 ) we get now

( 3 ) Let X , Y be k-functors and Y' c Y a closed subfunctor. If X admits an open covering ( X j ) j E J with afine schemes such that each k [ X j ] is free as a k- module, then .,doh(X, Y ' ) is closed in J t i u ~ ( X , Y ) .

If X is a scheme, then X is called locally free if and only if there is an open covering as above.

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1 Schemes - Les-Mathematiques.net 1 Schemes It is the purpose of this first chapter to give the necessary introduction to schemes following the functorial approach of [DG]. This approach