Chapter TwoCHAPTER 2 the language of schemes, there is no harm in taking an algebraic variety to be a quasi-projective variety; the proofs of the algebraic de Rham theorem are exactly

arX

iv:1

302.

5834

v3 [

mat

h.A

G]

11 J

an 2

014

ch2˙elzein˙tu˙11jan˙edited January 14, 2014 6x9

Chapter Two

From Sheaf Cohomology to the Algebraic de Rham

Theorem

by Fouad El Zein and Loring W. Tu

INTRODUCTION

The concepts of homology and cohomology trace their origin to the work of Poincare inthe late nineteenth century. They attach to a topological space algebraic structures suchas groups or rings that are topological invariants of the space. There are actually manydifferent theories, for example, simplicial, singular, and de Rham theories. In 1931,Georges de Rham proved a conjecture of Poincare on a relationship between cycles andsmooth differential forms, which establishes for a smooth manifold an isomorphismbetween singular cohomology with real coefficients and de Rham cohomology.

More precisely, by integrating smooth forms over singular chains on a smooth man-ifold M , one obtains a linear map

Ak(M) → Sk(M,R)

from the vector spaceAk(M) of smoothk-forms onM to the vector spaceSk(M,R)of real singulark-cochains onM . The theorem of de Rham asserts that this linear mapinduces an isomorphism

H∗dR(M)

∼→ H∗(M,R)

between the de Rham cohomologyH∗dR(M) and the singular cohomologyH∗(M,R),

under which the wedge product of classes of closed smooth differential forms corre-sponds to the cup product of classes of cocycles. Using complex coefficients, there issimilarly an isomorphism

h∗(A•(M,C)

) ∼→ H∗(M,C),

whereh∗(A•(M,C)

)denotes the cohomology of the complexA•(M,C) of smooth

C-valued forms onM .By an algebraic variety, we will mean a reduced separated scheme of finite type

over an algebraically closed field [14, Vol. 2, Ch. VI, Sec. 1.1, p. 49]. In fact, the fieldthroughout the article will be the field of complex numbers. For those not familiar with

http://arxiv.org/abs/1302.5834v3

70


CHAPTER 2

the language of schemes, there is no harm in taking an algebraic variety to be a quasi-projective variety; the proofs of the algebraic de Rham theorem are exactly the same inthe two cases.

Let X be a smooth complex algebraic variety with the Zariski topology. A reg-ular function on an open setU ⊂ X is a rational function that is defined at everypoint of U . A differential k-form on X is algebraic if locally it can be written as∑

fI dgi1 ∧ · · · ∧ dgik for some regular functionsfI , gij . With the complex topol-ogy, the underlying set of the smooth varietyX becomes a complex manifoldXan. Byde Rham’s theorem, the singular cohomologyH∗(Xan,C) can be computed from thecomplex of smoothC-valued differential forms onXan. Grothendieck’s algebraic deRham theorem asserts that the singular cohomologyH∗(Xan,C) can in fact be com-puted from the complexΩ•

algof sheaves of algebraic differential forms onX . Since

algebraic de Rham cohomology can be defined over any field, Grothendieck’s theoremlies at the foundation of Deligne’s theory of absolute Hodgeclasses (see Chapter?? inthis volume).

In spite of its beauty and importance, there does not seem to be an accessible ac-count of Grothendieck’s algebraic de Rham theorem in the literature. Grothendieck’spaper [7], invoking higher direct images of sheaves and a theorem of Grauert–Remmert,is quite difficult to read. An impetus for our work is to give anelementary proof ofGrothendieck’s theorem, elementary in the sense that we useonly tools from standardtextbooks as well as some results from Serre’s groundbreaking FAC and GAGA papers([12] and [13]).

This article is in two parts. In Part I, comprising Sections 1through 6, we proveGrothendieck’s algebraic de Rham theorem more or less from scratch for a smoothcomplex projective varietyX , namely, that there is an isomorphism

H∗(Xan,C) ≃ H∗(X,Ω•

alg)

between the complex singular cohomology ofXan and the hypercohomology of thecomplexΩ•

algof sheaves of algebraic differential forms onX . The proof, relying

mainly on Serre’s GAGA principle and the technique of hypercohomology, necessi-tates a discussion of sheaf cohomology, coherent sheaves, and hypercohomology, andso another goal is to give an introduction to these topics. While Grothendieck’s theoremis valid as a ring isomorphism, to keep the account simple, weprove only a vector spaceisomorphism. In fact, we do not even discuss multiplicativestructures on hypercoho-mology. In Part II, comprising Sections 7 through 10, we develop more machinery,mainly theCech cohomology of a sheaf and theCech cohomology of a complex ofsheaves, as tools for computing hypercohomology. We prove that the general case ofGrothendieck’s theorem is equivalent to the affine case, andthen prove the affine case.

The reason for the two-part structure of our article is the sheer amount of back-ground needed to prove Grothendieck’s algebraic de Rham theorem in general. Itseems desirable to treat the simpler case of a smooth projective variety first, so thatthe reader can see a major landmark before being submerged inyet more machinery. Infact, the projective case is not necessary to the proof of thegeneral case, although thetools developed, such as sheaf cohomology and hypercohomology, are indispensable

THE ALGEBRAIC DE RHAM THEOREM BY F. EL ZEIN AND L. TU


71

to the general proof. A reader who is already familiar with these tools can go directlyto Part II.

Of the many ways to define sheaf cohomology, for example asCech cohomology,as the cohomology of global sections of a certain resolution, or as an example of aright-derived functor in an abelian category, each has its own merit. We have settledon Godement’s approach using his canonical resolution [5, Sec. 4.3, p. 167]. It has theadvantage of being the most direct. Moreover, its extensionto the hypercohomologyof a complex of sheaves gives at once theE2 terms of the standard spectral sequencesconverging to the hypercohomology.

What follows is a more detailed description of each section.In Part I, we recall inSection 1 some of the properties of sheaves. In Section 2, sheaf cohomology is definedas the cohomology of the complex of global sections of Godement’s canonical resolu-tion. In Section 3, the cohomology of a sheaf is generalized to the hypercohomologyof a complex of sheaves. Section 4 defines coherent analytic and algebraic sheavesand summarizes Serre’s GAGA principle for a smooth complex projective variety. Sec-tion 5 proves the holomorphic Poincare lemma and the analytic de Rham theorem forany complex manifold, and Section 6 proves the algebraic de Rham theorem for asmooth complex projective variety.

In Part II, we develop in Sections 7 and 8 theCech cohomology of a sheaf and of acomplex of sheaves. Section 9 reduces the algebraic de Rham theorem for an algebraicvariety to a theorem about affine varieties. Finally, in Section 10 we treat the affinecase.

We are indebted to George Leger for his feedback and to Jeffrey D. Carlson forhelpful discussions and detailed comments on the many drafts of the article. LoringTu is also grateful to the Tufts University Faculty ResearchAward Committee for aNew Directions in Research Award and to the National Center for Theoretical SciencesMathematics Division (Taipei Office) in Taiwan for hosting him during part of thepreparation of this manuscript.

PART I. SHEAF COHOMOLOGY, HYPERCOHOMOLOGY, AND THEPROJECTIVE CASE

2.1 SHEAVES

We assume a basic knowledge of sheaves as in [9, Ch. II, Sec. 1,pp. 60–69].

2.1.1 TheEtale Space of a Presheaf

Associated to a presheafF on a topological spaceX is another topological spaceEF ,called theetale spaceof F . Since the etale space is needed in the construction ofGodement’s canonical resolution of a sheaf, we give a brief discussion here. As a set,the etale spaceEF is the disjoint union

∐p∈X Fp of all the stalks ofF . There is a

natural projection mapπ : EF → X that mapsFp to p. A sectionof the etale spaceπ : EF → X overU ⊂ X is a maps : U → EF such thatπ s = idU , the identity

72


CHAPTER 2

map onU . For any open setU ⊂ X , elements ∈ F(U), and pointp ∈ U , let sp ∈ Fp

be the germ ofs atp. Then the elements ∈ F(U) defines a sections of the etale spaceoverU ,

s : U → EF ,

p 7→ sp ∈ Fp.

The collections(U) | U open inX, s ∈ F(U)

of subsets ofEF satisfies the conditions to be a basis for a topology onEF . Withthis topology, the etale spaceEF becomes a topological space. By construction, thetopological spaceEF is locally homeomorphic toX . For any elements ∈ F(U), thefunctions : U → EF is a continuous section ofEF . A sectiont of the etale spaceEF

is continuous if and only if every pointp ∈ X has a neighborhoodU such thatt = sonU for somes ∈ F(U).

Let F+ be the presheaf that associates to each open subsetU ⊂ X the abeliangroup

F+(U) := continuous sectionst : U → EF.

Under pointwise addition of sections, the presheafF+ is easily seen to be a sheaf,called thesheafificationor theassociated sheafof the presheafF . There is an obviouspresheaf morphismθ : F → F+ that sends a sections ∈ F(U) to the sections ∈F+(U).

EXAMPLE 2.1.1 For each open setU in a topological spaceX , letF(U) be the groupof all constantreal-valued functions onU . At each pointp ∈ X , the stalkFp isR. Theetale spaceEF is thusX ×R, but not with its usual topology. A basis forEF consistsof open sets of the formU × r for an open setU ⊂ X and a numberr ∈ R. Thus,the topology onEF = X × R is the product topology of the given topology onX andthe discrete topology onR. The sheafificationF+ is the sheafR of locally constantreal-valued functions.

EXERCISE 2.1.2 Prove that ifF is a sheaf, thenF ≃ F+. (Hint: the two sheafaxioms say precisely that for every open setU , the mapF(U) → F+(U) is one-to-oneand onto.)

2.1.2 Exact Sequences of Sheaves

From now on, we will consider only sheaves of abelian groups.A sequence of mor-phisms of sheaves of abelian groups

· · · −→ F1 d1−→ F2 d2−→ F3 d3−→ · · ·

on a topological spaceX is said to beexactatFk if Im dk−1 = ker dk; the sequenceis said to beexactif it is exact at everyFk. The exactness of a sequence of morphisms



73

of sheaves onX is equivalent to the exactness of the sequence of stalk maps at everypointp ∈ X (see [9, Exer. 1.2, p. 66]). An exact sequence of sheaves of the form

0 → E → F → G → 0 (2.1.1)

is said to be ashort exact sequence.It is not too difficult to show that the exactness of the sheaf sequence (2.1.1) over a

topological spaceX implies the exactness of the sequence of sections

0 → E(U) → F(U) → G(U) (2.1.2)

for every open setU ⊂ X , but that the last mapF(U) → G(U) need not be surjective.In fact, as we will see in Theorem 2.2.8, the cohomologyH1(U, E) is a measure of thenonsurjectivity of the mapF(U) → G(U) of sections.

Fix an open subsetU of a topological spaceX . To every sheafF of abelian groupson X , we can associate the abelian groupΓ(U,F) := F(U) of sections overU andto every sheaf mapϕ : F → G, the group homomorphismϕU : Γ(U,F) → Γ(U,G).This makesΓ(U, ) a functor from sheaves of abelian groups onX to abelian groups.

A functorF from the category of sheaves of abelian groups onX to the categoryof abelian groups is said to beexactif it maps a short exact sequence of sheaves

0 → E → F → G → 0

to a short exact sequence of abelian groups

0 → F (E) → F (F) → F (G) → 0.

If instead one has only the exactness of

0 → F (E) → F (F) → F (G), (2.1.3)

thenF is said to be aleft-exact functor. The sections functorΓ(U, ) is left exact butnot exact. (By Proposition 2.2.2 and Theorem 2.2.8, the nextterm in the exact sequence(2.1.3) is the first cohomology groupH1(U, E).)

2.1.3 Resolutions

Recall thatR is the sheaf of locally constant functions with values inR andAk isthe sheaf of smoothk-forms on a manifoldM . The exterior derivatived : Ak(U) →Ak+1(U), asU ranges over all open sets inM , defines a morphism of sheavesd : Ak →Ak+1.

PROPOSITION 2.1.3 On any manifoldM of dimensionn, the sequence of sheaves

0 → R → A0 d→ A1 d

→ · · ·d→ An → 0 (2.1.4)

is exact.

74


CHAPTER 2

PROOF. Exactness atA0 is equivalent to the exactness of the sequence of stalk

mapsRp → A0p

d→ A1

p for all p ∈ M . Fix a pointp ∈ M . Suppose[f ] ∈ A0p is

the germ of aC∞ function f : U → R, whereU is a neighborhood ofp, such thatd[f ] = [0] in A1

p. Then there is a neighborhoodV ⊂ U of p on whichdf ≡ 0. Hence,f is locally constant onV and[f ] ∈ Rp. Conversely, if[f ] ∈ Rp, thend[f ] = 0. Thisproves the exactness of the sequence (2.1.4) atA0.

Next, suppose[ω] ∈ Akp is the germ of a smoothk-form ω on some neighborhood

of p such thatd[ω] = 0 ∈ Ak+1p . This means there is a neighborhoodV of p on which

dω ≡ 0. By makingV smaller, we may assume thatV is contractible. By the Poincarelemma [3, Cor. 4.1.1, p. 35],ω is exact onV , sayω = dτ for someτ ∈ Ak−1(V ).Hence,[ω] = d[τ ] in Ak

p. This proves the exactness of the sequence (2.1.4) atAk fork > 0.

In general, an exact sequence of sheaves

0 → A → F0 → F1 → F2 → · · ·

on a topological spaceX is called aresolution of the sheafA. On a complex mani-fold M of complex dimensionn, the analogue of the Poincare lemma is the∂-Poincarelemma [6, p. 25], from which it follows that for each fixed integerp ≥ 0, the sheavesAp,q of smooth(p, q)-forms onM give rise to a resolution of the sheafΩp of holomor-phicp-forms onM :

0 → Ωp → Ap,0 ∂→ Ap,1 ∂

→ · · ·∂→ Ap,n → 0. (2.1.5)

The cohomology of theDolbeault complex

0 → Ap,0(M)∂→ Ap,1(M)

∂→ · · ·

∂→ Ap,n(M) → 0

of smooth(p, q)-forms onM is by definition theDolbeault cohomologyHp,q(M)of the complex manifoldM . (For (p, q)-forms on a complex manifold, see [6] orChapter??.)

2.2 SHEAF COHOMOLOGY

The de Rham cohomologyH∗dR(M) of a smoothn-manifoldM is defined to be the

cohomology of thede Rham complex

0 → A0(M) → A1(M) → A2(M) → · · · → An(M) → 0

of C∞-forms onM . De Rham’s theorem for a smooth manifoldM of dimensionngives an isomorphism between the real singular cohomologyHk(M,R) and the deRham cohomology ofM (see [3, Th. 14.28, p. 175; Th. 15.8, p. 191]). One obtainsthe de Rham complexA•(M) by applying the global sections functorΓ(M, ) to theresolution

0 → R → A0 → A1 → A2 → · · · → An → 0



75

of R, but omitting the initial termΓ(M,R). This suggests that the cohomology of asheafF might be defined as the cohomology of the complex of global sections of acertain resolution ofF . Now every sheaf has a canonical resolution: itsGodementresolution. Using the Godement resolution, we will obtain a well-defined cohomologytheory of sheaves.

2.2.1 Godement’s Canonical Resolution

Let F be a sheaf of abelian groups on a topological spaceX . In Section 2.1.1, wedefined the etale spaceEF of F . By Exercise 2.1.2, for any open setU ⊂ X , the groupF(U) may be interpreted as

F(U) = F+(U) = continuous sections ofπ : EF → X.

Let C0F(U) be the group of all (not necessarily continuous) sections ofthe etale spaceEF overU ; in other words,C0F(U) is the direct product

∏p∈U Fp. In the literature,

C0F is often called the sheaf ofdiscontinuous sectionsof the etale spaceEF of F .ThenF+ ≃ F is a subsheaf ofC0F and there is an exact sequence

0 → F → C0F → Q1 → 0, (2.2.1)

whereQ1 is the quotient sheafC0F/F . Repeating this construction yields exact se-quences

0 → Q1 →C0Q1 → Q2 → 0, (2.2.2)

0 → Q2 →C0Q2 → Q3 → 0, (2.2.3)

· · · .

The short exact sequences (2.2.1) and (2.2.2) can be splicedtogether to form alonger exact sequence

0 // F // C0F //

!! !!

C1F // Q2 // 0

Q1.

==③③③③③③③③

with C1F := C0Q1. Splicing together all the short exact sequences (2.2.1), (2.2.2),(2.2.3),. . . , and definingCkF := C0Qk results in the long exact sequence

0 → F → C0F → C1F → C2F → · · · ,

called theGodement canonical resolutionof F . The sheavesCkF are called theGode-ment sheavesof F . (The letter “C” stands for “canonical.”)

Next we show that the Godement resolutionF → C•F is functorial: a sheaf mapϕ : F → G induces a morphismϕ∗ : C

•F → C•G of their Godement resolutionssatisfying the two functorial properties: preservation ofthe identity and of composition.

76


CHAPTER 2

A sheaf morphism (sheaf map)ϕ : E → F induces a sheaf morphism

C0ϕ : C0E C0F//

∏Ep

∏Fp

and therefore a morphism of quotient sheaves

C0E/E C0F/F ,//

Q1E Q1

F

which in turn induces a sheaf morphism

C1ϕ : C0Q1E C0Q1

F .//

C1E C1F

By induction, we obtainCkϕ : CkE → CkF for all k. It can be checked that eachCk( )is a functor from sheaves to sheaves, called thekth Godement functor.

Moreover, the induced morphismsCkϕ fit into a commutative diagram

0 // E //

C0E //

C1E //

C2E //

· · ·

0 // F // C0F // C1F // C2F // · · · ,

so that collectively(Ckϕ)∞k=0 is a morphism of Godement resolutions.

PROPOSITION 2.2.1 If

0 → E → F → G → 0

is a short exact sequence of sheaves on a topological spaceX and Ck( ) is thekthGodement sheaf functor, then the sequence of sheaves

0 → CkE → CkF → CkG → 0

is exact.

We say that the Godement functorsCk( ) areexact functorsfrom sheaves to sheaves.



77

PROOF. For any pointp ∈ X , the stalkEp is a subgroup of the stalkFp withquotient groupGp = Fp/Ep. InterpretingC0E(U) as the direct product

∏p∈U Ep of

stalks overU , it is easy to verify that for any open setU ⊂ X ,

0 → C0E(U) → C0F(U) → C0G(U) → 0 (2.2.4)

is exact. In general, the direct limit of exact sequences is exact [2, Ch. 2, Exer. 19,p. 33]. Taking the direct limit of (2.2.4) over all neighborhoods of a pointp ∈ X , weobtain the exact sequence of stalks

0 → (C0E)p → (C0F)p → (C0G)p → 0

for all p ∈ X . Thus, the sequence of sheaves

0 → C0E → C0F → C0G → 0

is exact.LetQE be the quotient sheafC0E/E , and similarly forQF andQG . Then there is a

commutative diagram

0

0

0

0 // E

// C0E

// QE

// 0

0 // F

// C0F

// QF

// 0

0 // G

// C0G

// QG

// 0,

0 0 0

(2.2.5)

in which the three rows and the first two columns are exact. It follows by the ninelemma that the last column is also exact.1 TakingC0( ) of the last column, we obtainan exact sequence

0 // C0QE// C0QF

// C0QG// 0.

C1E C1F C1G

The Godement resolution is created by alternately takingC0 and taking quotients.We have shown that each of these two operations preserves exactness. Hence, theproposition follows by induction.

1To prove the nine lemma, view each column as a differential complex. Then the diagram (2.2.5) is ashort exact sequence of complexes. Since the cohomology groups of the first two columns are zero, the longexact cohomology sequence of the short exact sequence implies that the cohomology of the third column isalso zero [15, Th. 25.6, p. 285].

78


CHAPTER 2

2.2.2 Cohomology with Coefficients in a Sheaf

Let F be a sheaf of abelian groups on a topological spaceX . What is so special aboutthe Godement resolution ofF is that it is completely canonical. For any open setUin X , applying the sections functorΓ(U, ) to the Godement resolution ofF gives acomplex

0 → F(U) → C0F(U) → C1F(U) → C2F(U) → · · · . (2.2.6)

In general, thekth cohomologyof a complex

0 → K0 d→ K1 d

→ K2 → · · ·

will be denoted by

hk(K•) :=ker(d : Kk → Kk+1)

Im(d : Kk−1 → Kk).

We sometimes write a complex(K•, d) not as a sequence, but as a direct sumK• =⊕∞

k=0 Kk, with the understanding thatd : Kk → Kk+1 increases the degree by1

andd d = 0. Thecohomology ofU with coefficients in the sheafF , or thesheafcohomology ofF on U , is defined to be the cohomology of the complexC•F(U) =⊕

k≥0 CkF(U) of sections of the Godement resolution ofF (with the initial termF(U)

dropped from the complex (2.2.6)):

Hk(U,F) := hk(C•F(U)

).

PROPOSITION 2.2.2 LetF be a sheaf on a topological spaceX . For any open setU ⊂ X , we haveH0(U,F) = Γ(U,F).

PROOF. If0 → F → C0F → C1F → C2F → · · ·

is the Godement resolution ofF , then by definition

H0(U,F) = ker(d : C0F(U) → C1F(U)

).

In the notation of the preceding subsection,d : C0F(U) → C1F(U) is induced fromthe composition of sheaf maps

C0F ։ Q1 → C1F .

Thus,d : C0F(U) → C1F(U) is the composition of

C0F(U) → Q1(U) → C1F(U).

Note that the second mapQ1(U) → C1F(U) is injective, becauseΓ(U, ) is a left-exactfunctor. Hence,

H0(U,F) = ker(C0F(U) → C1F(U)

)

= ker(C0F(U) → Q1(U)

).



79

But from the exactness of

0 → F(U) → C0F(U) → Q1(U),

we see that

Γ(U,F) = F(U) = ker(C0F(U) → Q1(U)

)= H0(U,F).

2.2.3 Flasque Sheaves

Flasque sheaves are a special kind of sheaf with vanishing higher cohomology. AllGodement sheaves turn out to be flasque sheaves.

DEFINITION 2.2.3 A sheafF of abelian groups on a topological spaceX is flasque(French for “flabby”) if for every open setU ⊂ X , the restriction mapF(X) → F(U)is surjective.

For any sheafF , the Godement sheafC0F is clearly flasque becauseC0F(U) con-sists of all discontinuous sections of the etale spaceEF overU . In the notation of thepreceding subsection,CkF = C0Qk, so all Godement sheavesCkF are flasque.

PROPOSITION 2.2.4 (i) In a short exact sequence of sheaves

0 → Ei→ F

j→ G → 0 (2.2.7)

over a topological spaceX , if E is flasque, then for any open setU ⊂ X , thesequence of abelian groups

0 → E(U) → F(U) → G(U) → 0

is exact.

(ii) If E andF are flasque in(2.2.7), thenG is flasque.

(iii) If0 → E → L0 → L1 → L2 → · · · (2.2.8)

is an exact sequence of flasque sheaves onX , then for any open setU ⊂ X thesequence of abelian groups of sections

0 → E(U) → L0(U) → L1(U) → L2(U) → · · · (2.2.9)

is exact.

PROOF. (i) To simplify the notation, we will usei to denoteiU : E(U) → F(U)for all U ; similarly, j = jU . As noted in Section 2.1.2, the exactness of

0 → E(U)i→ F(U)

j→ G(U) (2.2.10)

80


CHAPTER 2

is true in general, whetherE is flasque or not. To prove the surjectivity ofj fora flasqueE , let g ∈ G(U). SinceF → G is surjective as a sheaf map, all stalkmapsFp → Gp are surjective. Hence, every pointp ∈ U has a neighborhoodUα ⊂ U on which there exists a sectionfα ∈ F(Uα) such thatj(fα) = g|Uα

.

Let V be the largest union⋃

α Uα on which there is a sectionfV ∈ F(V ) suchthat j(fV ) = g|V . We claim thatV = U . If not, then there are a setUα notcontained inV andfα ∈ F(Uα) such thatj(fα) = g|Uα

. OnV ∩ Uα, writing jfor jV ∩Uα

, we havej(fV − fα) = 0.

By the exactness of the sequence (2.2.10) atF(V ∩ Uα),

fV − fα = i(eV,α) for someeV,α ∈ E(V ∩ Uα).

SinceE is flasque, one can find a sectioneU ∈ E(U) such thateU |V ∩Uα= eV,α.

OnV ∩ Uα,fV = i(eV,α) + fα.

If we modify fα tofα = i(eU ) + fα onUα,

thenfV = fα onV ∩ Uα, andj(fα) = g|Uα. By the gluing axiom for the sheaf

F , the elementsfV andfα piece together to give an elementf ∈ F(V ∪ Uα)such thatj(f) = g|V ∪Uα

. This contradicts the maximality ofV . Hence,V = Uandj : F(U) → G(U) is onto.

(ii) SinceE is flasque, for any open setU ⊂ X the rows of the commutative diagram

0 // E(X)

α

// F(X)

β

jX // G(X)

γ

// 0

0 // E(U) // F(U)jU // G(U) // 0

are exact by (i), whereα, β, andγ are the restriction maps. SinceF is flasque,the mapβ : F(X) → F(U) is surjective. Hence,

jU β = γ jX : F(X) → G(X) → G(U)

is surjective. Therefore,γ : G(X) → G(U) is surjective. This proves thatG isflasque.

(iii) The long exact sequence (2.2.8) is equivalent to a collection of short exact se-quences

0 → E →L0 → Q0 → 0, (2.2.11)

0 → Q0 →L1 → Q1 → 0, (2.2.12)

· · · .



81

In (2.2.11), the first two sheaves are flasque, soQ0 is flasque by (ii). Similarly,in (2.2.12), the first two sheaves are flasque, soQ1 is flasque. By induction, allthe sheavesQk are flasque.

By (i), the functorΓ(U, ) transforms the short exact sequences of sheaves intoshort exact sequences of abelian groups

0 → E(U) →L0(U) → Q0(U) → 0,

0 → Q0(U) →L1(U) → Q1(U) → 0,

· · · .

These short exact sequences splice together into the long exact sequence (2.2.9).

COROLLARY 2.2.5 Let E be a flasque sheaf on a topological spaceX . For everyopen setU ⊂ X and everyk > 0, the cohomologyHk(U, E) = 0.

PROOF. Let0 → E → C0E → C1E → C2E → · · ·

be the Godement resolution ofE . It is an exact sequence of flasque sheaves. By Propo-sition 2.2.4(iii), the sequence of groups of sections

0 → E(U) → C0E(U) → C1E(U) → C2E(U) → · · ·

is exact. It follows from the definition of sheaf cohomology that

Hk(U, E) =

E(U) for k = 0,

0 for k > 0.

A sheafF on a topological spaceX is said to beacycliconU ⊂ X if Hk(U,F) =0 for all k > 0. Thus, a flasque sheaf onX is acyclic on every open set ofX .

EXAMPLE 2.2.6 Let X be an irreducible complex algebraic variety with the Zariskitopology. Recall that the constant sheafC over X is the sheaf of locally constantfunctions onX with values inC. Because any two open sets in the Zariski topology ofX have a nonempty intersection, the only continuous sectionsof the constant sheafCover any open setU are the constant functions. Hence,C is flasque. By Corollary 2.2.5,Hk(X,C) = 0 for all k > 0.

COROLLARY 2.2.7 Let U be an open subset of a topological spaceX . ThekthGodement sections functorΓ(U, Ck( )), which assigns to a sheafF on X the groupΓ(U, CkF) of sections ofCkF overU , is an exact functor from sheaves onX to abeliangroups.

82


CHAPTER 2

PROOF. Let0 → E → F → G → 0

be an exact sequence of sheaves. By Proposition 2.2.1, for any k ≥ 0,

0 → CkE → CkF → CkG → 0

is an exact sequence of sheaves. SinceCkE is flasque, by Proposition 2.2.4(i),

0 → Γ(U, CkE) → Γ(U, CkF) → Γ(U, CkG) → 0

is an exact sequence of abelian groups. Hence,Γ(U, Ck( )

)is an exact functor from

sheaves to groups.

Although we do not need it, the following theorem is a fundamental property ofsheaf cohomology.

THEOREM 2.2.8 A short exact sequence

0 → E → F → G → 0

of sheaves of abelian groups on a topological spaceX induces a long exact sequencein sheaf cohomology,

· · · → Hk(X, E) → Hk(X,F) → Hk(X,G) → Hk+1(X, E) → · · · .

PROOF. Because the Godement sections functorΓ(X, Ck( )

)is exact, from the

given short exact sequence of sheaves one obtains a short exact sequence of complexesof global sections of Godement sheaves

0 → C•E(X) → C•F(X) → C•G(X) → 0.

The long exact sequence in cohomology [15, Sec. 25] associated to this short exactsequence of complexes is the desired long exact sequence in sheaf cohomology.

2.2.4 Cohomology Sheaves and Exact Functors

As before, a sheaf will mean a sheaf of abelian groups on a topological spaceX . Acomplex of sheavesL• onX is a sequence of sheaves

0 → L0 d→ L1 d

→ L2 d→ · · ·

onX such thatd d = 0. Denote the kernel and image sheaves ofL• by

Zk := Zk(L•) := ker(d : Lk → Lk+1

),

Bk := Bk(L•) := Im(d : Lk−1 → Lk

).

Then thecohomology sheafHk := Hk(L•) of the complexL• is the quotient sheaf

Hk := Zk/Bk.



83

For example, by the Poincare lemma, the complex

0 → A0 → A1 → A2 → · · ·

of sheaves ofC∞-forms on a manifoldM has cohomology sheaves

Hk = Hk(A•) =

R for k = 0,

0 for k > 0.

PROPOSITION 2.2.9 LetL• be a complex of sheaves on a topological spaceX . Thestalk of its cohomology sheafHk at a pointp is thekth cohomology of the complexL•

p

of stalks.

PROOF. Since

Zkp = ker

(dp : L

kp → Lk+1

p

)and Bk

p = Im(dp : L

k−1p → Lk

p

)

(see [9, Ch. II, Exer. 1.2(a), p. 66]), one can also compute the stalk of the cohomologysheafHk by computing

Hkp = (Zk/Bk)p = Zk

p /Bkp = hk(L•

p),

the cohomology of the sequence of stalk maps ofL• atp.

Recall that amorphismϕ : F• → G• of complexes of sheaves is a collection ofsheaf mapsϕk : Fk → Gk such thatϕk+1

d = d ϕk for all k. A morphismϕ : F• → G• of complexes of sheaves induces morphismsϕk : Hk(F•) → Hk(G•)of cohomology sheaves. The morphismϕ : F• → G• of complexes of sheaves iscalled aquasi-isomorphismif the induced morphismsϕk : Hk(F•) → Hk(G•) ofcohomology sheaves are isomorphisms for allk.

PROPOSITION 2.2.10 Let L• =⊕

k≥0 Lk be a complex of sheaves on a topolog-

ical spaceX . If T is an exact functor from sheaves onX to abelian groups, then itcommutes with cohomology:

T(Hk(L•)

)= hk

(T (L•)

).

PROOF. We first prove thatT commutes with cocycles and coboundaries. Apply-ing the exact functorT to the exact sequence

0 → Zk → Lk d→ Lk+1

results in the exact sequence

0 → T (Zk) → T (Lk)d→ T (Lk+1),

which proves that

Zk(T (L•)

):= ker

(T (Lk)

d→ T (Lk+1)

)= T (Zk).

84


CHAPTER 2

(By abuse of notation, we write the differential ofT (L•) also asd, instead ofT (d).)The differentiald : Lk−1 → Lk factors into a surjectionLk−1

։ Bk followed byan injectionBk → Lk:

Lk−1 d //

"" ""

Lk.

Bk.

==⑤⑤⑤⑤⑤⑤⑤⑤

Since an exact functor preserves surjectivity and injectivity, applyingT to the diagramabove yields a commutative diagram

T (Lk−1)d //

$$ $$

T (Lk),

T (Bk)::

::

which proves that

Bk(T (L•)

):= Im

(T (Lk−1)

d→ T (Lk)

)= T (Bk).

Applying the exact functorT to the exact sequence of sheaves

0 → Bk → Zk → Hk → 0

gives the exact sequence of abelian groups

0 → T (Bk) → T (Zk) → T (Hk) → 0.

Hence,

T(Hk(L•)

)= T (Hk) =

T (Zk)

T (Bk)=

Zk(T (L•)

)

Bk(T (L•)

) = hk(T (L•)

).

2.2.5 Fine Sheaves

We have seen that flasque sheaves on a topological spaceX are acyclic on any opensubset ofX . Fine sheaves constitute another important class of such sheaves.

A sheaf mapf : F → G over a topological spaceX induces at each pointx ∈ Xa group homomorphismfx : Fx → Gx of stalks. Thesupportof the sheaf morphismfis defined to be

supp f = x ∈ X | fx 6= 0.

If two sheaf maps over a topological spaceX agree at a point, then they agree in aneighborhood of that point, so the set where two sheaf maps agree is open inX . Sincethe complementX − supp f is the subset ofX where the sheaf mapf agrees with thezero sheaf map, it is open and thereforesupp f is closed.



85

DEFINITION 2.2.11 Let F be a sheaf of abelian groups on a topological spaceXandUα a locally finite open cover ofX . A partition of unity of F subordinate toUα is a collectionηα : F → F of sheaf maps such that

(i) supp ηα ⊂ Uα;

(ii) for each pointx ∈ X , the sum∑

ηα,x = idFx, the identity map on the stalkFx.

Note that althoughα may range over an infinite index set, the sum in (ii) is a finitesum, becausex has a neighborhood that meets only finitely many of theUα’s andsupp ηα ⊂ Uα.

DEFINITION 2.2.12 A sheafF on a topological spaceX is said to befine if for everylocally finite open coverUα ofX , the sheafF admits a partition of unity subordinateto Uα.

PROPOSITION 2.2.13 The sheafAk of smoothk-forms on a manifoldM is a finesheaf onM .

PROOF. LetUα be a locally finite open cover ofM . Then there is aC∞ partitionof unity ρα onM subordinate toUα [15, App. C, p. 346]. (This partition of unityρα is a collection of smoothR-valued functions, not sheaf maps.) For any open setU ⊂ M , defineηα,U : Ak(U) → Ak(U) by

ηα,U (ω) = ραω.

If x /∈ Uα, thenx has a neighborhoodU disjoint fromsupp ρα. Hence,ρα vanishesidentically onU andηα,U = 0, so that the stalk mapηα,x : Ak

x → Akx is the zero map.

This proves thatsupp ηα ⊂ Uα.For anyx ∈ M , the stalk mapηα,x is multiplication by the germ ofρα, so

∑α ηα,x

is the identity map on the stalkAkx. Hence,ηα is a partition of unity of the sheafAk

subordinate toUα.

Let R be a sheaf of commutative rings on a topological spaceX . A sheafF ofabelian groups onX is called asheaf ofR-modules(or simply anR-module) if forevery open setU ⊂ X , the abelian groupF(U) has anR(U)-module structure andmoreover, for allV ⊂ U , the restriction mapF(U) → F(V ) is compatible with themodule structure in the sense that the diagram

R(U) ×F(U)

// F(U)

R(V )×F(V ) // F(V )

commutes.A morphismϕ : F → G of sheaves ofR-modules overX is a sheaf morphism

such that for each open setU ⊂ X , the group homomorphismϕU : F(U) → G(U) isanR(U)-module homomorphism.

86


CHAPTER 2

If A0 is the sheaf ofC∞ functions on a manifoldM , then the sheafAk of smoothk-forms onM is a sheaf ofA0-modules. By a proof analogous to that of Proposi-tion 2.2.13, any sheaf ofA0-modules over a manifold is a fine sheaf. In particular, thesheavesAp,q of smooth(p, q)-forms on a complex manifold are all fine sheaves.

2.2.6 Cohomology with Coefficients in a Fine Sheaf

A topological spaceX is paracompactif every open cover ofX admits a locally finiteopen refinement. In working with fine sheaves, one usually hasto assume that thetopological space is paracompact, in order to be assured of the existence of a locallyfinite open cover. A common and important class of paracompact spaces is the class oftopological manifolds [16, Lem. 1.9, p. 9].

A fine sheaf is generally not flasque. For example,f(x) = secx is aC∞ functionon the open intervalU = ] − π/2, π/2[ that cannot be extended to aC∞ function onR. This shows thatA0(R) → A0(U) is not surjective. Thus, the sheafA0 of C∞

functions is a fine sheaf that is not flasque.While flasque sheaves are useful for defining cohomology, finesheaves are more

prevalent in differential topology. Although fine sheaves need not be flasque, theyshare many of the properties of flasque sheaves. For example,on a manifold, Proposi-tion 2.2.4 and Corollary 2.2.5 remain true if the sheafE is fine instead of flasque.

PROPOSITION 2.2.14 (i) In a short exact sequence of sheaves

0 → E → F → G → 0 (2.2.13)

of abelian groups over a paracompact spaceX , if E is fine, then the sequence ofabelian groups of global sections

0 → E(X)i→ F(X)

j→ G(X) → 0

is exact.In (ii) and (iii), assume that every open subset ofX is paracompact (a manifold is anexample of such a spaceX).

(ii) If E is fine andF is flasque in(2.2.13), thenG is flasque.

(iii) If0 → E → L0 → L1 → L2 → · · ·

is an exact sequence of sheaves onX in whichE is fine and all theLk are flasque,then for any open setU ⊂ X , the sequence of abelian groups

0 → E(U) → L0(U) → L1(U) → L2(U) → · · ·

is exact.

PROOF. To simplify the notation,iU : E(U) → F(U) will generally be denoted byi. Similarly, “fα onUαβ” will mean fα|Uαβ

. As in Proposition 2.2.4(i), it suffices to



87

show that ifE is a fine sheaf, thenj : F(X) → G(X) is surjective. Letg ∈ G(X).SinceFp → Gp is surjective for allp ∈ X , there exist an open coverUα of X andelementsfα ∈ F(Uα) such thatj(fα) = g|Uα

. By the paracompactness ofX , we mayassume that the open coverUα is locally finite. OnUαβ := Uα ∩ Uβ ,

j(fα|Uαβ− fβ|Uαβ

) = j(fα)|Uαβ− j(fβ)|Uαβ

= g|Uαβ− g|Uαβ

= 0.

By the exactness of the sequence

0 → E(Uαβ)i→ F(Uαβ)

j→ G(Uαβ),

there is an elementeαβ ∈ E(Uαβ) such that onUαβ ,

fα − fβ = i(eαβ).

Note that on the triple intersectionUαβγ := Uα ∩ Uβ ∩ Uγ , we have

i(eαβ + eβγ) = fα − fβ + fβ − fγ = i(eαγ).

SinceE is a fine sheaf, it admits a partition of unityηα subordinate toUα. Wewill now view an element ofE(U) for any open setU as a continuous section of theetale spaceEE overU . Then the sectionηγ(eαγ) ∈ E(Uαγ) can be extended by zeroto a continuous section ofEE overUα:

ηγeαγ(p) =

(ηγeαγ)(p) for p ∈ Uαγ ,

0 for p ∈ Uα − Uαγ .

(Proof of the continuity ofηγeαγ : OnUαγ , ηγeαγ is continuous. Ifp ∈ Uα−Uαγ , thenp /∈ Uγ , sop /∈ supp ηγ . Sincesupp ηγ is closed, there is an open setV containingp such thatV ∩ supp ηγ = ∅. Thus,ηγeαγ = 0 on V , which proves thatηγeαγ iscontinuous atp.)

To simplify the notation, we will omit the overbar and writeηγeαγ ∈ E(Uα) alsofor the extension by zero ofηγeαγ ∈ E(Uαγ). Let eα be the locally finite sum

eα =∑

γ

ηγeαγ ∈ E(Uα).

On the intersectionUαβ,

i(eα − eβ) = i(∑

γ

ηγeαγ −∑

γ

ηγeβγ

)= i(∑

γ

ηγ(eαγ − eβγ))

= i(∑

γ

ηγeαβ

)= i(eαβ) = fα − fβ .

Hence, onUαβ,fα − i(eα) = fβ − i(eβ).

88


CHAPTER 2

By the gluing sheaf axiom for the sheafF , there is an elementf ∈ F(X) such thatf |Uα

= fα − i(eα). Then

j(f)|Uα= j(fα) = g|Uα

for all α.

By the uniqueness sheaf axiom for the sheafG, we havej(f) = g ∈ G(X). This provesthe surjectivity ofj : F(X) → G(X).(ii), (iii) Assuming that every open subsetU of X is paracompact, we can apply (i) toU . Then the proofs of (ii) and (iii) are the same as in Proposition 2.2.4(ii), (iii).

The analogue of Corollary 2.2.5 forE a fine sheaf then follows as before. Theupshot is the following theorem.

THEOREM 2.2.15 LetX be a topological space in which every open subset is para-compact. Then a fine sheaf onX is acyclic on every open subsetU .

REMARK 2.2.16 Sheaf cohomology can be characterized uniquely by a set of axioms[16, Def. 5.18, pp. 176–177]. Both the sheaf cohomology in terms of Godement’s reso-lution and theCech cohomology of a paracompact Hausdorff space satisfy these axioms[16, pp. 200–204], so at least on a paracompact Hausdorff space, sheaf cohomology isisomorphic toCech cohomology. Since theCech cohomology of a triangularizablespace with coefficients in the constant sheafZ is isomorphic to its singular cohomol-ogy with integer coefficients [3, Th. 15.8, p. 191], the sheafcohomologyHk(M,Z) ofa manifoldM is isomorphic to the singular cohomologyHk(M,Z). In fact, the sameargument shows that one may replaceZ byR or byC.

2.3 COHERENT SHEAVES AND SERRE’S GAGA PRINCIPLE

Given two sheavesF andG onX , it is easy to show that the presheafU 7→ F(U) ⊕G(U) is a sheaf, called thedirect sumof F andG and denoted byF ⊕ G. We writethe direct sum ofp copies ofF asF⊕p. If U is an open subset ofX , therestrictionF|U of the sheafF to U is the sheaf onU defined by(F|U )(V ) = F(V ) for everyopen subsetV of U . LetR be a sheaf of commutative rings on a topological spaceX .A sheafF of R-modules onM is locally free of rankp if every pointx ∈ M has aneighborhoodU on which there is a sheaf isomorphismF|U ≃ R|⊕p

U .Given a complex manifoldM , letOM be its sheaf of holomorphic functions. When

understood from the context, the subscriptM is usually suppressed andOM is simplywritten O. A sheaf ofO-modules on a complex manifold is also called ananalyticsheaf.

EXAMPLE 2.3.1 On a complex manifoldM of complex dimensionn, the sheafΩk

of holomorphick-forms is an analytic sheaf. It is locally free of rank(nk

), with local

framedzi1 ∧ · · · ∧ dzik for 1 ≤ i1 < · · · < ik ≤ n.

EXAMPLE 2.3.2 The sheafO∗ of nowhere-vanishing holomorphic functions withpointwise multiplication on a complex manifoldM is notan analytic sheaf, since mul-tiplying a nowhere-vanishing functionf ∈ O∗(U) by the zero function0 ∈ O(U) willresult in a function not inO∗(U).



89

Let R be a sheaf of commutative rings on a topological spaceX , letF be a sheafof R-modules onX , and letf1, . . . , fn be sections ofF over an open setU in X . Foranyr1, . . . , rn ∈ R(U), the map

R⊕n(U) → F(U),

(r1, . . . , rn) 7→∑

rifi

defines a sheaf mapϕ : R⊕n|U → F|U overU . The kernel ofϕ is a subsheaf of(R|U )

⊕n called thesheaf of relationsamongf1, . . . , fn, denoted byS(f1, . . . , fn).We say thatF|U is generated byf1, . . . , fn if ϕ : R⊕n → F is surjective overU .

A sheafF of R-modules onX is said to beof finite type if every x ∈ X hasa neighborhoodU on whichF is generated by finitely many sectionsf1, . . . , fn ∈F(U). In particular, then, for everyy ∈ U , the valuesf1,y, . . . , fn,y ∈ Fy generate thestalkFy as anRy-module.

DEFINITION 2.3.3 A sheafF ofR-modules on a topological spaceX is coherentif

(i) F is of finite type; and

(ii) for any open setU ⊂ X and any collection of sectionsf1, . . . , fn ∈ F(U), thesheafS(f1, . . . , fn) of relations amongf1, . . . , fn is of finite type overU .

THEOREM 2.3.4 (i) The direct sum of finitely many coherent sheaves is coherent.

(ii) The kernel, image, and cokernel of a morphism of coherent sheaves are coherent.

PROOF. For a proof, see Serre [12, Subsec. 13, Ths. 1 and 2, pp. 208–209].

A sheafF of R-modules on a topological spaceX is said to belocally finitelypresentedif everyx ∈ X has a neighborhoodU on which there is an exact sequenceof the form

R| ⊕qU → R| ⊕p

U → F|U → 0;

in this case, we say thatF has afinite presentationor thatF is finitely presentedonU . If F is a coherent sheaf ofR-modules onX , then it is locally finitely presented.

Remark. Having a finite presentation locally is a consequence of coherence, but isnot equivalent to it. Having a finite presentation means thatfor one setof generatorsof F , the sheaf of relations among them is finitely generated. Coherence is a strongercondition in that it requires the sheaf of relations amongany setof elements ofF to befinitely generated.

A sheafR of rings onX is clearly a sheaf ofR-modules of finite type. For it to becoherent, for any open setU ⊂ X and any sectionsf1, . . . , fn, the sheafS(f1, . . . , fn)of relations amongf1, . . . , fn must be of finite type.

EXAMPLE 2.3.5 If OM is the sheaf of holomorphic functions on a complex manifoldM , thenOM is a coherent sheaf ofOM -modules (Oka’s theorem [4, Sec. 5]).

90


CHAPTER 2

EXAMPLE 2.3.6 If OX is the sheaf of regular functions on an algebraic varietyX ,thenOX is a coherent sheaf ofOX -modules (Serre [12, Sec. 37, Prop. 1]).

A sheaf ofOX -modules on an algebraic variety is called analgebraic sheaf.

EXAMPLE 2.3.7 On a smooth varietyX of dimensionn, the sheafΩk of algebraick-forms is an algebraic sheaf. It is locally free of rank

(nk

)[14, Ch. III, Th. 2, p. 200].

THEOREM 2.3.8 LetR be a coherent sheaf of rings on a topological spaceX . Thena sheafF ofR-modules onX is coherent if and only if it is locally finitely presented.

PROOF. (⇒) True for any coherent sheaf ofR-modules, whetherR is coherent ornot.(⇐) Suppose there is an exact sequence

R⊕q → R⊕p → F → 0

on an open setU in X . SinceR is coherent, by Theorem 2.3.4 so areR⊕p, R⊕q, andthe cokernelF of R⊕q → R⊕p.

Since the structure sheavesOX or OM of an algebraic varietyX or of a complexmanifoldM are coherent, an algebraic or analytic sheaf is coherent if and only if it islocally finitely presented.

EXAMPLE 2.3.9 A locally free analytic sheafF over a complex manifoldM is auto-matically coherent since every pointp has a neighborhoodU on which there is an exactsequence of the form

0 → O⊕pU → F|U → 0,

so thatF|U is finitely presented.

For our purposes, we define aStein manifold to be a complex manifold that isbiholomorphic to a closed submanifold ofCN (this is not the usual definition, but isequivalent to it [11, p. 114]). In particular, a complex submanifold ofCN defined byfinitely many holomorphic functions is a Stein manifold. Oneof the basic theoremsabout coherent analytic sheaves is Cartan’s theorem B.

THEOREM 2.3.10(Cartan’s theorem B) A coherent analytic sheafF is acyclic on aStein manifoldM , i.e.,Hq(M,F) = 0 for all q ≥ 1.

For a proof, see [8, Th. 14, p. 243].Let X be a smooth quasi-projective variety defined over the complex numbers and

endowed with the Zariski topology. The underlying set ofX with the complex topologyis a complex manifoldXan. Similarly, if U is a Zariski open subset ofX , let Uan bethe underlying set ofU with the complex topology. Since Zariski open sets are open inthe complex topology,Uan is open inXan.

Denote byOXanthe sheaf of holomorphic functions onXan. If F is a coherent

algebraic sheaf onX , thenX has an open coverU by Zariski open sets such that oneach open setU there is an exact sequence

O⊕qU → O⊕p

U → F|U → 0



91

of algebraic sheaves. Moreover,Uan is an open cover ofXan and the morphismO⊕q

U → O⊕pU of algebraic sheaves induces a morphismO⊕q

Uan→ O⊕p

Uanof analytic

sheaves. Hence, there is a coherent analytic sheafFan on the complex manifoldXan

defined byO⊕q

Uan→ O⊕p

Uan→ Fan|Uan

→ 0.

(Rename the open coverUan asUαα∈A. A section ofFan over an open setV ⊂Xan is a collection of sectionssα ∈ (Fan|Uα

)(Uα ∩ V ) that agree on all pairwiseintersections(Uα ∩ V ) ∩ (Uβ ∩ V ).)

In this way one obtains a functor( )an from the category of smooth complex quasi-projective varieties and coherent algebraic sheaves to thecategory of complex mani-folds and analytic sheaves. Serre’s GAGA (“Geometrie algebrique et geometrie analy-tique”) principle [13] asserts that for smooth complex projective varieties, the functor( )an is an equivalence of categories and moreover, for allq, there are isomorphisms ofcohomology groups

Hq(X,F) ≃ Hq(Xan,Fan), (2.3.1)

where the left-hand side is the sheaf cohomology ofF onX endowed with the Zariskitopology and the right-hand side is the sheaf cohomology ofFan onXan endowed withthe complex topology.

When X is a smooth complex quasi-projective variety, to distinguish betweensheaves of algebraic and sheaves of holomorphic forms, we write Ωp

algfor the sheaf

of algebraicp-forms onX andΩpan for the sheaf of holomorphicp-forms onXan (for

the definition of algebraic forms, see the introduction to this chapter). Ifz1, . . . , zn arelocal parameters forX [14, Ch. II, Sec. 2.1, p. 98], then bothΩp

algandΩp

an are locallyfree with framedzi1 ∧ · · · ∧ dzip, whereI = (i1, . . . , ip) is a strictly increasingmultiindex between1 andn inclusive. (For the algebraic case, see [14, Vol. 1, Ch. III,Sec. 5.4, Th. 4, p. 203].) Hence, locally there are sheaf isomorphisms

0 → O(np)U → Ωp

alg|U → 0 and 0 → O

(np)Uan

→ Ωpan|Uan

→ 0,

which show thatΩpan is the coherent analytic sheaf associated to the coherent algebraic

sheafΩpalg

.

Let k be a field. Anaffine closed setin kN is the zero set of finitely many polyno-mials onkN , and anaffine varietyis an algebraic variety biregular to an affine closedset. The algebraic analogue of Cartan’s theorem B is the following vanishing theoremof Serre for an affine variety [12, Sec. 44, Cor. 1, p. 237].

THEOREM 2.3.11(Serre) A coherent algebraic sheafF on an affine varietyX isacyclic onX , i.e.,Hq(X,F) = 0 for all q ≥ 1.

2.4 THE HYPERCOHOMOLOGY OF A COMPLEX OF SHEAVES

This section requires some knowledge of double complexes and their associated spec-tral sequences. One possible reference is [3, Chs. 2 and 3]. The hypercohomology ofa complexL• of sheaves of abelian groups on a topological spaceX generalizes the

92


CHAPTER 2

cohomology of a single sheaf. To define it, first form the double complex of globalsections of the Godement resolutions of the sheavesLq:

K =⊕

p,q

Kp,q =⊕

p,q

Γ(X, CpLq).

This double complex comes with two differentials: a horizontal differential

δ : Kp,q → Kp+1,q

induced from the Godement resolution and a vertical differential

d : Kp,q → Kp,q+1

induced from the complexL•. Since the differentiald : Lq → Lq+1 induces a mor-phism of complexesC•Lq → C•Lq+1, whereC• is the Godement resolution, the ver-tical differential in the double complexK commutes with the horizontal differential.ThehypercohomologyH∗(X,L•) of the complexL• is the total cohomology of thedouble complex, i.e., the cohomology of the associated single complex

K• =⊕

Kk =⊕

k

⊕

p+q=k

Kp,q

with differentialD = δ + (−1)pd:

Hk(X,L•) = Hk

D(K•).

If the complex of sheavesL• consists of a single sheafL0 = F in degree0,

0 → F → 0 → 0 → · · · ,

then the double complex⊕

Kp,q =⊕

Γ(X, CpLq) has nonzero entries only in thezeroth row, which is simply the complex of sections of the Godement resolution ofF :

K =

0 1 2

Γ(X, C0F) Γ(X, C1F) Γ(X, C2F)

0 0 0

0 0 0

p

q

In this case, the associated single complex is the complexΓ(X, C•F) of global sec-tions of the Godement resolution ofF , and the hypercohomology ofL• is the sheafcohomology ofF :

Hk(X,L•) = hk

(Γ(X, C•F)

)= Hk(X,F). (2.4.1)

It is in this sense that hypercohomology generalizes sheaf cohomology.



93

2.4.1 The Spectral Sequences of Hypercohomology

Associated to any double complex(K, d, δ) with commuting differentialsd and δare two spectral sequences converging to the total cohomologyH∗

D(K). One spectralsequence starts withE1 = Hd andE2 = HδHd. By reversing the roles ofd andδ, weobtain a second spectral sequence withE1 = Hδ andE2 = HdHδ (see [3, Ch. II]).By the usualspectral sequence of a double complex, we will mean the first spectralsequence, with the vertical differentiald as the initial differential.

In the category of groups, theE∞ term is the associated graded group of the totalcohomologyH∗

D(K) relative to a canonically defined filtration and is not necessarilyisomorphic toH∗

D(K) because of the extension phenomenon in group theory.Fix a nonnegative integerp and letT = Γ

(X, Cp( )

)be the Godement sections

functor that associates to a sheafF on a topological spaceX the group of sectionsΓ(X, CpF) of the Godement sheafCpF . SinceT is an exact functor by Corollary 2.2.7,by Proposition 2.2.10 it commutes with cohomology:

hq(T (L•)

)= T

(Hq(L•)

), (2.4.2)

whereHq := Hq(L•) is the qth cohomology sheaf of the complexL• (see Sec-tion 2.2.4). For the double complexK =

⊕Γ(X, CpLq), theE1 term of the first

spectral sequence is the cohomology ofK with respect to the vertical differentiald.Thus,Ep,q

1 = Hp,qd is theqth cohomology of thepth columnKp,• = Γ

(X, Cp(L•)

)of

K:

Ep,q1 = Hp,q

d = hq(Kp,•) = hq(Γ(X, CpL•)

)

= hq(T (L•)

)(definition ofT )

= T(Hq(L•)

)(by (2.4.2))

= Γ(X, CpHq) (definition ofT ).

Hence, theE2 term of the first spectral sequence is

Ep,q2 = Hp,q

δ (E1) = Hp,qδ H•,•

d = hpδ(H

•,qd ) = hp

δ

(Γ(X, C•Hq)

)= Hp(X,Hq) .

(2.4.3)Note that theqth row of the double complex

⊕Kp,q =

⊕Γ(X, CpLq) calculates

the sheaf cohomology ofLq onX . Thus, theE1 term of the second spectral sequenceis

Ep,q1 = Hp,q

δ = hpδ(K

•,q) = hpδ

(Γ(X, C•Lq)

)= Hp(X,Lq) (2.4.4)

and theE2 term is

Ep,q2 = Hp,q

d (E1) = Hp,qd H•,•

δ = hqd

(Hp,•

δ

)= hq

d

(Hp(X,L•)

).

THEOREM 2.4.1 A quasi-isomorphismF• → G• of complexes of sheaves of abeliangroups over a topological spaceX (see p. 83) induces a canonical isomorphism inhypercohomology:

H∗(X,F•)

∼→ H

∗(X,G•).

94


CHAPTER 2

PROOF. By the functoriality of the Godement sections functors, a morphismF• →G• of complexes of sheaves induces a homomorphismΓ(X, CpFq) → Γ(X, CpGq)that commutes with the two differentialsd andδ and hence induces a homomorphismH∗(X,F•) → H∗(X,G•) in hypercohomology.

Since the spectral sequence construction is functorial, the morphismF• → G• alsoinduces a morphismEr(F

•) → Er(G•) of spectral sequences and a morphism of the

filtrationsFp

(HD(KF•)

)→ Fp

(HD(KG•)

)

on the hypercohomology ofF• andG•. We will shorten the notationFp

(HD(KF•)

)

toFp(F•).

By definition, the quasi-isomorphismF• → G• induces an isomorphism of coho-mology sheavesH∗(F•)

∼→ H∗(G•), and by (2.4.3) an isomorphism of theE2 terms

of the first spectral sequences ofF• and ofG•:

Ep,q2 (F•) = Hp

(X,Hq(F•)

) ∼→ Hp

(X,Hq(G•)

)= Ep,q

2 (G•).

An isomorphism of theE2 terms induces an isomorphism of theE∞ terms:

⊕

p

Fp(F•)

Fp+1(F•)= E∞(F•)

∼→ E∞(G•) =

⊕

p

Fp(G•)

Fp+1(G•).

We claim that in fact, the canonical homomorphismH∗(X,F•) → H∗(X,G•) is

an isomorphism. Fix a total degreek and letF kp (F

•) = Fp(F•) ∩Hk(X,F•). Since

K•,•(F•) =⊕

Γ(X, CpFq)

is a first-quadrant double complex, the filtrationF kp (F

•)p onHk(X,F•) is finite in

length:

Hk(X,F•) = F k

0 (F•) ⊃ F k

1 (F•) ⊃ · · · ⊃ F k

k (F•) ⊃ F k

k+1(F•) = 0.

A similar finite filtrationF kp (G

•)p exists onHk(X,G•).SupposeF k

p (F•) → F k

p (G•) is an isomorphism. We will prove thatF k

p−1(F•) →

F kp−1(G

•) is an isomorphism. In the commutative diagram

0 // F kp (F

•) //

F kp−1(F

•) //

F kp−1(F

•)/F kp (F

•) //

0

0 // F kp (G

•) // F kp−1(G

•) // F kp−1(G

•)/F kp (G

•) // 0,

the two outside vertical maps are isomorphisms, by the induction hypothesis and be-causeF• → G• induces an isomorphism of the associated graded groups. By the fivelemma, the middle vertical mapF k

p−1(F•) → F k

p−1(G•) is also an isomorphism. By

induction on the filtration subscriptp, asp moves fromk + 1 to 0, we conclude that

Hk(X,F•) = F k

0 (F•) → F k

0 (G•) = H

k(X,G•)

is an isomorphism.



95

THEOREM 2.4.2 If L• is a complex of acyclic sheaves of abelian groups on a topo-logical spaceX , then the hypercohomology ofL• is isomorphic to the cohomology ofthe complex of global sections ofL•:

Hk(X,L•) ≃ hk

(L•(X)

),

whereL•(X) denotes the complex

0 → L0(X) → L1(X) → L2(X) → · · · .

PROOF. Let K be the double complexK =⊕

Kp,q =⊕

CpLq(X). BecauseeachLq is acyclic onX , in the second spectral sequence ofK, by (2.4.4) theE1 termis

Ep,q1 = Hp(X,Lq) =

Lq(X) for p = 0,

0 for p > 0.

E1 = Hδ =

0 1 2

L0(X) 0 0

L1(X) 0 0

L2(X) 0 0

p

q

Hence,

Ep,q2 = Hp,q

d Hδ =

hq(L•(X)

)for p = 0,

0 for p > 0.

Therefore, the spectral sequence degenerates at theE2 term and

Hk(X,L•) ≃ E0,k

2 = hk(L•(X)

).

2.4.2 Acyclic Resolutions

Let F be a sheaf of abelian groups on a topological spaceX . A resolution

0 → F → L0 → L1 → L2 → · · ·

of F is said to beacycliconX if each sheafLq is acyclic onX , i.e.,Hk(X,Lq) = 0for all k > 0.

If F is a sheaf onX , we will denote byF• the complex of sheaves such thatF0 = F andFk = 0 for k > 0.

THEOREM 2.4.3 If 0 → F → L• is an acyclic resolution of the sheafF on atopological spaceX , then the cohomology ofF can be computed from the complex ofglobal sections ofL•:

Hk(X,F) ≃ hk(L•(X)

).

96


CHAPTER 2

PROOF. The resolution0 → F → L• may be viewed as a quasi-isomorphism ofthe two complexes

0 // F

// 0

// 0

// · · ·

0 // L0 // L1 // L2 // · · · ,

since

H0(top row) = H0(F•) = F ≃ Im(F → L0) = ker(L0 → L1) = H0(bottom row)

and the higher cohomology sheaves of both complexes are zero. By Theorem 2.4.1,there is an induced morphism in hypercohomology

Hk(X,F•) ≃ H

k(X,L•).

The left-hand side is simply the sheaf cohomologyHk(X,F) by (2.4.1). By Theo-rem 2.4.2, the right-hand side ishk(L•(X)). Hence,

Hk(X,F) ≃ hk(L•(X)

).

So in computing sheaf cohomology, any acyclic resolution ofF on a topologicalspaceX can take the place of the Godement resolution.

Using acyclic resolutions, we can give simple proofs of de Rham’s and Dolbeault’stheorems.

EXAMPLE 2.4.4(De Rham’s theorem) By the Poincare lemma ([3, Sec. 4, p. 33], [6,p. 38]), on aC∞ manifoldM the sequence of sheaves

0 → R → A0 → A1 → A2 → · · · (2.4.5)

is exact. Since eachAk is fine and hence acyclic onM , (2.4.5) is an acyclic resolutionof R. By Theorem 2.4.3,

H∗(M,R) ≃ h∗(A•(M)

)= H∗

dR(M).

Because the sheaf cohomologyH∗(M,R) of a manifold is isomorphic to the real sin-gular cohomology ofM (Remark 2.2.16), de Rham’s theorem follows.

EXAMPLE 2.4.5(Dolbeault’s theorem) According to the∂-Poincare lemma [6, pp. 25and 38], on a complex manifoldM the sequence of sheaves

0 → Ωp → Ap,0 ∂→ Ap,1 ∂

→ Ap,2 → · · ·

is exact. As in the previous example, because each sheafAp,q is fine and hence acyclic,by Theorem 2.4.3,

Hq(M,Ωp) ≃ hq(Ap,•(M)

)= Hp,q(M).

This is the Dolbeault isomorphism for a complex manifoldM .



97

2.5 THE ANALYTIC DE RHAM THEOREM

The analytic de Rham theorem is the analogue of the classicalde Rham theorem fora complex manifold, according to which the singular cohomology withC coefficientsof any complex manifold can be computed from its sheaves of holomorphic forms.Because of the holomorphic Poincare lemma, the analytic deRham theorem is fareasier to prove than its algebraic counterpart.

2.5.1 The Holomorphic Poincare Lemma

Let M be a complex manifold andΩkan the sheaf of holomorphick-forms onM . Lo-

cally, in terms of complex coordinatesz1, . . . , zn, a holomorphic form can be writtenas∑

aI dzi1 ∧ · · · ∧ dzin , where theaI are holomorphic functions. Since for a holo-morphic functionaI ,

daI = ∂aI + ∂aI =∑

i

∂aI∂zi

dzi +∑

i

∂aI∂zi

dzi =∑

i

∂aI∂zi

dzi,

the exterior derivatived maps holomorphic forms to holomorphic forms. Note thataIis holomorphic if and only if∂aI = 0.

THEOREM 2.5.1(Holomorphic Poincare lemma)On a complex manifoldM of com-plex dimensionn, the sequence

0 → C → Ω0an

d→ Ω1

an

d→ · · · → Ωn

an → 0

of sheaves is exact.

PROOF. We will deduce the holomorphic Poincare lemma from the smooth Poincarelemma and the∂-Poincare lemma by a double complex argument. The double com-plex

⊕Ap,q of sheaves of smooth(p, q)-forms has two differentials∂ and ∂. These

differentials anticommute because

0 = d d = (∂ + ∂)(∂ + ∂) = ∂2 + ∂∂ + ∂∂ + ∂2

= ∂∂ + ∂∂.

The associated single complex⊕

AkC

, whereAkC

=⊕

p+q=k Ap,q with differential

d = ∂ + ∂, is simply the usual complex of sheaves of smoothC-valued differentialforms onM . By the smooth Poincare lemma,

Hkd(A

•C) =

C for k = 0,

0 for k > 0.

By the ∂-Poincare lemma, the sequence

0 → Ωpan → Ap,0 ∂

→ Ap,1 ∂→ · · · → Ap,n → 0

is exact for eachp and so theE1 term of the usual spectral sequence of the doublecomplex

⊕Ap,q is

98


CHAPTER 2

E1 = H∂ =

0 1 2

Ω0an Ω1

an Ω2an

0 0 0

0 0 0

p

q

.

Hence, theE2 term is given by

Ep,q2 =

Hp

d(Ω•an) for q = 0,

0 for q > 0.

Since the spectral sequence degenerates at theE2 term,

Hkd(Ω

•an) = E2 = E∞ ≃ Hk

d(A•C) =

C for k = 0,

0 for k > 0,

which is precisely the holomorphic Poincare lemma.

2.5.2 The Analytic de Rham Theorem

THEOREM 2.5.2 LetΩkan be the sheaf of holomorphick-forms on a complex manifold

M . Then the singular cohomology ofM with complex coefficients can be computed asthe hypercohomology of the complexΩ•

an:

Hk(M,C) ≃ Hk(M,Ω•

an).

PROOF. LetC• be the complex of sheaves that isC in degree0 and zero otherwise.The holomorphic Poincare lemma may be interpreted as a quasi-isomorphism of thetwo complexes

0 // C

// 0

// 0

// · · ·

0 // Ω0an

// Ω1an

// Ω2an

// · · ·,

since

H0(C•) = C ≃ Im(C → Ω0an)

= ker(Ω0an → Ω1

an) (by the holomorphic Poincare lemma)

= H0(Ω•an)

and the higher cohomology sheaves of both complexes are zero.By Theorem 2.4.1, the quasi-isomorphismC• ≃ Ω•

an induces an isomorphism

H∗(M,C•) ≃ H

∗(M,Ω•an) (2.5.1)



99

in hypercohomology. SinceC• is a complex of sheaves concentrated in degree0, by(2.4.1) the left-hand side of (2.5.1) is the sheaf cohomology Hk(M,C), which is iso-morphic to the singular cohomologyHk(M,C) by Remark 2.2.16.

In contrast to the sheavesAk andAp,q in de Rham’s theorem and Dolbeault’s the-orem, the sheavesΩ•

an are generally neither fine nor acyclic, because in the analyticcategory there is no partition of unity. However, whenM is a Stein manifold, the com-plexΩ•

an is a complex of acyclic sheaves onM by Cartan’s theorem B. It then followsfrom Theorem 2.4.2 that

Hk(M,Ω•

an) ≃ hk(Ω•

an(M)).

This proves the following corollary of Theorem 2.5.2.

COROLLARY 2.5.3 The singular cohomology of a Stein manifoldM with coefficientsin C can be computed from the holomorphic de Rham complex:

Hk(M,C) ≃ hk(Ω•

an(M)).

2.6 THE ALGEBRAIC DE RHAM THEOREM FOR A PROJECTIVEVARIETY

LetX be a smooth complex algebraic variety with the Zariski topology. The underlyingset ofX with the complex topology is a complex manifoldXan. LetΩk

algbe the sheaf of

algebraick-forms onX , andΩkan the sheaf of holomorphick-forms onXan. According

to the holomorphic Poincare lemma (Theorem 2.5.1), the complex of sheaves

0 → C → Ω0an

d→ Ω1

an

d→ Ω2

an

d→ · · · (2.6.1)

is exact. However, there is no Poincare lemma in the algebraic category; the complex

0 → C → Ω0alg

→ Ω1alg

→ Ω2alg

→ · · ·

is in general not exact.

THEOREM 2.6.1 (Algebraic de Rham theorem for a projective variety)If X is asmooth complex projective variety, then there is an isomorphism

Hk(Xan,C) ≃ Hk(X,Ω•

alg)

between the singular cohomology ofXan with coefficients inC and the hypercohomol-ogy ofX with coefficients in the complexΩ•

alg of sheaves of algebraic differential formsonX .

PROOF. By Theorem 2.4.1, the quasi-isomorphismC• → Ω•an of complexes of

sheaves induces an isomorphism in hypercohomology

H∗(Xan,C

•) ≃ H∗(Xan,Ω

•an). (2.6.2)

100


CHAPTER 2

In the second spectral sequence converging toH∗(Xan,Ω•an), by (2.4.4) theE1 term is

Ep,q1,an = Hp(Xan,Ω

qan).

By (2.4.4) theE1 term in the second spectral sequence converging to the hypercoho-mologyH∗(X,Ω•

alg) is

Ep,q1,alg

= Hp(X,Ωqalg).

SinceX is a smooth complex projective variety, Serre’s GAGA principle (2.3.1)applies and gives an isomorphism

Hp(X,Ωqalg) ≃ Hp(Xan,Ω

qan).

The isomorphismE1,alg

∼→ E1,an induces an isomorphism inE∞. Hence,

H∗(X,Ω•

alg) ≃ H

∗(Xan,Ω•an). (2.6.3)

Combining (2.4.1), (2.6.2), and (2.6.3) gives

H∗(Xan,C) ≃ H∗(Xan,C

•) ≃ H∗(Xan,Ω

•an) ≃ H

∗(X,Ω•alg).

Finally, by the isomorphism between sheaf cohomology and singular cohomology (Re-mark 2.2.16), we may replace the sheaf cohomologyH∗(Xan,C) by a singular coho-mology group:

H∗(Xan,C) ≃ H∗(X,Ω•

alg).

PART II. CECH COHOMOLOGY AND THE ALGEBRAIC DE RHAMTHEOREM IN GENERAL

The algebraic de Rham theorem (Theorem 2.6.1) in fact does not require the hypothesisof projectivity onX . In this section we will extend it to an arbitrary smooth algebraicvariety defined overC. In order to carry out this extension, we will need to developtwo more machineries: theCech cohomology of a sheaf and theCech cohomology ofa complex of sheaves.Cech cohomology provides a practical method for computingsheaf cohomology and hypercohomology.

2.7 CECH COHOMOLOGY OF A SHEAF

Cech cohomology may be viewed as a generalization of the Mayer–Vietoris sequencefrom two open sets to arbitrarily many open sets.



101

2.7.1 Cech Cohomology of an Open Cover

Let U = Uαα∈A be an open cover of the topological spaceX indexed by a linearlyordered set A, andF a presheaf of abelian groups onX . To simplify the notation, wewill write the (p+ 1)-fold intersectionUα0

∩ · · · ∩ UαpasUα0···αp

. Define thegroupof Cechp-cochainsonU with values in the presheafF to be the direct product

Cp(U,F) :=∏

α0<···<αp

F(Uα0···αp).

An elementω of Cp(U,F) is then a function that assigns to each finite set of indicesα0, . . . , αp an elementωα0...αp

∈ F(Uα0...αp). We will write ω = (ωα0...αp

), wherethe subscripts range over allα0 < · · · < αp. In particular, the subscriptsα0, . . . , αp

must all be distinct. Define theCech coboundary operator

δ = δp : Cp(U,F) → Cp+1(U,F)

by the alternating sum formula

(δω)α0...αp+1=

p+1∑

i=0

(−1)iωα0···αi···αp+1,

whereαi means to omit the indexαi; moreover, the restriction ofωα0···αi···αp+1from

Uα0···αi···αp+1toUα0...αp+1

is suppressed in the notation.

PROPOSITION 2.7.1 If δ is theCech coboundary operator, thenδ2 = 0.

PROOF. Basically, this is true because in(δ2ω)α0···αp+2, we omit two indicesαi,

αj twice with opposite signs. To be precise,

(δ2ω)α0···αp+2=∑

(−1)i(δω)α0···αi···αp+2

=∑

j<i

(−1)i(−1)jωα0···αj ···αi···αp+2

+∑

j>i

(−1)i(−1)j−1ωα0···αi···αj ···αp+2

= 0.

It follows from Proposition 2.7.1 thatC•(U,F) :=⊕∞

p=0 Cp(U,F) is a cochain

complex with differentialδ. The cohomology of the complex(C∗(U,F), δ),

Hp(U,F) :=ker δpIm δp−1

=p-cocycles

p-coboundaries,

is called theCech cohomologyof the open coverU with values in the presheafF .

102


CHAPTER 2

2.7.2 Relation BetweenCech Cohomology and Sheaf Cohomology

In this subsection we construct a natural map from theCech cohomology of a sheafon an open cover to its sheaf cohomology. This map is based on aproperty of flasquesheaves.

L EMMA 2.7.2 SupposeF is a flasque sheaf of abelian groups on a topological spaceX , andU = Uα is an open cover ofX . Then the augmentedCech complex

0 → F(X) →∏

α

F(Uα) →∏

α<β

F(Uαβ) → · · ·

is exact.

In other words, for a flasque sheafF onX ,

Hk(U,F) =

F(X) for k = 0,

0 for k > 0.

PROOF. [5, Th. 5.2.3(a), p. 207].

Now supposeF is any sheaf of abelian groups on a topological spaceX andU =Uα is an open cover ofX . LetK•,• =

⊕Kp,q be the double complex

Kp,q = Cp(U, CqF) =∏

α0<···<αp

CqF(Uα0···αp).

We augment this complex with an outside bottom row (q = −1) and an outside leftcolumn (p = −1):

0 F(X)∏

F(Uα)∏

F(Uαβ)

0 C0F(X)∏

C0F(Uα)∏

C0F(Uαβ)

0 C1F(X)∏

C1F(Uα)∏

C1F(Uαβ)

OO

//p

q

//

//

//

//

//

//

//

//

//

//

//

//

OO OO OO

ǫ ǫ

OO OO OO

OO OO OO

(2.7.1)

Note that theqth row of the double complexK•,• is theCech cochain complex ofthe Godement sheafCqF and thepth column is the complex of groups for computingthe sheaf cohomology

∏α0<···<αp

H∗(Uα0···αp,F).

By Lemma 2.7.2, each row of the augmented double complex (2.7.1) is exact.Hence, theE1 term of the second spectral sequence of the double complex is



103

E1 = Hδ =

0 1 2

C0F(X) 0 0

C1F(X) 0 0

C2F(X) 0 0

p

q

and theE2 term is

E2 = HdHδ =

0 1 2

H0(X,F) 0 0

H1(X,F) 0 0

H2(X,F) 0 0

p

q

.

So the second spectral sequence of the double complex (2.7.1) degenerates at theE2

term and the cohomology of the associated single complexK• of⊕

Kp,q is

HkD(K•) ≃ Hk(X,F).

In the augmented complex (2.7.1), by the construction of Godement’s canonicalresolution, theCech complexC•(U,F) injects into the complexK• via a cochain map

ǫ : Ck(U,F) → Kk,0 → Kk,

which gives rise to an induced map

ǫ∗ : Hk(U,F) → HkD(K•) = Hk(X,F) (2.7.2)

in cohomology.

DEFINITION 2.7.3 A sheafF of abelian groups on a topological spaceX is acyclicon an open coverU = Uα ofX if the cohomology

Hk(Uα0···αp,F) = 0

for all k > 0 and all finite intersectionsUα0···αpof open sets inU.

THEOREM 2.7.4 If a sheafF of abelian groups is acyclic on an open coverU =Uα of a topological spaceX , then the induced mapǫ∗ : Hk(U,F) → Hk(X,F) isan isomorphism.

PROOF. BecauseF is acyclic on each intersectionUα0···αp, the cohomology of the

pth column of (2.7.1) is∏

H0(Uα0···αp,F) =

∏F(Uα0···αp

), so that theE1 term ofthe usual spectral sequence is

104


CHAPTER 2

E1 = Hd =

0 1 2

∏F(Uα0

)∏

F(Uα0α1)

∏F(Uα0α1α2

)

0 0 0

0 0 0

p

q

,

and theE2 term is

E2 = HδHd =

0 1 2

H0(U,F) H1(U,F) H2(U,F)

0 0 0

0 0 0

p

q

.

Hence, the spectral sequence degenerates at theE2 term and there is an isomorphism

ǫ∗ : Hk(U,F) ≃ HkD(K•) ≃ Hk(X,F).

Remark. Although we used a spectral sequence argument to prove Theorem 2.7.4,in the proof there is no problem with the extension of groups in theE∞ term, sincealong each antidiagonal

⊕p+q=k E

p,q∞ there is only one nonzero box. For this reason,

Theorem 2.7.4 holds for sheaves of abelian groups, not just for sheaves of vector spaces.

2.8 CECH COHOMOLOGY OF A COMPLEX OF SHEAVES

Just as the cohomology of a sheaf can be computed using aCech complex on an opencover (Theorem 2.7.4), the hypercohomology of a complex of sheaves can also becomputed using theCech method.

Let (L•, dL) be a complex of sheaves on a topological spaceX , andU = Uα anopen cover ofX . To define theCech cohomology ofL• onU, letK =

⊕Kp,q be the

double complexKp,q = Cp(U,Lq)

with its two commuting differentialsδ anddL. We will call K theCech–sheaf doublecomplex. TheCech cohomologyH∗(U,L•) of L• is defined to be the cohomology ofthe single complex

K• =⊕

Kk, whereKk =⊕

p+q=k

Cp(U,Lq) anddK = δ + (−1)pdL,

associated to theCech–sheaf double complex.



105

2.8.1 The Relation BetweenCech Cohomology and Hypercohomology

There is an analogue of Theorem 2.7.4 that allows us to compute hypercohomologyusing an open cover.

THEOREM 2.8.1 If L• is a complex of sheaves of abelian groups on a topologicalspaceX such that each sheafLq is acyclic on the open coverU = Uα of X , thenthere is an isomorphismHk(U,L•) ≃ Hk(X,L•) between theCech cohomology ofL• on the open coverU and the hypercohomology ofL• onX .

TheCech cohomology of the complexL• is the cohomology of the associated sin-gle complex of the double complex

⊕p,q C

p(U,Lq) =⊕

p,q

∏α Lq(Uα0···αp

), whereα = (α0 < · · · < αp). The hypercohomology of the complexL• is the cohomology ofthe associated single complex of the double complex

⊕q,r C

rLq(X). To compare thetwo, we form the triple complex with terms

Np,q,r = Cp(U, CrLq)

and three commuting differentials: theCech differentialδC , the differentialdL of thecomplexL•, and the Godement differentialdC .

LetN•,•,• be any triple complex with three commuting differentialsd1, d2, andd3of degrees(1, 0, 0), (0, 1, 0), and(0, 0, 1), respectively. SummingNp,q,r overp andq,or overq andr, one can form two double complexes fromN•,•,•:

Nk,r =⊕

p+q=k

Np,q,r

with differentialsδ = d1 + (−1)pd2, d = d3,

andN ′p,ℓ =

⊕

q+r=ℓ

Np,q,r

with differentialsδ′ = d1, d′ = d2 + (−1)qd3.

PROPOSITION 2.8.2 For any triple complexN•,•,•, the two associated double com-plexesN•,• andN ′•,• have the same associated single complex.

PROOF. Clearly, the groups

Nn =⊕

k+r=n

Nk,r =⊕

p+q+r=n

Np,q,r

andN ′n =

⊕

p+ℓ=n

N ′p,ℓ =⊕

p+q+r=n

Np,q,r

106


CHAPTER 2

are equal. The differentialD for N• =⊕

n Nn is

D = δ + (−1)kd = d1 + (−1)pd2 + (−1)p+qd3.

The differentialD′ for N ′• =⊕

n N′n is

D′ = δ′ + (−1)pd′ = d1 + (−1)p(d2 + (−1)qd3

)= D.

Thus, any triple complexN•,•,• has an associated single complexN• whose co-homology can be computed in two ways, either from the double complex(N•,•, D) orfrom the double complex(N ′•,•, D′).

We now apply this observation to theCech–Godement–sheaf triple complex

N•,•,• =⊕

Cp(U, CrLq)

of the complexL• of sheaves. Thekth column of the double complexN•,• =⊕

Nk,r

is ⊕p+q=k

∏α0<···<αp

Cr+1Lq(Uα0···αp)

⊕p+q=k

∏α0<···<αp

CrLq(Uα0···αp)

⊕p+q=k

∏α0<···<αp

C0Lq(Uα0···αp),

...

OO

OO

OO

where the vertical differentiald is the Godement differentialdC . SinceL• is acyclic onthe open coverU = Uα, this column is exact except in the zeroth row, and the zerothrow of the cohomologyHd is

⊕

p+q=k

∏

α0<···<αp

Lq(Uα0···αp) =

⊕

p+q=k

Cp(U,Lq) =⊕

p+q=k

Kp,q = Kk,

the associated single complex of theCech–sheaf double complex. Thus, theE1 termof the first spectral sequence ofN•,• is

E1 = Hd =

0 1 2

K0 K1 K2

0 0 0

0 0 0

k

r

,



107

and so theE2 term is

E2 = Hδ(Hd) = H∗dK

(K•) = H∗(U,L•).

Although we are working with abelian groups, there are no extension issues, becauseeach antidiagonal inE∞ contains only one nonzero group. Thus, theE∞ term is

H∗D(N•) ≃ E2 = H∗(U,L•). (2.8.1)

On the other hand, theℓth row ofN ′•,• is

0 →⊕

q+r=ℓ

C0(U, CrLq) → · · · →⊕

q+r=ℓ

Cp(U, CrLq) →⊕

q+r=ℓ

Cp+1(U, CrLq) → · · · ,

which is theCech cochain complex of the flasque sheaf⊕

q+r=ℓ CrLq with differential

δ′ = δC . Thus, each row ofN ′•,• is exact except in the zeroth column, and the kernelof N ′0,ℓ → N ′1,ℓ is M ℓ =

⊕q+r=ℓ C

rLq(X). Hence, theE1 term of the secondspectral sequence is

E1 = Hδ′ =

0 1 2 3

M0 0 0 0

M1 0 0 0

M2 0 0 0

p

ℓ

.

TheE2 term isE2 = Hd′(Hδ′) = H∗

dM(M•) = H

∗(X,L•).

Since this spectral sequence forN•,• degenerates at theE2 term and converges toH∗

D′(N ′•), there is an isomorphism

E∞ = H∗D′(N ′•) ≃ E2 = H

∗(X,L•). (2.8.2)

By Proposition 2.8.2, the two groups in (2.8.1) and (2.8.2) are isomorphic. In thisway, one obtains an isomorphism between theCech cohomology and the hypercoho-mology of the complexL•:

H∗(U,L•) ≃ H∗(X,L•).

2.9 REDUCTION TO THE AFFINE CASE

Grothendieck proved his general algebraic de Rham theorem by reducing it to the spe-cial case of an affine variety. This section is an exposition of his ideas in [7].

THEOREM 2.9.1(Algebraic de Rham theorem)LetX be a smooth algebraic varietydefined over the complex numbers, andXan its underlying complex manifold. Thenthe singular cohomology ofXan with C coefficients can be computed as the hyperco-homology of the complexΩ•

alg of sheaves of algebraic differential forms onX with itsZariski topology:

Hk(Xan,C) ≃ Hk(X,Ω•

alg).

108


CHAPTER 2

By the isomorphismHk(Xan,C) ≃ Hk(Xan,Ω•an) of the analytic de Rham the-

orem, Grothendieck’s algebraic de Rham theorem is equivalent to an isomorphism inhypercohomology

Hk(X,Ω•

alg) ≃ Hk(Xan,Ω

•an).

The special case of Grothendieck’s theorem for an affine variety is especially interest-ing, since it does not involve hypercohomology.

COROLLARY 2.9.2(The affine case) LetX be a smooth affine variety defined overthe complex numbers and

(Ω•

alg(X), d)

the complex of algebraic differential forms onX . Then the singular cohomology withC coefficients of its underlying complex man-ifold Xan can be computed as the cohomology of its complex of algebraicdifferentialforms:

Hk(Xan,C) ≃ hk(Ω•

alg(X)).

It is important to note that the left-hand side is the singular cohomology of thecomplex manifoldXan, not of the affine varietyX . In fact, in the Zariski topology, aconstant sheaf on an irreducible variety is always flasque (Example 2.2.6), and henceacyclic (Corollary 2.2.5), so thatHk(X,C) = 0 for all k > 0 if X is irreducible.

2.9.1 Proof that the General Case Implies the Affine Case

Assume Theorem 2.9.1. It suffices to prove that for a smooth affine complex vari-etyX , the hypercohomologyHk(X,Ω•

alg) reduces to the cohomology of the complexΩ•

alg(X). SinceΩqalg is a coherent algebraic sheaf, by Serre’s vanishing theoremfor an

affine variety (Theorem 2.3.11),Ωqalg is acyclic onX . By Theorem 2.4.2,

Hk(X,Ω•

alg) ≃ hk(Ω•

alg(X)).

2.9.2 Proof that the Affine Case Implies the General Case

Assume Corollary 2.9.2. The proof is based on the facts that every algebraic varietyX has anaffine open cover, an open coverU = Uα in which everyUα is an affineopen set, and that the intersection of two affine open sets is affine open. The existenceof an affine open cover for an algebraic variety follows from the elementary fact thatevery quasi-projective variety has an affine open cover; since an algebraic variety bydefinition has an open cover by quasi-projective varieties,it necessarily has an opencover by affine varieties.

SinceΩ•alg is a complex of locally free and hence coherent algebraic sheaves, by

Serre’s vanishing theorem for an affine variety (Theorem 2.3.11),Ω•alg is acyclic on an

affine open cover. By Theorem 2.8.1, there is an isomorphism

H∗(U,Ω•alg) ≃ H

∗(X,Ω•alg) (2.9.1)

between theCech cohomology ofΩ•alg on the affine open coverU and the hyperco-

homology ofΩ•alg onX . Similarly, by Cartan’s theorem B (because a complex affine



109

variety with the complex topology is Stein) and Theorem 2.8.1, the corresponding state-ment in the analytic category is also true: ifUan := (Uα)an, then

H∗(Uan,Ω•an) ≃ H

∗(Xan,Ω•an). (2.9.2)

TheCech cohomologyH∗(U,Ω•alg) is the cohomology of the single complex as-

sociated to the double complex⊕

Kp,qalg =

⊕Cp(U,Ωq

alg). TheE1 term of the usualspectral sequence of this double complex is

Ep,q1,alg

= Hp,qd = hq

d(Kp,•) = hq

d

(Cp(U,Ω•

alg))

= hqd

( ∏

α0<···<αp

Ω•alg(Uα0···αp

))

=∏

α0<···<αp

hqd

(Ω•

alg(Uα0···αp))

=∏

α0<···<αp

Hq(Uα0···αp,an,C) (by Corollary 2.9.2).

A completely similar computation applies to the usual spectral sequence of thedouble complex

⊕Kp,q

an =⊕

p,q Cp(Uan,Ω

qan) converging to theCech cohomology

H∗(Uan,Ω•an): theE1 term of this spectral sequence is

Ep,q1,an =

∏

α0<···<αp

hqd

(Ω•

an(Uα0···αp,an))

=∏

α0<···<αp

Hq(Uα0···αp,an,C) (by Corollary 2.5.3).

The isomorphism inE1 terms,

E1,alg

∼→ E1,an,

commutes with theCech differentiald1 = δ and induces an isomorphism inE∞ terms,

E∞,alg∼ // E∞,an

H∗(U,Ω•alg) H∗(Uan,Ω

•an).

Combined with (2.9.1) and (2.9.2), this gives

H∗(X,Ω•

alg) ≃ H∗(Xan,Ω

•an),

which, as we have seen, is equivalent to the algebraic de Rhamtheorem (Theorem 2.9.1)for a smooth complex algebraic variety.

110


CHAPTER 2

2.10 THE ALGEBRAIC DE RHAM THEOREM FOR AN AFFINE VARIETY

It remains to prove the algebraic de Rham theorem in the form of Corollary 2.9.2 for asmooth affine complex varietyX . This is the most difficult case and is in fact the heartof the matter. We give a proof that is different from Grothendieck’s in [7].

A normal crossing divisoron a smooth algebraic variety is a divisor that is locallythe zero set of an equation of the formz1 · · · zk = 0, wherez1, . . . , zN are local pa-rameters. We first describe a standard procedure by which anysmooth affine varietyX may be assumed to be the complement of a normal crossing divisor D in a smoothcomplex projective varietyY . Let X be the projective closure ofX ; for example, ifXis defined by polynomial equations

fi(z1, . . . , zN) = 0

in CN , thenX is defined by the equations

fi

(Z1

Z0

, . . . ,ZN

Z0

)= 0

in CPN , whereZ0, . . . , ZN are the homogeneous coordinates onCPN and zi =Zi/Z0. In general,X will be a singular projective variety. By Hironaka’s resolutionof singularities, there is a surjective regular mapπ : Y → X from a smooth projec-tive varietyY to X such thatπ−1(X − X) is a normal crossing divisorD in Y andπ|Y−D : Y − D → X is an isomorphism. Thus, we may assume thatX = Y − D,with an inclusion mapj : X → Y .

Let ΩkYan

(∗D) be the sheaf of meromorphick-forms onYan that are holomorphiconXan with poles of any order≥ 0 alongDan (order0 means no poles) and letAk

Xan

be the sheaf ofC∞ complex-valuedk-forms onXan. By abuse of notation, we usej also to denote the inclusionXan → Yan. The direct image sheafj∗Ak

Xanis by

definition the sheaf onYan defined by(j∗A

kXan

)(V ) = Ak

Xan(V ∩Xan)

for any open setV ⊂ Yan. Since a section ofΩkYan

(∗D) overV is holomorphic onV ∩ Xan and therefore smooth there, the sheafΩk

Yan(∗D) of meromorphic forms is a

subsheaf of the sheafj∗AkXan

of smooth forms. The main lemma of our proof, due toHodge and Atiyah [10, Lem. 17, p. 77], asserts that the inclusion

Ω•Yan

(∗D) → j∗A•Xan

(2.10.1)

of complexes of sheaves is a quasi-isomorphism. This lemma makes essential use ofthe fact thatD is a normal crossing divisor. Since the proof of the lemma is quitetechnical, in order not to interrupt the flow of the exposition, we postpone it to the endof the chapter.

By Theorem 2.4.1, the quasi-isomorphism (2.10.1) induces an isomorphism

Hk(Yan,Ω

•Yan

(∗D))≃ H

k(Yan, j∗A•Xan

) (2.10.2)



111

in hypercohomology. If we can show that the right-hand side is Hk(Xan,C) and theleft-hand side ishk

(Ω•

alg(X)), the algebraic de Rham theorem for the affine varietyX

(Corollary 2.9.2),hk(Ω•

alg(X))≃ Hk(Xan,C), will follow.

2.10.1 The Hypercohomology of the Direct Image of a Sheaf of Smooth Forms

To deal with the right-hand side of (2.10.2), we prove a more general lemma valid onany complex manifold.

L EMMA 2.10.1 Let M be a complex manifold andU an open submanifold, withj : U → M the inclusion map. Denote the sheaf of smoothC-valuedk-forms onU byAk

U . Then there is an isomorphism

Hk(M, j∗A

•U ) ≃ Hk(U,C).

PROOF. Let A0 be the sheaf of smoothC-valued functions on the complex mani-foldM . For any open setV ⊂ M , there is anA0(V )-module structure on(j∗Ak

U )(V ) =Ak

U (U ∩ V ):

A0(V )×AkU (U ∩ V ) → Ak

U (U ∩ V ),

(f, ω) 7→ f · ω.

Hence,j∗AkU is a sheaf ofA0-modules onM . As such,j∗Ak

U is a fine sheaf onM(Section 2.2.5).

Since fine sheaves are acyclic, by Theorem 2.4.2,

Hk(M, j∗A

•U ) ≃ hk

((j∗A

•U )(M)

)

= hk(A•

U (U))

(definition ofj∗A•U )

= Hk(U,C) (by the smooth de Rham theorem).

Applying the lemma toM = Yan andU = Xan, we obtain

Hk(Yan, j∗A

•Xan

) ≃ Hk(Xan,C).

This takes care of the right-hand side of (2.10.2).

2.10.2 The Hypercohomology of Rational and Meromorphic Forms

Throughout this subsection, the smooth complex affine variety X is the complement ofa normal crossing divisorD in a smooth complex projective varietyY . LetΩq

Yan(nD)

be the sheaf of meromorphicq-forms onYan that are holomorphic onXan with polesof order≤ n alongDan. As before,Ωq

Yan(∗D) is the sheaf of meromorphicq-forms

onYan that are holomorphic onXan with at most poles (of any order) alongD. Sim-ilarly, Ωq

Y (∗D) andΩqY (nD) are their algebraic counterparts, the sheaves of rational

112


CHAPTER 2

q-forms onY that are regular onX with poles alongD of arbitrary order or order≤ nrespectively. Then

ΩqYan

(∗D) = lim−→n

ΩqYan

(nD) and ΩqY (∗D) = lim

−→n

ΩqY (nD).

Let ΩqX andΩq

Y be the sheaves of regularq-forms onX andY , respectively; theyare what would be writtenΩq

alg if there is only one variety. Similarly, letΩqXan

andΩq

Yanbe sheaves of holomorphicq-forms onXan andYan, respectively. There is an-

other description of the sheafΩqY (∗D) that will prove useful. Since a regular form on

X = Y − D that is not defined onD can have at most poles alongD (no essentialsingularities), ifj : X → Y is the inclusion map, then

j∗ΩqX = Ωq

Y (∗D).

Note that the corresponding statement in the analytic category is not true: ifj : Xan →Yan now denotes the inclusion of the corresponding analytic manifolds, then in general

j∗ΩqXan

6= ΩqYan

(∗D)

because a holomorphic form onXan that is not defined alongDan may have an essentialsingularity onDan.

Our goal now is to prove that the hypercohomologyH∗(Yan,Ω

•Yan

(∗D))

of thecomplexΩ•

Yan(∗D) of sheaves of meromorphic forms onYan is computable from the

algebraic de Rham complex onX :

Hk(Yan,Ω

•Yan

(∗D))≃ hk

(Γ(X,Ω•

alg)).

This will be accomplished through a series of isomorphisms.First, we prove something akin to a GAGA principle for hypercohomology. The

proof requires commuting direct limits and cohomology, forwhich we shall invokethe following criterion. A topological space is said to benoetherianif it satisfies thedescending chain condition for closed sets: any descendingchainY1 ⊃ Y2 ⊃ · · ·of closed sets must terminate after finitely many steps. As shown in a first course inalgebraic geometry, affine and projective varieties are noetherian [9, Exa. 1.4.7, p. 5;Exer.1.7(b), p. 8; Exer. 2.5(a), p. 11].

PROPOSITION 2.10.2(Commutativity of direct limit with cohomology) Let(Fα) bea direct system of sheaves on a topological spaceZ. The natural map

lim−→

Hk(Z,Fα) → Hk(Z, lim−→

Fα)

is an isomorphism if

(i) Z is compact; or

(ii) Z is noetherian.



113

PROOF. For (i), see [10, Lem. 4, p. 61]. For (ii), see [9, Ch. III, Prop. 2.9, p. 209]or [5, Ch. II, remark after Th. 4.12.1, p. 194].

PROPOSITION 2.10.3 In the notation above, there is an isomorphism in hypercoho-mology

H∗(Y,Ω•

Y (∗D))≃ H

∗(Yan,Ω

•Yan

(∗D)).

PROOF. SinceY is a projective variety and eachΩ•Y (nD) is locally free, we can

apply Serre’s GAGA principle (2.3.1) to get an isomorphism

Hp(Y,Ωq

Y (nD))≃ Hp

(Yan,Ω

qYan

(nD)).

Next, take the direct limit of both sides asn → ∞. Since the projective varietyY isnoetherian and the complex manifoldYan is compact, by Proposition 2.10.2, we obtain

Hp(Y, lim

−→n

ΩqY (nD)

)≃ Hp

(Yan, lim−→

n

ΩqYan

(nD)),

which isHp(Y,Ωq

Y (∗D))≃ Hp

(Yan,Ω

qYan

(∗D)).

Now the two cohomology groupsHp(Y,Ωq

Y (∗D))

andHp(Yan,Ω

qYan

(∗D))

aretheE1 terms of the second spectral sequences of the hypercohomologies ofΩ•

Y (∗D)andΩ•

Yan(∗D), respectively (see (2.4.4)). An isomorphism of theE1 terms induces an

isomorphism of theE∞ terms. Hence,

H∗(Y,Ω•

Y (∗D))≃ H

∗(Yan,Ω

•Yan

(∗D)).

PROPOSITION 2.10.4 In the notation above, there is an isomorphism

Hk(Y,Ω•

Y (∗D))≃ H

k(X,Ω•X)

for all k ≥ 0.

PROOF. If V is an affine open set inY , thenV is noetherian and so by Proposi-tion 2.10.2(ii), forp > 0,

Hp(V,Ωq

Y (∗D))= Hp

(V, lim

−→n

ΩqY (nD)

)

≃ lim−→n

Hp(V,Ωq

Y (nD))

= 0,

the last equality following from Serre’s vanishing theorem(Theorem 2.3.11), sinceV isaffine andΩq

Y (nD) is locally free and therefore coherent. Thus, the complex ofsheavesΩ•

Y (∗D) is acyclic on any affine open coverU = Uα of Y . By Theorem 2.8.1, itshypercohomology can be computed from itsCech cohomology:

Hk(Y,Ω•

Y (∗D))≃ Hk

(U,Ω•

Y (∗D)).

114


CHAPTER 2

Recall that ifj : X → Y is the inclusion map, thenΩ•Y (∗D) = j∗Ω

•X . By def-

inition, the Cech cohomologyHk(U,Ω•

Y (∗D))

is the cohomology of the associatedsingle complex of the double complex

Kp,q = Cp(U,Ωq

Y (∗D))= Cp(U, j∗Ω

qX)

=∏

α0<···<αp

Ωq(Uα0···αp∩X). (2.10.3)

Next we compute the hypercohomologyHk(X,Ω•X). The restrictionU|X := Uα∩

X of U toX is an affine open cover ofX . SinceΩqX is locally free [14, Ch. III, Th. 2,

p. 200], by Serre’s vanishing theorem for an affine variety again,

Hp(Uα ∩X,ΩqX) = 0 for all p > 0.

Thus, the complex of sheavesΩ•X is acyclic on the open coverU|X of X . By Theo-

rem 2.8.1,H

k(X,Ω•X) ≃ Hk(U|X ,Ω•

X).

TheCech cohomologyHk(U|X ,Ω•X) is the cohomology of the single complex associ-

ated to the double complex

Kp,q = Cp(U|X ,ΩqX)

=∏

α0<···<αp

Ωq(Uα0···αp∩X). (2.10.4)

Comparing (2.10.3) and (2.10.4), we get an isomorphism

Hk(Y,Ω•

Y (∗D))≃ H

k(X,Ω•X)

for everyk ≥ 0.

Finally, becauseΩqX is locally free, by Serre’s vanishing theorem for an affine va-

riety still again,Hp(X,ΩqX) = 0 for all p > 0. Thus,Ω•

X is a complex of acyclicsheaves onX . By Theorem 2.4.2, the hypercohomologyHk(X,Ω•

X) can be computedfrom the complex of global sections ofΩ•

X :

Hk(X,Ω•

X) ≃ hk(Γ(X,Ω•

X))= hk

(Ω•

alg(X)

). (2.10.5)

Putting together Propositions 2.10.3 and 2.10.4 with (2.10.5), we get the desiredinterpretation

Hk(Yan,Ω

•Yan

(∗D))≃ hk

(Ω•

alg(X)

)

of the left-hand side of (2.10.2). Together with the interpretation of the right-hand sideof (2.10.2) asHk(Xan,C), this gives Grothendieck’s algebraic de Rham theorem foran affine variety,

Hk(Xan,C) ≃ hk(Ω•

alg(X)

).



115

2.10.3 Comparison of Meromorphic and Smooth Forms

It remains to prove that (2.10.1) is a quasi-isomorphism. Wewill reformulate the lemmain slightly more general terms. LetM be a complex manifold of complex dimensionn, let D be a normal crossing divisor inM , and letU = M − D be the complementof D in M , with j : U → M the inclusion map. Denote byΩq

M (∗D) the sheaf ofmeromorphicq-forms onM that are holomorphic onU with at most poles alongD,and byAq

U := AqU ( ,C) the sheaf of smoothC-valuedq-forms onU . For eachq, the

sheafΩqM (∗D) is a subsheaf ofj∗A

qU .

L EMMA 2.10.5(Fundamental lemma of Hodge and Atiyah ([10, Lem. 17, p. 77]))The inclusionΩ•

M (∗D) → j∗A•U of complexes of sheaves is a quasi-isomorphism.

PROOF. We remark first that this is alocal statement. Indeed, the main advantageof using sheaf theory is to reduce the global statement of thealgebraic de Rham theoremfor an affine variety to a local result. The inclusionΩ•

M (∗D) → j∗A•U of complexes

induces a morphism of cohomology sheavesH∗(Ω•

M (∗D))→ H∗(j∗A

•U ). It is a

general fact in sheaf theory that a morphism of sheaves is an isomorphism if and onlyif its stalk maps are all isomorphisms [9, Prop. 1.1, p. 63], so we will first examinethe stalks of the sheaves in question. There are two cases:p ∈ U andp ∈ D. Forsimplicity, letΩq

p := (ΩqM )p be the stalk ofΩq

M atp ∈ M and letAqp := (Aq

U )p be thestalk ofAq

U atp ∈ U .Case 1:At a pointp ∈ U , the stalk ofΩq

M (∗D) isΩqp, and the stalk ofj∗A

qU isAq

p.Hence, the stalk maps of the inclusionΩ•

M (∗D) → j∗A•U atp are

0 // Ω0p

//

Ω1p

//

Ω2p

//

· · ·

0 // A0p

// A1p

// A2p

// · · · .

(2.10.6)

Being a chain map, (2.10.6) induces a homomorphism in cohomology. By the holo-morphic Poincare lemma (Theorem 2.5.1), the cohomology ofthe top row of (2.10.6)is

hk(Ω•p) =

C for k = 0,

0 for k > 0.

By the complex analogue of the smooth Poincare lemma ([3, Sec. 4, p. 33] and [6,p. 38]), the cohomology of the bottom row of (2.10.6) is

hk(A•p) =

C for k = 0,

0 for k > 0.

Since the inclusion map (2.10.6) takes1 ∈ Ω0p to 1 ∈ A0

p, it is a quasi-isomorphism.By Proposition 2.2.9, forp ∈ U ,

Hk(Ω•

M (∗D))p≃ hk

((Ω•

M (∗D))p)= hk(Ω•

p)

116


CHAPTER 2

andHk(j∗A

•U )p ≃ hk

((j∗A

•U )p)= hk(A•

p).

Therefore, by the preceding paragraph, atp ∈ U the inclusionΩ•M (∗D) → j∗A

•U

induces an isomorphism of stalks

Hk(Ω•

M (∗D))p≃ Hk(j∗A

•U )p (2.10.7)

for all k > 0.Case 2:Similarly, we want to show that (2.10.7) holds forp /∈ U , i.e., forp ∈ D.

Note that to show the stalks of these sheaves atp are isomorphic, it is enough to showthe spaces of sections are isomorphic over a neighborhood basis of polydisks.

Choose local coordinatesz1, . . . , zn so thatp = (0, . . . , 0) is the origin andD isthe zero set ofz1 · · · zk = 0 on some coordinate neighborhood ofp. Let P be thepolydiskP = ∆n := ∆× · · · ×∆ (n times), where∆ is a small disk centered at theorigin inC, say of radiusǫ for someǫ > 0. ThenP ∩ U is thepolycylinder

P ∗ := P ∩ U = ∆n ∩ (M −D)

= (z1, . . . , zn) ∈ ∆n | zi 6= 0 for i = 1, . . . , k

= (∆∗)k ×∆n−k,

where∆∗ is the punctured disk∆− 0 in C. Note thatP ∗ has the homotopy type ofthe torus(S1)k. For 1 ≤ i ≤ k, let γi be a circle wrapping once around theith ∆∗.Then a basis for the homology ofP ∗ is given by the submanifolds

∏i∈J γi for all the

various subsetsJ ⊂ [1, k].Since on the polydiskP ,

(j∗A•U )(P ) = A•

U (P ∩ U) = A•(P ∗),

the cohomology of the complex(j∗A•U )(P ) is

h∗((j∗A

•U )(P )

)= h∗

(A•(P ∗)

)

= H∗(P ∗,C) ≃ H∗((S1)k,C

)

=∧([

dz1z1

], . . . ,

[dzkzk

]), (2.10.8)

the free exterior algebra on thek generators[dz1/z1], . . . , [dzk/zk]. Up to a constantfactor of2πi, this basis is dual to the homology basis cited above, as we can see byintegrating over products of loops.

For eachq, the inclusionΩqM (∗D) → j∗A

qU of sheaves induces an inclusion of

groups of sections over a polydiskP :

Γ(P,Ωq

M (∗D))→ Γ(P, j∗A

qU ).

As q varies, the inclusion of complexes

i : Γ(P,Ω•

M (∗D))→ Γ(P, j∗A

•U )



117

induces a homomorphism in cohomology

i∗ : h∗(Γ(P,Ω•

M (∗D)))→ h∗

(Γ(P, j∗A

•U ))=∧([

dz1z1

], . . . ,

[dzkzk

]).

(2.10.9)Since eachdzj/zj is a closed meromorphic form onP with poles alongD, it definesa cohomology class inh∗

(Γ(P,Ω•

M (∗D))). Therefore, the mapi∗ is surjective. If we

could showi∗ were an isomorphism, then by taking the direct limit over allpolydisksP containingp, we would obtain

H∗(Ω•

M (∗D))p≃ H∗(j∗A

•U )p for p ∈ D, (2.10.10)

which would complete the proof of the fundamental lemma (Lemma 2.10.5).We now compute the cohomology of the complexΓ

(P,Ω•

M (∗D)).

PROPOSITION 2.10.6 Let P be a polydisk∆n in Cn, andD the normal crossingdivisor defined inP by z1 · · · zk = 0. The cohomology ringh∗

(Γ(P,Ω•(∗D))

)is

generated by[dz1/z1], . . . , [dzk/zk].

PROOF. The proof is by induction on the numberk of irreducible components ofthe singular setD. Whenk = 0, the divisorD is empty and meromorphic forms onPwith poles alongD are holomorphic. By the holomorphic Poincare lemma,

h∗(Γ(P,Ω•)

)= H∗(P,C) = C.

This proves the base case of the induction.The induction step is based on the following lemma.

L EMMA 2.10.7 LetP be a polydisk∆n, andD the normal crossing divisor definedby z1 · · · zk = 0 in P . Letϕ ∈ Γ

(P,Ωq(∗D)

)be a closed meromorphicq-form onP

that is holomorphic onP ∗ := P − D with at most poles alongD. Then there existclosed meromorphic formsϕ0 ∈ Γ

(P,Ωq(∗D)

)andα1 ∈ Γ

(P,Ωq−1(∗D)

), which

have no poles alongz1 = 0, such that their cohomology classes satisfy the relation

[ϕ] = [ϕ0] +

[dz1z1

]∧ [α1].

PROOF. Our proof is an elaboration of the proof of Hodge and Atiyah [10, Lem. 17,p. 77]. We can writeϕ in the form

ϕ = dz1 ∧ α+ β,

where the meromorphic(q− 1)-formα and theq-formβ do not involvedz1. Next, weexpandα andβ as Laurent series inz1:

α = α0 + α1z−11 + α2z

−21 + · · ·+ αrz

−r1 ,

β = β0 + β1z−11 + β2z

−21 + · · ·+ βrz

−r1 ,

118


CHAPTER 2

whereαi andβi for 1 ≤ i ≤ r do not involvez1 or dz1 and are meromorphic inthe other variables, andα0, β0 are holomorphic inz1, are meromorphic in the othervariables, and do not involvedz1. Then

ϕ = (dz1 ∧ α0 + β0) +

(dz1 ∧

r∑

i=1

αiz−i1 +

r∑

i=1

βiz−i1

).

Setϕ0 = dz1 ∧ α0 + β0. By comparing the coefficients ofz−i1 dz1 andz−i

1 , wededuce from the conditiondϕ = 0,

dα1 = dα2 + β1 = dα3 + 2β2 = · · · = rβr = 0,dβ1 = dβ2 = dβ3 = · · · = dβr = 0,

anddϕ0 = 0.We can write

ϕ = ϕ0 +dz1z1

∧ α1 +

(dz1 ∧

r∑

i=2

αiz−i1 +

r∑

i=1

βiz−i1

). (2.10.11)

It turns out that the term within the parentheses in (2.10.11) is dθ for

θ = −α2

z1−

α3

2z21− · · · −

αr

(r − 1)zr−11

.

In (2.10.11), bothϕ0 andα1 are closed. Hence, the cohomology classes satisfy therelation

[ϕ] = [ϕ0] +

[dz1z1

]∧ [α1].

Sinceϕ0 andα1 are meromorphic forms which do not have poles alongz1 = 0,their singularity set is contained in the normal crossing divisor defined byz2 · · · zk = 0,which hask − 1 irreducible components. By induction, the cohomology classes ofϕ0

andα1 are generated by[dz2/z2], . . . , [dzk/zk]. Hence,[ϕ] is a graded-commutativepolynomial in[dz1/z1], . . . , [dzk/zk]. This completes the proof of Proposition 2.10.6.

PROPOSITION 2.10.8 Let P be a polydisk∆n in Cn, andD the normal crossingdivisor defined byz1 · · · zk = 0 in P . Then there is a ring isomorphism

h∗(Γ(P,Ω•(∗D))

)≃∧([

dz1z1

], . . . ,

[dzkzk

]).

PROOF. By Proposition 2.10.6, the graded-commutativealgebrah∗(Γ(P,Ω•(∗D))

)

is generated by[dz1/z1], . . . , [dzk/zk]. It remains to show that these generators sat-isfy no algebraic relations other than those implied by graded commutativity. Letωi = dzi/zi andωI := ωi1···ir := ωi1 ∧ · · · ∧ ωir . Any linear relation among the



119

cohomology classes[ωI ] in h∗(Γ(P,Ω•(∗D))

)would be, on the level of forms, of the

form ∑cIωI = dξ (2.10.12)

for some meromorphic formξ with at most poles alongD. But by restriction toP −D,this would give automatically a relation inΓ(P, j∗A

qU ). Sinceh∗

(Γ(P, j∗A

qU ))=∧

([ω1], . . . , [ωk]) is freely generated by[ω1], . . . , [ωk] (see (2.10.8)), the only possiblerelations (2.10.12) are all implied by graded commutativity.

Since the inclusionΩ•M (∗D) → j∗A

∗U induces an isomorphism

H∗(Ω•

M (∗D))p≃ H∗(j∗A

•U )p

of stalks of cohomology sheaves for allp, the inclusionΩ•M (∗D) → j∗A

∗U is a quasi-

isomorphism. This completes the proof of Lemma 2.10.5.


Bibliography

[1] Alberto Arabia and Zoghman Mebkhout. Introduction a lacohomologie de deRham. Preprint, 260 pages, 1997.

[2] Michael F. Atiyah and Ian G. Macdonald.Introduction to Commutative Algebra.Addison–Wesley, Reading, Massachusetts, 1969.

[3] Raoul Bott and Loring W. Tu.Differential Forms in Algebraic Topology, thirdcorrected printing. Springer, New York, 1995.

[4] Henri Cartan. Varietes analytiques complexes et cohomologie.Colloque de Brux-elles, 41–55, 1953.

[5] Roger Godement.Topologie algebrique et theorie des faisceaux. Hermann, Paris,1958.

[6] Phillip Griffiths and Joe Harris.Principles of Algebraic Geometry. John Wiley,New York, 1978.

[7] Alexandre Grothendieck. On the de Rham cohomology of algebraic varieties.Publ. Math. IHES, 29: 95–103, 1966.

[8] Robert Gunning and Hugo Rossi.Analytic Functions of Several Complex Vari-ables. Prentice-Hall, Englewood Cliffs, NJ, 1965.

[9] Robin Hartshorne.Algebraic Geometry, volume 52 ofGraduate Texts in Mathe-matics. Springer, New York, 1977.

[10] W. V. D. Hodge and Michael F. Atiyah. Integrals of the second kind on an alge-braic variety.Annals of Math., 62:56–91, 1955.

[11] Lars Hormander.An Introduction to Complex Analysis in Several Variables, thirdedition. North-Holland, New York, 1990.

[12] Jean-Pierre Serre. Faisceaux algebriques coherents.Annals of Math., 61:197–278,1955.

[13] Jean-Pierre Serre. Geometrie analytique et geometrie algebrique. Ann. Inst.Fourier, 6:1–42, 1956.

[14] Igor R. Shafarevich.Basic Algebraic Geometry, two volumes, second edition.Springer, New York, 1994.

BIBLIOGRAPHY


121

[15] Loring W. Tu. An Introduction to Manifolds, second edition. Universitext.Springer, New York, 2011.

[16] Frank W. Warner.Foundations of Differentiable Manifolds and Lie groups, vol-ume 94 ofGraduate Texts in Mathematics. Springer, New York, 1983.

Documents

Chapter TwoCHAPTER 2 the language of schemes, there is no harm in taking an algebraic variety to be a quasi-projective variety; the proofs of the algebraic de Rham theorem are exactly