Cohomology and hypercohomology

5.1 Definition of cohomology

Let X be a topological space. We list some basic definitions concerningsheaves on X. If F is a sheaf of R-modules on X, we denote by Γ(X;F )the R-module of global sections of F . For a complex (C·, d) of sheaves (ofR-modules) on X, the convention will always be that d has degree +1. Wewill usually just write C· when the differential d is clear from the context.By convention, all complexes of sheaves will be bounded below.Finally, for a complex of sheaves as above, Hi(C·, d), or just HiC· when dis clear from the context, denotes the ith cohomology sheaf of the complex.

Definition 1. (i) Let (C·, d) and (D·, d) be two complexes of sheaves onX. A quasi-isomorphism f : C· → D· is a morphism of complexesof sheaves which induces an isomorphism on the cohomology sheaves.The complexes (C·, d) and (D·, d) are quasi-isomorphic if there existssome quasi-isomorphism f : C· → D·.

(ii) Two morphisms f : C· → D·, g : C· → D· are chain homotopic if thereexists an R-module homomorphism T : C· → D· of degree −1 suchthat f − g = dT + Td.

(iii) A sheaf I on X is injective if it is an injective object in the abeliancategory of sheaves of R-modules on X: given sheaves F and G on Xand a commutative diagram

0 −−−−→ F −−−−→ G

f

yI

there exists a map f : G→ I making the diagram commute.

(iv) A resolution I · of a sheaf F is a complex I · with Ik = 0 for k < 0,together with an injection F → I0, such that

0→ F → I0 → I1 → · · ·

is exact. In particular, the map F [0] → I · is a quasi-isomorphism,where F [0] denotes the complex with a single term in degree 0 con-sisting of F . An injective resolution I · of F is a resolution such thatIk is injective for all k.

1

(v) More generally, if C· is a complex of sheaves, then an injective reso-lution of C· is an injective quasi-isomorphism f : C· → I ·, where Ik isinjective for all k.

Lemma 2. (i) Let F be a sheaf on X. Then an injective resolution (I ·, d)of F exists.

(ii) If I · is a resolution of F and J · is an injective resolution of F , thenthere exists a quasi-isomorphism f : I · → J · which is compatible withthe two inclusions F → I0, F → J0.

(iii) If I · and J · are two injective resolutions of F and f : I · → J · andg : I · → J · are two quasi-isomorphisms compatible with the inclusionsof F in degree 0, then f and g are chain homotopic.

Definition 3. If F is a sheaf of R-modules on X, let I · be an injectiveresolution of F and define

H i(X;F ) = H i(Γ(X; I ·), d).

It is independent of the choice of injective resolution I · by the above lemma,in the sense that if J · is another such, then there is a canonical isomorphismH i(Γ(X; I ·), d) ∼= H i(Γ(X; J ·), d). Finally the sheaf cohomology groupsH i(X;F ) satisfy the usual properties. In particular, H0(X;F ) = Γ(X;F )and, given a short exact sequence 0→ F ′ → F → F ′′ → 0 of sheaves on X,there is a long exact cohomology sequence

· · · → H i−1(X;F ′′)δ−→ H i(X;F ′)→ H i(X;F )→ H i(X;F ′′)

δ−→ H i+1(X;F ′)→ · · · ,

where the connecting homomorphisms δ are also functorial in a natural sensewith respect to commutative diagrams of short exact sequences of complexesof sheaves on X.

Remark 4. The cohomology groups Hk(X;F ) are the derived functors ofthe global section functor Γ.

In practice, it is hard to work with injective resolutions, and we needother methods for computing cohomology.

Definition 5. Let K be a sheaf on X. Then K is acyclic if H i(X;K) = 0for all i > 0. An acyclic resolution (K·, d) of a sheaf F is a resolutionF → K· such that Ki is acyclic for every i.

2

Lemma 6. Let (K·, d) be an acyclic resolution of a sheaf F . Then, for alli ≥ 0,

H i(X;F ) ∼= H i(Γ(X;K·), d).

Proof. Since there is an exact sequence

0→ F → K0 → K1,

and the global section functor is left exact, we see that

H0(X;F ) = Kerd : H0(X;K0)→ H0(X;K1).

Also, using the exact sequence

0→ F → K0 → K0/F → 0

and the vanishing of H1(X;K0), we see that

H1(X;F ) = Cokerd : H0(X;K0)→ H0(X;K0/F ).

Moreover, using the exact sequence

0→ K0/F → K1 → K2,

we see that

H0(X;K0/F ) = Kerd : H0(X;K1)→ H0(X;K2).

Thus H1(X;F ) is isomorphic to

Kerd : H0(X;K1)→ H0(X;K2)/ Imd : H0(X;K0)→ H0(X;K1)= H1(Γ(X;K·), d).

Finally, for k ≥ 2, we argue by induction on k, assuming inductively thatthe result has been proved for all k − 1 ≥ 1 and all sheaves G. Then, giventhe sheaf F , there is the induced acyclic resolution of K0/F :

0→ K0/F → K1 → K2 → · · · .

Moreover, the long exact cohomology sequence associated to

0→ F → K0 → K0/F → 0

and the vanishing of H i(X;K0) for i ≥ 1 shows that, for k ≥ 2,

Hk(X;F ) ∼= Hk−1(X;K0/F ) ∼= Hk−1(Γ(X;K·+1), d) = Hk(Γ(X;K·), d).

This completes the inductive step.

3

5.2 Some examples of acyclic sheaves

To make use of Lemma 6, we need to produce some examples of acyclicsheaves. Of course, an injective sheaf I is acyclic, because the complex I[0]which is I in degree zero and zero elsewhere is an injective resolution of I.Hence, H i(X; I) = 0 for all i > 0. The following gives another very broadclass of acyclic sheaves.

Definition 7. A sheaf F is flasque if, for every pair of open subsets U ⊆ Vof X, the restriction morphism Γ(V ;F )→ Γ(U ;F ) is surjective.

The method of constructing injective resolutions given in Hartshorneshows that, given an arbitrary sheaf F , there exists an injective resolution(I ·, d) of F such that all of the Ik are flasque. However, this is a somewhatsuperfluous argument since the following is easy to show directly:

Lemma 8. An injective sheaf is flasque.

A Zorn’s lemma argument shows:

Lemma 9. Let F be flasque and suppose that there is an exact sequence ofsheaves

0→ F → A→ B → 0.

Then the induced map Γ(X;A)→ Γ(X;B) is surjective, i.e.

0→ H0(X;F )→ H0(X;A)→ H0(X;B)→ 0

is exact.

Straightforward arguments also show:

Lemma 10. (i) Given an exact sequence

0→ F ′ → F → F ′′ → 0,

if F ′ and F are flasque, then so F ′′.

(ii) Suppose that0→ F 0 → F 1 → · · ·

is an exact sequence of flasque sheaves. Then the associated sequenceof global sections

0→ Γ(X;F 0)→ Γ(X;F 1)→ · · ·

is also exact.

4

Finally, we have:

Lemma 11. An flasque sheaf is acyclic.

Proof. There exists an injective sheaf I and an injective homomorphismF → I. Thus we have an exact sequence

0→ F → I → I/F → 0.

By Lemma 9, the map H0(X; I) → H0(X; I/F ) is surjective and thusH1(X;F ) = 0 since H1(X; I) = 0. Now assume inductively that we haveshown, for all k−1 ≥ 1, and for all flasque sheaves G, that Hk−1(X;G) = 0.For k ≥ 2, we have an isomorphism

Hk(X;F ) ∼= Hk−1(X; I/F ).

By (i) of Lemma 10, I/F is flasque. Hence, by the inductive assumption,Hk−1(X; I/F ) = 0 for all k ≥ 2. Hence Hk(X;F ) = 0 as well, completingthe inductive step.

The usefulness of flasque sheaves is that it enables us to construct acanonical flasque resolution (due to Godement).

Definition 12. Let F be a sheaf on X, and define C0Go(F ), the sheaf of

germs of discontinuous sections of F , by: for every open subset U of X,

C0Go(F )(U) =

∏x∈U

Fx.

If V ⊆ U are two open subsets, there is the obvious restriction map

C0Go(F )(U)→ C0

Go(F )(V ),

making C0Go(F ) a sheaf (not just a presheaf) of R-modules on X. Clearly,

C0Go(F ) is flasque. There is a natural injection of sheaves F → C0

Go(F ),which associates to a section s ∈ F (U) the element (sx) ∈

∏x∈U Fx. Then

we defineC1

Go(F ) = C0Go(C0

Go(F )/F ),

with a natural map

d : C0Go(F )→ C0

Go(F )/F → C0Go(C0

Go(F )/F ) = C1Go(F ).

Inductively, suppose that we have defined CrGo(F ) and the map Cr−1Go (F )→

CrGo(F ). Then set

Cr+1Go (F ) = C0

Go(CrGo(F )/ ImCr−1Go (F )).

5

We then have the composition

dGo : CrGo(F )→ CrGo(F )/ ImCr−1Go (F )→ Cr+1

Go (F ),

with the first map surjective with kernel ImCr−1Go (F ) and the second map

injective. It follows that (C·Go(F ), dGo) is exact except in dimension zero,

and is a resolution of F by flasque sheaves. We will refer to it as the canonicalflasque resolution or Godement resolution of F .

We can view (C·Go(·), dGo) as a functor from the abelian category of

sheaves on X to the category of complexes on X, because a homomorphismf : F → G defines a morphism of complexes f : C·

Go(F ) → C·Go(G) in a

functorial way. Then:

Lemma 13. (i) The functor (C·Go(·), dGo) is exact. In other words, given

an exact sequenceF ′ → F → F ′′

of sheaves on X, the sequence of complexes

C·Go(F ′)→ C·

Go(F )→ C·Go(F ′′)

is exact.

(ii) If α : F ′ → F is a homomorphism of sheaves and α : C·Go(F ′) →

C·Go(F ) is the induced homomorphism, then

Ker α = C·Go(Kerα) and Im α = C·

Go(Imα).

(iii) The functor (C·Go(·), dGo) commutes with taking cohomology. In other

words, given a complex of sheaves on X

· · · δ−→ F i−1 δ−→ F iδ−→ F i+1 δ−→ · · · ,

and if δ is the induced map C·Go(F i) → C·

Go(F i+1), then the corre-sponding cohomology sheaves satisfy

Hi(C·Go(F ·), δ) = C·

Go(HiF ·).

Proof. (i) Clearly the sequence

C0Go(F ′)→ C0

Go(F )→ C0Go(F ′′)

6

is exact. To prove (i), given the inductive construction of C·Go(F ), it suffices

to show that the sequence

C0Go(F ′)/F ′ → C0

Go(F )/F → C0Go(F ′′)/F ′′

is exact. This is a straightforward exercise.

(ii) We have an exact sequence

0→ Kerα→ F ′α−→ F.

Using (i), the sequence

0→ C·Go(Kerα)→ C·

Go(F ′)α−→ C·

Go(F )

is also exact. Thus Ker α = C·Go(Kerα). As for Imα, we have a commutative

diagramF ′ −−−−→ Imα −−−−→ 0

α

y yF

=−−−−→ F

Thus, using (i) again, there is a corresponding diagram

C·Go(F ′) −−−−→ C·

Go(Imα) −−−−→ 0

α

y yC·

Go(F )=−−−−→ C·

Go(F )

It then follows easily that the image of C·Go(Imα) is contained in Im α and

that the corresponding homomorphism C·Go(Imα)→ Im α is both injective

and surjective, hence an isomorphism.

(iii) This follows easily from (ii).

Here (ii) and (iii) are general properties of exact functors.The other class of acyclic sheaves we will consider are the fine sheaves.

Definition 14. A sheaf K on X is fine if, for every open cover Uα of X,there exists a family ϕα : K → K of endomorphisms of K, satisfying

1. The support of ϕα is contained in Uα. In other words, there is a closedsubset Sα of Uα such that ϕα|X − Sα = 0.

2. (Local finiteness). For all x ∈ X, there exists an open neighborhoodV of x such that there are only finitely many α with ϕα|V 6= 0.

7

3.∑

α ϕα = Id. (Here the sum is meaningful by (2).)

The primary example of fine sheaves is given by:

Example 15. If M is a C∞ manifold and C∞M = A0M denotes the sheaf of

rings of C∞ functions on M , then, for every open cover Uα of M , thereexists a C∞ partition of unity fα subordinate to Uα. Then, if K is anysheaf of C∞M -modules, multiplication by fα defines an endomorphism of Ksatisfying the conditions of the above definition. Thus, such a K is a finesheaf.

In particular, the sheaves AkM of C∞ differential forms on M , as well asthe sheaves Ap,qM in case M is a complex manifold, are fine sheaves.

Proposition 16. A fine sheaf K is acyclic.

Proof. Consider the Godement resolution C·Go(K). We need to show that

the complex Γ(X;C·Go(K)) has no cohomology in positive degrees. Given

k > 0, let ξ ∈ KerΓ(X;CkGo(K)) → Γ(X;Ck+1Go (K)). By local exactness,

there exists an open cover Uα of X such that, for all α, there existsan ηα ∈ Ck−1

Go (K)(Uα) such that dGoηα = ξ|Uα. Let ϕα : K → K be afamily of endomorphisms of K satisfying (1)–(3) of Definition 14 for theopen cover Uα. Since C·

Go is functorial, the ϕα lift to endomorphisms ϕαof the complex C·

Go(K), and it is easy to check that the ϕα satisfy the sameconditions as ϕα. Define

η =∑α

ϕα(ηα).

Here, ϕα(ηα) is a priori only defined on Uα, but by the support condition itextends over X. Also, locally around every point x of X, the sum

∑α ϕα(ηα)

is actually a finite sum, and so the section η is well defined.Then we compute (again with the understanding that this computation

is meaningful on all sufficiently small open sets):

dGoη =∑α

dGoϕα(ηα) =∑α

ϕα(dGoηα)

=∑α

ϕα(ξ) = ξ.

Thus Hk(Γ(X;C·Go(K)), dGo) = 0 for k > 0, so that Hk(X;K) = 0 for

k > 0 and K is acyclic.

Corollary 17 (de Rham isomorphism). If M is a C∞ manifold, thenHk(M ;C) = Hk(A·(M), d). More generally, if V is a flat vector bundleon M , then Hk(M ;V ) = Hk(A·(M ;V ), d).

8

Proof. By the Poincare lemma, (A·M , d) is a resolution of C. Example 15

shows that the AkM are fine sheaves, hence acyclic. By Lemma 6, Hk(M ;C)is computed by taking the cohomology of the complex of global sectionsΓ(M ;A·

M ) = A·(M). The case of a flat vector bundle is similar.

Corollary 18 (Dolbeault isomorphism). If M is a complex manifold, thenHq(M ; Ωp

M ) = Hp,q

∂(M). More generally, if V is a holomorphic vector bun-

dle on M , then Hq(M ; ΩpM ⊗ V ) = Hp,q

∂(M ;V ).

Proof. The proof is the same as the proof of the de Rham isomorphism,using the ∂-Poincare lemma instead of the usual Poincare lemma.

5.3 Cech cohomology and Leray’s theorem

Let Uαα∈I be an open cover of X, where we assume the index set I to betotally ordered, and let F be a sheaf on X. We define the Cech cohomologyof F with respect to the cover Uα in the usual way: for each finite sequenceα0 < α1 < · · · < αk, define

Uα0,α1,...,αk = Uα0 ∩ Uα1 ∩ · · · ∩ Uαk ,

and define the Cech complex C·(Uα;F ) via

Ck(Uα;F ) =∏

α0<α1<···<αk

Γ(Uα0,α1,...,αk , F ).

We define the Cech differential δ : Ck(Uα;F ) → Ck+1(Uα;F ) by: for asection t = (tα0,α1,...,αk) of Ck(Uα, F ),

δtα0,α1,...,αk+1=

k+1∑i=0

(−1)itα0,...,αi,...,αk+1|Uα0,α1,...,αk+1

,

where the notation αi means that this term is omitted. It is straightforwardto check that δ δ = 0. Then we define H ·(Uα;F ) to be the cohomologyof the complex (Ck(Uα;F ), δ). By the sheaf axioms, it is easy to checkthat H0(Uα;F ) = Γ(X;F ).

There is also a sheaf version: define

Ck(Uα;F ) =∏

α0<α1<···<αk

(jα0,α1,...,αk)∗(F |Uα0,α1,...,αk),

where jα0,α1,...,αk : Uα0,α1,...,αk → X is the inclusion. There is also a differ-ential δ : Ck(Uα;F ) → Ck+1(Uα;F ) which is defined by analogy withδ : Ck(Uα;F ) → Ck+1(Uα;F ). A combinatorial argument using thesheaf axioms shows:

9

Proposition 19. The complex (Ck(Uα;F ), δ) is a resolution of F .

Note that Γ(X; C·(Uα;F )) = C·(Uα;F ) (compatibly with δ).

Lemma 20. If F is flasque, then Hk(Uα;F ) = 0 for k > 0.

Proof. Let U be an open subset of X, and let i : U → X be the inclusion.An easy argument shows that, if F is a flasque sheaf on X, then F |U isa flasque sheaf on U , and that, if G is a flasque sheaf on U , then i∗G isa flasque sheaf on X. Thus, the sheaves Ck(Uα;F ) are flasque, so that(C·(Uα;F ), δ) is a flasque resolution of F . It follows that

Hk(X;F ) = Hk(Γ(Ck(Uα;F ), δ)) = Hk(Uα;F ).

Thus Hk(Uα;F ) = 0 for k > 0.

By (ii) of Lemma 2, if I · is an injective resolution of F , there is aninduced morphism of complexes Ck(Uα;F ) → I ·. Taking global sections,there is a morphism of complexes C·(Uα, F ) → Γ(X; I ·) and hence aninduced homomorphism

H ·(Uα;F )→ H ·(X;F ).

With an appropriate notion of refinements, we can define

H ·(X;F ) = lim−→Uα

H ·(Uα;F ).

The homomorphisms H ·(Uα;F )→ H ·(X;F ) fit together to give a homo-morphism H ·(X;F )→ H ·(X;F ).

Theorem 21. For every open cover, H0(Uα;F ) ∼= H0(X;F ). Moreover,H1(X;F ) ∼= H1(X;F ) via the natural homomorphism. If X is paracompact,then, for every k, Hk(X;F ) ∼= Hk(X;F ).

The really computationally useful version of Cech cohomology is thefollowing:

Theorem 22 (Leray’s theorem). Let F be a sheaf on X and let Uα bean open cover such that, for every sequence α0 < α1 < · · · < αk and forevery k > 0, Hk(Uα0,α1,...,αk ;F ) = 0. Then H ·(Uα;F ) → H ·(X;F ) is anisomorphism.

10

Proof. Choose a flasque resolution K· of F or more generally any resolu-tion which has the property that K· and K·|Uα0,α1,...,αk are acyclic for allα0, α1, . . . , αk, and consider the double complex (C·(Uα;K·), δ, d), whereδ is the Cech differential and d is induced by the differential on K·. Forthe associated single complex (s(C·(Uα;K·), D), with D = δ + (−1)pd inbidegree (p, q), we can consider the two spectral sequences associated to thedouble complex C·(Uα;K·). For the one with Ep,q1 given by taking thecohomology first with respect to δ, we get

Ep,q1 = Hq(Uα;Kp) =

0, if q > 0;

Γ(X;Kp), if q = 0,

where we have used Lemma 20 to conclude that Hq(Uα;Kp) = 0 for q > 0.Thus the E2 = E∞ term is Hp(Γ(X;K·), (−1)pd) = Hp(X;F ).

Looking at the other spectral sequence, the E1 term is the d-cohomologyof the complex

C·(Uα;K0)→ C·(Uα;K1)→ · · · .

Since K·|Uα0,α1,...,αk is flasque, it is an acyclic resolution of F |Uα0,α1,...,αk .ThenH ·(Uα0,α1,...,αk ;F ) is the cohomology of the complex Γ(Uα0,α1,...,αk ;K·).But then the d-cohomology of Cp(Uα;K0) in degree q is∏

α0<α1<···<αpHq(Uα0,α1,...,αp , F ).

Since Hq(Uα0,α1,...,αp ;F ) = 0 for q > 0, the d-cohomology of the complexC·(Uα;K·) is just C·(Uα;F ) and the E2 term is H ·(Uα;F ). Compar-ing, we see that H ·(Uα;F ) ∼= H ·(X;F ).

Remark 23. (i) Note that, with our sign conventions, the proof identifiesH ·(Uα;F ) with the cohomology of the complex (Γ(X;K·), (−1)pd), and infact there is an inclusion (Γ(X;K·), (−1)pd) → (s(C·(Uα;K·), D) whichis a quasi-isomorphism by the above proof. Of course, the cohomologyof (Γ(X;K·), (−1)pd) is the same as the cohomology of (Γ(X;K·), d) andindeed the complexes (Γ(X;K·), (−1)pd) and (Γ(X;K·), d) are isomorphicby the map Γ(X;K·)→ Γ(X;K·) which is (−1)p(p−1)/2 Id in degree p.

(ii) Even without the hypothesis that Hk(Uα0,α1,...,αk ;F ) = 0 for k > 0, theproof exhibits a homomorphism

Ep,02 = Hp(Uα;F )→ Ep,0∞ → Hp(X;F )

which is the homomorphism described in the remarks before the statementof Theorem 21.

11

Definition 24. Given a sheaf F on X, an open cover which satisfies thehypotheses of Leray’s theorem will be called a Leray cover for F .

To use Leray’s theorem, we need in general a class of sheaves and Leraycovers for all of the sheaves in the given class. There are three broad classeswhere this applies:

1. The class of constant sheaves and contractible open subsets (for exam-ple open subsets which are homeomorphic to convex subsets of Rn).

2. The class of quasicoherent sheaves and affine open subsets.

3. The class of coherent analytic sheaves and Stein spaces (spaces biholo-morphic to closed analytic subspaces of CN for some N , for examplepolydisks or more generally strictly pseudoconvex domains).

As another application, recall that we previously wanted to compare aclass in H1(C;O∗C) which came from a class in H1(C;O∗C) with a corre-

sponding class in H0,1

∂(C), where C was a compact Riemann surface. Let

us discuss a very general procedure of which this is a special case: X is atopological space, Uα is an open cover of X, F is a sheaf on X, and (A·, d)is an acyclic resolution of F . More precisely, we assume that A·|Uα0,α1,...,αk

is acyclic for every open subset Uα0,α1,...,αk . Note that Leray’s theorem thenimplies that

Hq(Uα;Ak) = 0

for all q > 0 and all k. We want to pass from an element of H1(Uα;F ) toan element of H1(Γ(X;A·), d).

Claim 25. The recipe for doing so is as follows: given an element hαβ ∈C1(Uα;F ) inducing a class ξ ∈ H1(Uα;F ), let tαβ ∈ C1(Uα;A0) beits image, and write tαβ = sα−sβ for some s = (sα) ∈ C0(Uα;A0), whichis possible since H1(Uα;A0) = 0. Then ξ corresponds to the image of theelement dsα = dsβ ∈ H0(X;A1) = H0(X;A1) in H1(Γ(X;A·), d), at leastup to a sign.

Proof. Consider the double complex (C·(Uα;A·), δ, d), where δ is the Cechdifferential and d is induced by the differential on A·. We have the associatedsingle complex (s(C·(Uα;A·), D)), with D = δ+(−1)pd. We will examinethis complex directly, as a concrete way of understanding the associatedspectral sequences. Note that, for (cα) ∈ C0(Uα;A0),

D(cα) = (cα − cβ, dcα) ∈ C1(Uα;A0)⊕ C0(Uα;A1),

12

and, for (bαβ, ϕα) ∈ C1(Uα;A0)⊕ C0(Uα;A1),

D(bαβ, ϕα) = (δ(bαβ),−dbαβ + (ϕα − ϕβ),−dϕα)

∈ C2(Uα;A0)⊕ C1(Uα;A1)⊕ C0(Uα;A2).

We can then see directly that

H1(s(C·(Uα;A·), D) = H1(Γ(X;A·), d) = H1(X;F )

as follows: if D(bαβ, ϕα) = 0, then δ(bαβ) = 0. Since H1(Uα;A0) = 0, wecan write bαβ = δ(cα) for some (cα) ∈ C0(Uα;A0), unique up to adding aσ ∈ H0(X;A0). Thus, after modifying (bαβ, ϕα) by D(cα), we can assumethat bαβ = 0 for all α, β, and that ϕα satisfies: ϕα = ϕβ for all α, β, henceϕα is the restriction of ϕ ∈ Γ(X;A1) such that dϕ = 0. Hence there isa representative for the class (bαβ, ϕα) of the form (0, ϕ), and it is uniqueup to adding Dσ = (0, dσ) for an arbitrary σ ∈ Γ(X;A0). This identifiesH1(s(C·(Uα;A·)), D) with H1(Γ(X;A·), d).

Going back to the original construction, given a class ξ ∈ H1(Uα;F )represented by hαβ ∈ C1(Uα;F ), with image tαβ ∈ C1(Uα;A0) and suchthat tαβ = sα − sβ for some s = (sα) ∈ C0(Uα;A0), we have

D(sα) = tαβ + dsα ∈ C1(Uα;A0)⊕ C0(Uα;A1).

Also Dtαβ = dsα − dsβ = 0, and likewise D(dsα) = 0. Thus, tαβ and −dsαboth define classes in H1(s(C·(Uα;A·), D), and these classes are equal.Tracing through the various identifications shows that tαβ is the image ofξ ∈ H1(Uα;F ) under the map H1(Uα;F ) → H1(X;F ), and that it isequal to the class corresponding to −dsα.

5.4 Hypercohomology

We begin by collecting some basic results about injective resolutions of com-plexes.

Proposition 26. Let K· be a complex of sheaves on X, let f : K· → L·

be an injective quasi-isomorphism, and let g : K· → I · be a morphism ofcomplexes, where I · is injective. Then there exists an extension of g, i.e. amorphism g : L· → I · with g = g f , and any two such extensions of g arechain homotopic.

Recall that an injective resolution of a complex K· is an injective quasi-isomorphism of complexes f : K· → I ·, where I · is injective.

13

Proposition 27. Let K· be a complex of sheaves on X.

(i) An injective resolution f : K· → I · exists.

(ii) If f : K· → I · and g : K· → J · are two injective resolutions, then I ·

and J · are quasi-isomorphic, via an quasi-isomorphism φ : I · → J ·

such that φ f = g.

(iii) With I ·, J ·, f , g as in (ii), any two quasi-isomorphisms φ1 : I · → J ·

and φ2 : I · → J · such that φif = g, i = 1, 2, are chain homotopic.

Definition 28. For K· a complex of sheaves on X, we define the hypercoho-mology H·(X;K·) to be the cohomology of the complex (Γ(X; I ·), d), whereI · is an injective resolution of K·. By the previous proposition, H·(X;K·)does not depend on the choice of an injective resolution, up to a canonicalisomorphism. Its properties are similar to those of ordinary cohomology:it is functorial, and given a short exact sequence of complexes of sheaves,there is a long exact sequence of hypercohomology groups, with functorialconnecting homomoorphisms.

For example, if K· = K[0] is a single sheaf K in dimension 0, thenH·(X;K[0]) = H ·(X;K).

As before, we would like to compute hypercohomology by using acyclicsheaves.

Proposition 29. Let K· be a complex of sheaves on X, and let f : K· → L·

be a quasi-isomorphism, where the complex L· is acyclic. We call such aquasi-isomorphism an acyclic resolution of K·. Then

H·(X;K·) ∼= H ·(Γ(X;L·), d).

Note: in the above we do not need to assume that f is injective. It isthen easy to show:

Corollary 30. If f : K· → L· is a quasi-isomorphism of complexes ofsheaves, then f induces an isomorphism

H·(X;K·) ∼= H·(X;L·).

We can also use the Godement resolutions of each Ki to construct anacyclic resolution of the complex K·. In fact, we get a double complex(C·

Go(K·), dGo, d), where dGo is the differential CiGo(K·)→ Ci+1Go (K·) coming

from the Godement resolution for K· and d : C·Go(Kj)→ C·

Go(Kj+1) is thedifferential induced by the functoriality of the Godement resolution. Notethat, in this case, dGo and d commute, so if we want to form the associatedsingle complex (s(C·

Go(K·)), d), we have to use D = dGo + (−1)pd.

14

Lemma 31. The associated single complex (s(C·Go(K·)), D) is an acyclic

resolution of K·.

Proof. Quite generally, we have the following lemma (whose statement andproof could have come much earlier):

Lemma 32. Suppose that K· is a complex, that A·,· is a double complex,and that K· → s(A·,·) is a morphism of complexes such that, for each p,(Ap,·, d′′) is a resolution of Kp. Then K· → s(A·,·) is a quasi-isomorphism.

Proof. Working in the abelian category of sheaves on X, we look at thespectral sequence of the double complex s(A·,·) whose E1 term is Ep,q1 =Hqd′′(A

p,·). By assumption, this is 0 for q > 0 and H0d′′(A

p,·) = Kp. ViewingK· as a filtered complex of sheaves with the trivial filtration in degree zero,the morphism K· → s(A·,·) is then a morphism of filtered complexes induc-ing an isomorphism on the E1 terms of the corresponding spectral sequences.Hence K· → s(A·,·) is a quasi-isomorphism.

Returning to the proof of Lemma 31, we see that K· → s(C·Go(K·)) is a

resolution of K·, and it is acyclic since the terms of C·Go(K·) are flasque.

Corollary 33. H·(X;K·) ∼= H ·(Γ(X; s(C·Go(K·))), D).

Looking now at the double complex of global sections Γ(X; s(C·Go(K·))),

with its two differentials dGo and d, we have the following:

Theorem 34. There are two spectral sequences abutting to H·(X;K·). Thefirst has E1 term given by

Ep,q1 = Hq(X;Kp),

and the differential d1 is induced by d : Kp → Kp+1. The second spectralsequence has E2 term given by

Ep,q2 = Hp(X;Hq(K·)).

Proof. The first statement follows by looking at the spectral sequence whichhas Ep,q0 = Γ(X;CqGo(Kp)) and d0 = dGo. Then Ep,q1 = Hq(X;Kp).

For the second spectral sequence, consider the spectral sequence whichhas Ep,q0 = Γ(X;CpGo(Kq)) and d0 = d (up to a sign). For the E1 term, wehave to compute the cohomology of the complex (Γ(X;CpGo(K·)), d). Aneasy argument using Lemma 13 shows that

Hq(Γ(X;CpGo(K·), d)) = Γ(X;Hq(CpGo(K·), d)) = Γ(X;CpGo(HqK·)),

15

with the differential equal to dGo. Then the E2 term satisfies

Ep,q2 = Hp(X;Hq(K·))

as claimed.

Remark 35. We can recover the above theorem in the general context ofinjective resolutions by using Cartan-Eilenberg resolutions.

There is also a Cech version of hypercohomology: given a complex K·

of sheaves and an open cover Uα of X, we can define H·(Uα;K·) bytaking the single complex associated to the double complex C·(Uα;K·)with the two commuting differentials δ and d (δ is the Cech differential andd is the differential induced by the differential on K·) and total differentialD = δ + (−1)pd. Note that, as in Remark 23, the inclusion Γ(X;K·) ⊆C0(Uα;K·) defines an inclusion of complexes

(Γ(X;K·), (−1)pd) ⊆ (s(C·(Uα;K·)), D.

A straightforward spectral sequence argument shows:

Proposition 36. If Uα is a Leray cover for Kp for every p, then

H·(Uα;K·) ∼= H·(X;K·).

Finally, we discuss the hypercohomology of filtered complexes of sheaves.Suppose that K· is a complex of sheaves on X, with a decreasing filtrationF ·K· by subcomplexes of of sheaves. Then there is an induced filtration ofthe Godement resolution, F ·C·

Go(K·), by defining

F ·C·Go(K·) = ImC·

Go(F ·K·)→ C·Go(K·).

Note that C·Go preserves injections. Also, by the exactness of the functor

C·Go,

grpF C·Go(K·) = C·

Go(grpF K·).

There is then an induced filtration on Γ(X; s(C·Go(K·))) and hence a spectral

sequence with

Ep,q1 = Hp+q(X; grpF K·) =⇒ Hp+q(X;K·).

Of course, we can also work this out via injective resolutions of the appro-priate kind (filtered injective resolutions, cf. Stacks 13.26).

16

Example 37. There are two general filtrations on a complex K· of sheaves:

1. The trivial or naive or stupid filtration (French: filtration bete):

σ≥pKq =

0; if q < p;

Kq; if q ≥ p.

Thus the associated graded complex grpσK· is the complex Kp[−p]with a single nonzero term Kp in degree p (and hence all differentialsare 0).

2. The canonical filtration, which is an increasing filtration defined by:

τ≤pKq =

0, if q > p;

Kerd : Kp → Kp+1, if q = p;

Kq; if q < p.

Thus the associated graded complex grτpK· is the complex

Kp−1/Ker d→ Ker d,

where the first Ker d is the kernel of d : Kp−1 → Kp and the second isthe kernel of d : Kp → Kp+1, and the two terms are located in degreesp − 1 and p respectively. Note that the map Kerd : Kp → Kp+1 →(HpK·)[−p] defines a quasi-isomorphism from grτpK

· to the complex(HpK·)[−p] (viewed as a complex with the single nonzero term HpK·

in degree p). In addition, if f : K· → L· is a quasi-isomorphism,then f induces a morphism τ≤pK

q → τ≤pLq and hence grτpK

· →grτpK

·, which is a quasi-isomorphism for every p because it induces theisomorphism HpK·[−p] → HpL·[−p]. By definition, the morphism offiltered complexes τ≤·K

· → τ≤·L· is then a filtered quasi-isomorphism.

This is the reason for calling the filtration τ≤p canonical. By contrast,the naive filtration does not have this property.

Each of the above filtrations defines a corresponding spectral sequence.The spectral sequence arising from the filtration σ≥pK· has

Ep,q1 = Hp+q(X;Kp[−p]) = Hq(X;Kp).

In fact, it is easy using the definitions to identify this spectral sequence withthe first hypercohomology spectral sequence of Theorem 34.

17

For the canonical filtration, we first have to reindex by −p because τ≤pK·

is an increasing filtration. Once we do so, we get a spectral sequence withE1 term

Ep,q1 = Hp+q(X; grτ−pK·) = Hp+q(X; (H−p)K·[p]) = H2p+q(X;H−pK·).

Up to a shift in indices, this is the same as the E2 term of the secondhypercohomology spectral sequence. Note d1 : Ep,q1 → Ep+1,q

1 is a map

H2p+q(X;H−pK·)→ H2p+2+q(X;H−p−1K·),

which can be identified with d2 for the second hypercohomology spectralsequence. And in fact one can show that the two spectral sequences agreeafter an appropriate shift.

5.5 Applications to complex manifolds and schemes

Let M be a complex manifold. By the holomorphic Poincare lemma, thecomplex of sheaves (Ω·

M , d) is a resolution of C. Thus:

Theorem 38. For all k, Hk(M ;C) ∼= Hk(M ; Ω·M ).

By Theorem 34, there is a spectral sequence with Ep,q1 = Hq(M ; ΩpM )

abutting to Hk(M ; Ω·M ) = Hk(M ;C). Of course, we already know of such a

spectral sequence, the Hodge spectral sequence. The following propositionstates, somewhat informally, that these are the same spectral sequence:

Proposition 39. There is a natural identification of the hypercohomologyspectral sequence for Hk(M ; Ω·

M ) with the Hodge spectral sequence.

Proof. We have seen that we can identify the hypercohomology spectralsequence with the spectral sequence arising from the filtration σ≥pΩ·

M . Onthe other hand, by Lemma 32, we have a quasi-isomorphism of complexes

Ω·M → s(A·,·

M ),

and the ∂-Poincare lemma implies that this induces a quasi-isomorphism offiltered complexes

(Ω·M , σ

≥p)→ (s(A·,·M ), ′F ),

where ′F ps(A·,·M ) =

⊕r≥pA

r,·. Hence we get a morphism of spectral se-quences which is an isomorphism on the E1 terms, hence an isomorphismon Er for all r.

18

In particular, for M a smooth projective variety, or more generallythe compact complex manifold associated to a smooth scheme proper overSpecC, the above spectral sequence degenerates at E1.

Let X be a smooth scheme over SpecC, not necessarily proper, and letXan be the corresponding complex manifold. We can define the coherentalgebraic sheaves Ωi

X/C in the Zariski topology, as well as the de Rham dif-

ferential d. Note however that d is not OX -linear, for the sheaves ΩiX/C or

the analytic sheaves ΩiXan . We can then consider the “algebraic” hyperco-

homology, i.e. the hypercohomology H·Zar(X; Ω·

X/C) computed with respectto the Zariski topology.

For formal reasons, we have a homomorphism of hypercohomology groups

H·Zar(X; Ω·

X/C)→ H·(Xan; Ω·Xan).

In fact, there is a morphism of filtered complexes Ω·X/C → π∗Ω

·Xan , where

π : Xan → X is the identity map, but where Xan has the usual (Haus-dorff) topology but X has the Zariski topology. Then we have a morphismH·

Zar(X;π∗Ω·Xan) → H·(Xan; Ω·

Xan), which we shall describe in more detailin remarks on the Leray spectral sequence. The filtrations on all complexesare the trivial filtration σ≥p. Thus there is a morphism of the correspondingspectral sequences. For the E1 term, this morphism is the correspondingmorphism

HqZar(X; Ωp

X/C)→ Hq(Xan; ΩpXan).

As a consequence, we obtain:

Theorem 40. Suppose that X is a smooth projective variety, or more gen-erally is proper over SpecC. Then the natural map

H·Zar(X; Ω·

X/C)→ H·(Xan; Ω·Xan)

is an isomorphism.

Proof. In this case, by GAGA for X projective, and by Grothendieck’s gen-eralization to the proper case, Hq

Zar(X; ΩpX/C) → Hq(Xan; Ωp

Xan) is an iso-morphism for every p, q. Hence the map on the E1 terms of the hyperco-homology spectral sequences is an isomorphism, so the sane is true for theabutments.

In fact, Theorem 40 holds without any assumption of properness. Themain point is the following:

19

Theorem 41 (Grothendieck’s algebraic de Rham theorem). Let X be asmooth affine variety over C. Then

Hk(Xan;C) ∼= Hk(Γ(X; Ω·X/C), d).

Note that Theorem 40 implies in particular that, in case X is properover SpecC, the hypercohomology spectral sequence for H·

Zar(X; Ω·X/C) de-

generates at E1.One way to interpret Theorem 40 is that de Rham cohomology can be

defined purely algebraically. In particular, suppose that k is a subfield ofC, for example a number field, and that X is a scheme defined over k, andfor simplicity proper over Spec k. Let XC = X ×Spec k SpecC. Then we canconsider the finite dimensional k-vector space H·

Zar(X; Ω·X/k), and general

results imply that

H·Zar(X; Ω·

X/k)⊗k C ∼= H·Zar(XC; Ω·

X/C) ∼= H·(XanC ; Ω·

XanC

) ∼= H ·(XC;C).

In other words, Hk(XC;C) is the complexification of a k-vector space. Buteven if k = Q, the rational structure induced on Hk(XC;C) via this isomor-phism is very different from the rational structure given by Hk(XC;C) ∼=Hk(XC;Q) ⊗Q C. This difference accounts for the mysterious powers of2π√−1 in our discussion of Tate twists and polarizations. For example,

suppose that L = OX(D) is the line bundle on X associated to the divisorD, defined over k. Then the integral cohomology class of D, or equivalentlyc1(L), is given as follows: take the transition functions hαβ of L with respectto an open cover Uα of Xan

C , where we can assume that the Uα ∩ Uβ aresimply connected. Then

c1(L) =1

2π√−1

δ log hαβ,

where δ is the Cech coboundary, a class in H2(Uα;Z) which maps to anelement of H2(Xan

C ;Z). However, using instead a cover by Zariski opensubsets Vij such that L|Vij is (algebraically) trivial, with transition func-tions gij ∈ O∗X , then clearly dgij/gij ∈ H1(X; Ω1

X/k). In fact, there is an

analogue c(L) of the first Chern class c1(L) with values in H2Zar(X; Ω·

X/k),

more precisely in F 1H2Zar(X; Ω·

X/k), which we can see as follows: the map

O∗X → Ω1X/k defined by h 7→ dh/h gives rise to a morphism of complexes

O∗X [−1]→ σ≥1Ω·X/k.

20

Here as usual O∗X [−1] means the complex with a single term in degree one,and hence trivial differential, and we get a morphism of complexes becausedh/h is a closed form. There is thus an induced map on hypercohomology

c : H2Zar(X;O∗X [−1]) = H1

Zar(X;O∗X)→ H2Zar(X;σ≥1Ω·

X/k)→ F 1H2Zar(X; Ω·

X/k),

which is compatible with the natural map F 1H2Zar(X; Ω·

X/k)→ H1(X; Ω1X/k).

However, the image of c(L) in F 1H2Zar(XC; Ω·

XC/C) ∼= H2(X;C) only agrees

with the integral class c1(L) after multiplying by the factor of 1/2π√−1,

which measures the difference between the k-structure and the integral (orrational) structure on H2(Xan

C ;C).

5.6 The Leray spectral sequence

In this section, we describe the Leray spectral sequence, one of the mostimportant examples of the Grothendieck spectral sequence for the derivedfunctors of a composition of two functors.

Let f : X → Y be a continuous map of topological spaces. If F is a sheafon X, we have the direct image sheaf f∗F . The functor f∗ from sheaveson X to sheaves on Y is left exact, and we define the higher direct imagesheaves Rif∗ by: choose an injective resolution F → I ·. Then Rif∗F isthe ith cohomology sheaf of the complex f∗I

·. As before, this definition isindependent of the choice of an injective resolution, and the Rif∗ are thederived functors of f∗. A straightforward argument (cf. Hartshorne) showsthat Rif∗F is the sheaf associated to the presheaf on Y defined by:

U 7→ H i(f−1(U);F ).

As in the case of cohomology, we would like to compute higher directimages using a broader class of sheaves than just injective ones.

Definition 42. A sheaf K is f∗-acyclic if Rif∗K = 0 for all i > 0. Forexample, an injective sheaf is f∗-acyclic. An f∗-acyclic resolution F → K·

of F is a resolution such that Ki is f∗-acyclic for all i.

The argument of Lemma 6 then shows:

Lemma 43. Let K· be an f∗-acyclic resolution of F . Then, for all i ≥ 0,Rif∗F = Hif∗K·.

Similarly, the arguments of Lemma 11 show:

21

Lemma 44. If K is a flasque sheaf on X, then K is f∗-acyclic. Thus aflasque resolution is f∗-acyclic.

Also, it is an easy exercise to show:

Lemma 45. If K is a flasque sheaf on X, then f∗K is a flasque sheaf onY .

Theorem 46 (Leray spectral sequence). Let F be a sheaf on X. There isa spectral sequence abutting to H ·(X;F ) with E2 term given by

Ep,q2 = Hp(Y ;Rqf∗F ).

Proof. Choose a flasque resolution C· of F , for example the Godement res-olution, and consider the hypercohomology H(Y ; f∗C

·) of the direct imagecomplex f∗C

·. The first hypercohomology spectral sequence has E1 term

Ep,q1 = Hq(Y ; f∗Cp).

By Lemma 45, f∗Cp is flasque for all p. Thus Hq(Y ; f∗C

p) = 0 for q > 0. SoEp,q1 = 0 if q > 0, and Ep,01 = H0(Y ; f∗C

p) = H0(X;Cp). The differential

d1 : Ep,01 → Ep+1,01 is the differential induced by d : Cp → Cp+1. Thus,

Ep,02 = Hp(X;F ) = Ep,0∞ , the spectral sequence degenerates at E2, and thefiltration on Hp(Y ; f∗C

·) is trivial. Hence Hp(Y ; f∗C·) ∼= Hp(X;F ).

The second spectral sequence for H·(Y ; f∗C·) has E2 term

Ep,q2 = Hp(Y ;Hqf∗Cp).

By Lemmas 43 and 44, Hqf∗Cp ∼= Rqf∗F . Putting the results about thetwo spectral sequences together gives the statement of the theorem.

Remark 47. The edge homomorphisms induce homomorphisms Ep,02 =

Hp(Y ; f∗F ) → Hp(X;F ) and Hp(X;F ) → E0,p2 = H0(Y ;Rpf∗F ). In fact,

we can see the homomorphismHp(Y ; f∗F )→ Hp(X;F ) quite directly. Moregenerally, let G be a sheaf on Y . Then there are homomorphisms of coho-mology groups Hp(Y ;G)→ Hp(X; f−1G) defined as follows. First, it is easyto check that f−1C·

Go(G) ∼= C·Go(f−1G). Moreover, f−1 is an exact functor,

so we get: f−1C·Go(G) ∼= C·

Go(f−1G) is a flasque resolution of f−1G. Thehomomorphism Γ(Y ;G)→ Γ(X; f−1G) extends to a morphism

Γ(Y ;C·Go(G))→ Γ(X; f−1C·

Go(G)) ∼= Γ(X;C·Go(f−1G)).

Taking cohomology yields Hp(Y ;G) → Hp(X; f−1G). Finally, setting G =f∗F and using the canonical homomorphism f−1f∗F → F gives the desiredhomomorphism Hp(Y ; f∗F )→ Hp(X;F ). Moreover, if F is f∗-acyclic, thenHp(Y ; f∗F )→ Hp(X;F ) is an isomorphism.

22

More generally, if K· is a complex of sheaves on X, and using the Gode-ment resolution of K·, the arguments above show that there is a homomor-phism H·(Y ; f∗K

·)→ H·(X;K·). For future use, we note:

Lemma 48. If K· is a complex of f∗-acyclic sheaves on X, then the homo-morphism H·(Y ; f∗K

·)→ H·(X;K·) is an isomorphism.

Proof. The homomorphism H·(Y ; f∗K·) → H·(X;K·) induces a homomor-

phism on the first cohomology spectral sequences, which for the E1 pageis clearly Hq(Y ; f∗K

p) → Hq(X;Kp). By the above remark, since Kp isf∗-acyclic, Hq(Y ; f∗K

p) → Hq(X;Kp) is an isomorphism. Hence the mapH·(Y ; f∗K

·)→ H·(X;K·) is an isomorphism as well.

We can also generalize the above to hyperdirect images: Let K· be acomplex of sheaves on X. Choosing an injective resolution K· → I ·, wedefine

Rif∗K· = Hif∗I ·.

Arguments similar to those given for the proof of Theorem 34 show:

Theorem 49. There are two spectral sequences abutting to Rkf∗K·. Thefirst has E1 term given by

Ep,q1 = Rqf∗Kp,

and the differential d1 is induced by d : Kp → Kp+1. The second spectralsequence has E2 term given by

Ep,q2 = Rpf∗HqK·.

Similar but slightly more involved arguments prove the existence of theLeray spectral sequence for hypercohomology:

Theorem 50. Let K· be a complex of sheaves on X. There is a spectralsequence abutting to H·(X;K·) with E2 term given by

Ep,q2 = Hp(Y ;Rqf∗K).

5.7 Products

We give a very brief treatment of products in Cech cohomology and Cechhypercohomology.

23

Definition 51. Suppose that F1 and F2 are two sheaves on F and that wehave an R-bilinear map from F1×F2 to a third sheaf G, i.e. a homomorphismF1 ⊗ F2 → G. We denote the value on sections s1 of F1 and s2 of F2 bys1 · s2. Let Uα be an open cover of X. Define the cup product

` : Cn(Uα;F1)× Cm(Uα;F2)→ Cn+m(Uα;G)

by the formula

(s ` t)α0,α1,...,αk+1= sα0,α1,...,αn · tαn,α1,...,αn+m .

A straightforward calculation shows that

δ(s ` t) = δ(s) ` t+ (−1)ns ` δ(t).

Hence ` induces an R-bilinear map, also denoted `:

` : Hn(Uα;F1)× Hm(Uα;F2)→ Hn+m(Uα;G).

Now suppose we have a bilinear morphism of complexes of sheaves,ϕ : A· ⊗K· → L·. This means that we have a graded R-bilinear map ϕ, i.e.ϕ(Ap ⊗Kq) ⊆ Lp+q, which we again denote by ϕ(s, t) = s · t, satisfying

d(s · t) = ds · t+ (−1)ps · dt.

Equivalently, the induced map A· ⊗K· → L· is a morphism of complexes,where A·⊗K· has the tensor product differential d(s⊗t) = ds⊗t+(−1)ps⊗dt.A basic example of this is as follows: Given a complex K·, define the complexHom·(K·,K·) as follows:

Homk(K·,K·) = ϕ ∈ Hom(K·,K·) : ϕ(Kp) ⊆ Kp+k for all p.

Thus by definition d ∈ Hom1(K·,K·). Define the differential d by: if ϕ ∈Homk(K·,K·), then

dϕ = d ϕ+ (−1)k+1ϕ d ∈ Homk+1(K·,K·).

Equivalently, dϕ = [d, ϕ], where [·, ·] is the graded Lie bracket onHom·(K·,K·).We have the evaluation map

Hom·(K·,K·)⊗K· → K·,

which by definition satisfies

Homp(K·,K·)⊗Kq ⊆ Kp+q.

24

Also, for sections ϕ ∈ Homp(K·,K·) and s ∈ Kq,

(dϕ)(t) + (−1)pϕ(dt) = d(ϕ(t)) + (−1)p+1ϕ(dt) + (−1)pϕ(dt) = d(ϕ(t)).

Hence the evaluation morphism is a morphism of complexes in the abovesense.

Returning to the general picture of a morphism ϕ : A· ⊗ K· → L· ofcomplexes in the above sense, we would like to define analogous cup products

C·(Uα;A·)⊗ C·(Uα;K·)→ C·(Uα;L·) :

For s ∈ Cp(Uα;Aq) and t ∈ Cr(Uα;Ks), set

s ` t = (−1)rqs ` t.

Then ` : Cp(Uα;Aq)⊗ Cr(Uα;Ks)→ Cp+q(Uα;Lr+s) and one checksthat

D(s ` t) = D(s) ` t+ (−1)ns ` D(t),

in case s is homogeneous of degree n. Thus there is an induced cup product

H·(Uα;A·)⊗ H·(Uα;K·)→ H·(Uα;L·).

Returning to the original setting of products, suppose that F1 and F2

are sheaves on X and that there is a bilinear map F1 ⊗ F2 → G. It is ageneral fact that we can also define pairings of complexes for the Godementresolution, or more generally for injective resolutions, giving a cup product

Hn(X;F1)⊗Hm(X;F2)→ Hn+m(X;G).

Moreover, if Uα is a Leray cover for F1, F2, and G, then these cup productsagree under the isomorphisms. Finally, suppose that the cohomology of F1,F2, and G can be computed by complexes of sheaves A·, K·, L· which areacyclic with respect to the p-fold intersections of the open sets Uα andsuch that there is a pairing of complexes ϕ : A·⊗K· → L· as above which iscompatible with the inclusions F1 → A·, F2 → K·, G → L·. Then the cupproduct

H·(Uα;A·)⊗ H·(Uα;K·)→ H·(Uα;L·)

reduces to a pairing

H ·(Γ(X;A·), d)⊗H ·(Γ(X;K·), d)→ H ·(Γ(X;L·), d)

25

which is the induced pairing on global sections, and it is straightforward tocheck that it also agrees with the cup product

Hp(Uα;F1)× Hq(Uα;F2)→ Hp+q(Uα;G)

as defined above.For example, if M is a manifold, taking F1 = F2 = G = C and A· =

K· = L· = A·M with wedge product, we can identify wedge product on de

Rham cohomology with usual cup product on Cech cohomology (and onsingular cohomology with a little more work). A similar argument, for acomplex manifold M and a Stein cover Uα, identifies cup product

Hq(Uα; ΩpM )× Hs(Uα; Ωr

M )→ Hq+s(Uα; Ωp+rM )

with wedge product of forms

Hp,q

∂(M)⊗Hr,s

∂(M)→ Hp+r,q+s

∂(M).

We can similarly add a holomorphic vector bundles to the mix, but shallnot write down the general statement here.

26