Cohomology and hypercohomology

5.1 Definition of Let X be a . We list some basic definitions concerning sheaves on X. If F is a of R-modules on X, we denote by Γ(X; F ) the R- of global sections of F . For a complex (C·, d) of sheaves (of R-modules) on X, the convention will always be that d has degree +1. We will 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 d is 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 on X.A quasi-isomorphism f : C· → D· is a of complexes of sheaves which induces an isomorphism on the cohomology sheaves. The complexes (C·, d) and (D·, d) are quasi-isomorphic if there exists some quasi-isomorphism f : C· → D·.

(ii) Two f : C· → D·, g : C· → D· are chain homotopic if there exists an R- T : C· → D· of degree −1 such that f − g = dT + T d.

(iii) A sheaf I on X is injective if it is an in the abelian of sheaves of R-modules on X: given sheaves F and G on X and a commutative diagram 0 −−−−→ F −−−−→ G   fy I

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

(iv) A 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 that Ik 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 is injective 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 , then there exists a quasi-isomorphism f : I· → J · which is compatible with the two inclusions F → I0, F → J 0.

(iii) If I· and J · are two injective resolutions of F and f : I· → J · and g : I· → J · are two quasi-isomorphisms compatible with the inclusions of 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 injective resolution of F and define

Hi(X; F ) = Hi(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 isomorphism Hi(Γ(X; I·), d) =∼ Hi(Γ(X; J ·), d). Finally the groups Hi(X; F ) satisfy the usual properties. In particular, H0(X; F ) = Γ(X; F ) and, given a short exact 0 → F 0 → F → F 00 → 0 of sheaves on X, there is a long exact cohomology sequence

· · · → Hi−1(X; F 00) −→δ Hi(X; F 0) → Hi(X; F ) → Hi(X; F 00) −→δ Hi+1(X; F 0) → · · · , where the connecting homomorphisms δ are also functorial in a natural sense with respect to commutative diagrams of short exact of complexes of sheaves on X.

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

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

Definition 5. Let K be a sheaf on X. Then K is acyclic if Hi(X; K) = 0 for all i > 0. An acyclic resolution (K·, d) of a sheaf F is a resolution F → 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 all i ≥ 0, Hi(X; F ) =∼ Hi(Γ(X; K·), d). Proof. Since there is an 0 → F → K0 → K1, and the global section functor is left exact, we see that H0(X; F ) = Ker{d: 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 ) = Coker{d: H0(X; K0) → H0(X; K0/F ). Moreover, using the exact sequence 0 → K0/F → K1 → K2, we see that H0(X; K0/F ) = Ker{d: H0(X; K1) → H0(X; K2)}. Thus H1(X; F ) is isomorphic to Ker{d: H0(X; K1) → H0(X; K2)}/ Im{d: H0(X; K0) → H0(X; K1)} = H1(Γ(X; K·), d). Finally, for k ≥ 2, we argue by induction on k, assuming inductively that the result has been proved for all k − 1 ≥ 1 and all sheaves G. Then, given the 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 Hi(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 acyclic sheaves. Of course, an I is acyclic, because the complex I[0] which is I in degree zero and zero elsewhere is an injective resolution of I. Hence, Hi(X; I) = 0 for all i > 0. The following gives another very broad class of acyclic sheaves. Definition 7. A sheaf F is flasque if, for every pair of open subsets U ⊆ V of X, the restriction morphism Γ(V ; F ) → Γ(U; F ) is surjective. The method of constructing injective resolutions given in Hartshorne shows 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 somewhat superfluous 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 of sheaves 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 0 → F → F 00 → 0,

if F 0 and F are flasque, then so F 00.

(ii) Suppose that 0 → F 0 → F 1 → · · · is an exact sequence of flasque sheaves. Then the associated sequence of 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 homomorphism F → 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 thus H1(X; F ) = 0 since H1(X; I) = 0. Now assume inductively that we have shown, 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, completing the inductive step.

The usefulness of flasque sheaves is that it enables us to construct a canonical flasque resolution (due to Godement). 0 Definition 12. Let F be a sheaf on X, and define CGo(F ), the sheaf of germs of discontinuous sections of F , by: for every open subset U of X,

0 Y CGo(F )(U) = Fx. x∈U If V ⊆ U are two open subsets, there is the obvious restriction map

0 0 CGo(F )(U) → CGo(F )(V ), 0 making CGo(F ) a sheaf (not just a presheaf) of R-modules on X. Clearly, 0 0 CGo(F ) is flasque. There is a natural injection of sheaves F → CGo(F ), Q which associates to a section s ∈ F (U) the element (sx) ∈ x∈U Fx. Then we define 1 0 0 CGo(F ) = CGo(CGo(F )/F ), with a natural map

0 0 0 0 1 d: CGo(F ) → CGo(F )/F → CGo(CGo(F )/F ) = CGo(F ). r r−1 Inductively, suppose that we have defined CGo(F ) and the map CGo (F ) → r CGo(F ). Then set r+1 0 r r−1 CGo (F ) = CGo(CGo(F )/ Im CGo (F )).

5 We then have the composition

r r r−1 r+1 dGo : CGo(F ) → CGo(F )/ Im CGo (F ) → CGo (F ),

r−1 with the first map surjective with Im CGo (F ) and the second map · injective. It follows that (CGo(F ), dGo) is exact except in dimension zero, and is a resolution of F by flasque sheaves. We will refer to it as the canonical flasque resolution or Godement resolution of F .

· We can view (CGo(·), dGo) as a functor from the of sheaves on X to the category of complexes on X, because a homomorphism ˜ · · f : F → G defines a morphism of complexes f : CGo(F ) → CGo(G) in a functorial way. Then:

· Lemma 13. (i) The functor (CGo(·), dGo) is exact. In other words, given an exact sequence F 0 → F → F 00 of sheaves on X, the sequence of complexes

· 0 · · 00 CGo(F ) → CGo(F ) → CGo(F )

is exact.

0 · 0 (ii) If α: F → F is a homomorphism of sheaves and α˜ : CGo(F ) → · CGo(F ) is the induced homomorphism, then

· · Kerα ˜ = CGo(Ker α) and Imα ˜ = CGo(Im α).

· (iii) The functor (CGo(·), dGo) commutes with taking cohomology. In other words, given a complex of sheaves on X

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

˜ · i · i+1 and if δ is the induced map CGo(F ) → CGo(F ), then the corre- sponding cohomology sheaves satisfy

i · · ˜ · i · H (CGo(F ), δ) = CGo(H F ).

Proof. (i) Clearly the sequence

0 0 0 0 00 CGo(F ) → CGo(F ) → CGo(F )

6 · is exact. To prove (i), given the inductive construction of CGo(F ), it suffices to show that the sequence

0 0 0 0 0 00 00 CGo(F )/F → CGo(F )/F → CGo(F )/F is exact. This is a straightforward exercise. (ii) We have an exact sequence

0 → Ker α → F 0 −→α F.

Using (i), the sequence

· · 0 α˜ · 0 → CGo(Ker α) → CGo(F ) −→ CGo(F )

· is also exact. Thus Kerα ˜ = CGo(Ker α). As for Im α, we have a commutative diagram F 0 −−−−→ Im α −−−−→ 0   α  y y F −−−−→= F Thus, using (i) again, there is a corresponding diagram

· 0 · CGo(F ) −−−−→ CGo(Im α) −−−−→ 0     α˜y y · = · CGo(F ) −−−−→ CGo(F ) · It then follows easily that the image of CGo(Im α) is contained in Imα ˜ and · that the corresponding homomorphism CGo(Im α) → Imα ˜ 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 closed subset Sα of Uα such that ϕα|X − Sα = 0. 2. (Local finiteness). For all x ∈ X, there exists an open neighborhood V of x such that there are only finitely many α with ϕα|V 6= 0.

7 P 3. α ϕα = Id. (Here the sum is meaningful by (2).) The primary example of fine sheaves is given by: ∞ ∞ 0 Example 15. If M is a C and CM = AM denotes the sheaf of ∞ rings of C functions on M, then, for every open cover {Uα} of M, there ∞ exists a C partition of unity {fα} subordinate to {Uα}. Then, if K is any ∞ sheaf of CM -modules, multiplication by fα defines an endomorphism of K satisfying the conditions of the above definition. Thus, such a K is a fine sheaf. k ∞ In particular, the sheaves AM of C differential forms on M, as well as p,q the sheaves AM in case M is a complex manifold, are fine sheaves. Proposition 16. A fine sheaf K is acyclic. · Proof. Consider the Godement resolution CGo(K). We need to show that · the complex Γ(X; CGo(K)) has no cohomology in positive degrees. Given k k+1 k > 0, let ξ ∈ Ker{Γ(X; CGo(K)) → Γ(X; CGo (K)). By local exactness, there exists an open cover {Uα} of X such that, for all α, there exists k−1 an ηα ∈ CGo (K)(Uα) such that dGoηα = ξ|Uα. Let ϕα : K → K be a family of endomorphisms of K satisfying (1)–(3) of Definition 14 for the · open cover {Uα}. Since CGo is functorial, the ϕα lift to endomorphismsϕ ˜α · of the complex CGo(K), and it is easy to check that theϕ ˜α satisfy the same conditions as ϕα. Define X η = ϕ˜α(ηα). α

Here,ϕ ˜α(ηα) is a priori only defined on Uα, but by the support condition it P extends over X. Also, locally around every point x of X, the sum α ϕ˜α(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): X X dGoη = dGoϕ˜α(ηα) = ϕ˜α(dGoηα) α α X = ϕ˜α(ξ) = ξ. α k · k Thus H (Γ(X; CGo(K)), dGo) = 0 for k > 0, so that H (X; K) = 0 for k > 0 and K is acyclic.

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

8 · Proof. By the Poincar´elemma, (AM , d) is a resolution of C. Example 15 k k shows that the AM are fine sheaves, hence acyclic. By Lemma 6, H (M; C) is computed by taking the cohomology of the complex of global sections · · Γ(M; AM ) = A (M). The case of a flat vector bundle is similar. Corollary 18 (Dolbeault isomorphism). If M is a complex manifold, then Hq(M;Ωp ) = Hp,q(M). More generally, if V is a holomorphic vector bun- M ∂¯ dle on M, then Hq(M;Ωp ⊗ V ) = Hp,q(M; V ). M ∂¯ Proof. The proof is the same as the proof of the de Rham isomorphism, using the ∂¯-Poincar´elemma instead of the usual Poincar´elemma.

5.3 Cechˇ cohomology and Leray’s theorem

Let {Uα}α∈I be an open cover of X, where we assume the index set I to be totally ordered, and let F be a sheaf on X. We define the Cechˇ cohomology of F with respect to the cover {Uα} in the usual way: for each finite sequence α0 < α1 < ··· < αk, define

Uα0,α1,...,αk = Uα0 ∩ Uα1 ∩ · · · ∩ Uαk , · and define the Cechˇ complex Cˇ ({Uα}; F ) via ˇk Y C ({Uα}; F ) = Γ(Uα0,α1,...,αk ,F ). α0<α1<···<αk k k+1 We define the Cechˇ differential δ : Cˇ ({Uα}; F ) → Cˇ ({Uα}; F ) by: for a ˇk section t = (tα0,α1,...,αk ) of C ({Uα},F ), k+1 X i δtα0,α1,...,αk+1 = (−1) tα0,...,αˆi,...,αk+1 |Uα0,α1,...,αk+1 , i=0 where the notationα ˆi means that this term is omitted. It is straightforward · to check that δ ◦ δ = 0. Then we define Hˇ ({Uα}; F ) to be the cohomology k of the complex (Cˇ ({Uα}; F ), δ). By the sheaf axioms, it is easy to check 0 that Hˇ ({Uα}; F ) = Γ(X; F ). There is also a sheaf version: define ˇk Y C ({Uα}; F ) = (jα0,α1,...,αk )∗(F |Uα0,α1,...,αk ), α0<α1<···<αk where jα0,α1,...,αk : Uα0,α1,...,αk → X is the inclusion. There is also a differ- k k+1 ential δ : Cˇ ({Uα}; F ) → Cˇ ({Uα}; F ) which is defined by analogy with k k+1 δ : Cˇ ({Uα}; F ) → Cˇ ({Uα}; F ). A combinatorial argument using the sheaf axioms shows:

9 k Proposition 19. The complex (Cˇ ({Uα}; F ), δ) is a resolution of F .

· · Note that Γ(X; Cˇ ({Uα}; F )) = Cˇ ({Uα}; F ) (compatibly with δ).

k Lemma 20. If F is flasque, then Hˇ ({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 is a flasque sheaf on U, and that, if G is a flasque sheaf on U, then i∗G is k a flasque sheaf on X. Thus, the sheaves Cˇ ({Uα}; F ) are flasque, so that · (Cˇ ({Uα}; F ), δ) is a flasque resolution of F . It follows that

k k k k H (X; F ) = H (Γ(Cˇ ({Uα}; F ), δ)) = Hˇ ({Uα}; F ).

k Thus Hˇ ({Uα}; F ) = 0 for k > 0. By (ii) of Lemma 2, if I· is an injective resolution of F , there is an k · induced morphism of complexes Cˇ ({Uα}; F ) → I . Taking global sections, · · there is a morphism of complexes Cˇ ({Uα},F ) → Γ(X; I ) and hence an induced homomorphism

· · Hˇ ({Uα}; F ) → H (X; F ).

With an appropriate notion of refinements, we can define

Hˇ ·(X; F ) = lim Hˇ ·({U }; F ). −→ α {Uα}

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

0 ∼ 0 Theorem 21. For every open cover, Hˇ ({Uα}; F ) = H (X; F ). Moreover, Hˇ 1(X; F ) =∼ H1(X; F ) via the natural homomorphism. If X is paracompact, then, for every k, Hˇ k(X; F ) =∼ Hk(X; F ).

The really computationally useful version of Cechˇ cohomology is the following:

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

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 all · · α0, α1, . . . , αk, and consider the double complex (Cˇ ({Uα}; K ), δ, d), where δ is the Cechˇ differential and d is induced by the differential on K·. For · · p the associated single complex (s(Cˇ ({Uα}; K ),D), with D = δ + (−1) d in bidegree (p, q), we can consider the two spectral sequences associated to the ˇ· · p,q double complex C ({Uα}; K ). For the one with E1 given by taking the cohomology first with respect to δ, we get ( p,q q p 0, if q > 0; E = Hˇ ({Uα}; K ) = 1 Γ(X; Kp), if q = 0,

q p where we have used Lemma 20 to conclude that Hˇ ({Uα}; K ) = 0 for q > 0. p · p p Thus the E2 = E∞ term is H (Γ(X; K ), (−1) d) = H (X; F ). Looking at the other , the E1 term is the d-cohomology of the complex · 0 · 1 Cˇ ({Uα}; K ) → Cˇ ({Uα}; K ) → · · · . · Since K |Uα0,α1,...,αk is flasque, it is an acyclic resolution of F |Uα0,α1,...,αk . · · Then H (Uα0,α1,...,αk ; F ) is the cohomology of the complex Γ(Uα0,α1,...,αk ; K ). p 0 But then the d-cohomology of Cˇ ({Uα}; K ) in degree q is Y q H (Uα0,α1,...,αp ,F ). α0<α1<···<αp q Since H (Uα0,α1,...,αp ; F ) = 0 for q > 0, the d-cohomology of the complex · · · · Cˇ ({Uα}; K ) is just Cˇ ({Uα}; F ) and the E2 term is Hˇ ({Uα}; F ). Compar- · ∼ · ing, we see that Hˇ ({Uα}; F ) = H (X; F ). Remark 23. (i) Note that, with our sign conventions, the proof identifies · · p Hˇ ({Uα}; F ) with the cohomology of the complex (Γ(X; K ), (−1) d), and in · p · · fact there is an inclusion (Γ(X; K ), (−1) d) → (s(Cˇ ({Uα}; K ),D) which is a quasi-isomorphism by the above proof. Of course, the cohomology of (Γ(X; K·), (−1)pd) is the same as the cohomology of (Γ(X; K·), d) and indeed the complexes (Γ(X; K·), (−1)pd) and (Γ(X; K·), d) are isomorphic by the map Γ(X; K·) → Γ(X; K·) which is (−1)p(p−1)/2 Id in degree p. k (ii) Even without the hypothesis that H (Uα0,α1,...,αk ; F ) = 0 for k > 0, the proof exhibits a homomorphism p,0 ˇ p p,0 p E2 = H ({Uα}; F ) → E∞ → H (X; F ) which is the homomorphism described in the remarks before the statement of Theorem 21.

11 Definition 24. Given a sheaf F on X, an open cover which satisfies the hypotheses 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 Leray covers for all of the sheaves in the given class. There are three broad classes where this applies:

1. The class of constant sheaves and contractible open subsets (for exam- n ple open subsets which are homeomorphic to convex subsets of R ). 2. The class of quasicoherent sheaves and affine open subsets.

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

As another application, recall that we previously wanted to compare a 1 ∗ 1 ∗ class in H (C; OC ) which came from a class in H (C; OC ) 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 a · topological 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 then implies that q k Hˇ ({Uα}; A ) = 0 1 for all q > 0 and all k. We want to pass from an element of Hˇ ({Uα}; F ) to an element of H1(Γ(X; A·), d).

Claim 25. The recipe for doing so is as follows: given an element hαβ ∈ 1 1 1 0 Cˇ ({Uα}; F ) inducing a class ξ ∈ Hˇ ({Uα}; F ), let tαβ ∈ Cˇ ({Uα}; A ) be 0 0 its image, and write tαβ = sα − sβ for some s = (sα) ∈ Cˇ ({Uα}; A ), which 1 0 is possible since Hˇ ({Uα}; A ) = 0. Then ξ corresponds to the image of the 0 1 0 1 1 · element dsα = dsβ ∈ Hˇ (X; A ) = H (X; A ) in H (Γ(X; A ), d), at least up to a sign.

· · Proof. Consider the double complex (Cˇ ({Uα}; A ), δ, d), where δ is the Cechˇ differential and d is induced by the differential on A·. We have the associated · · p single complex (s(Cˇ ({Uα}; A ),D)), with D = δ + (−1) d. We will examine this complex directly, as a concrete way of understanding the associated 0 0 spectral sequences. Note that, for (cα) ∈ Cˇ ({Uα}; A ),

1 0 0 1 D(cα) = (cα − cβ, dcα) ∈ Cˇ ({Uα}; A ) ⊕ Cˇ ({Uα}; A ),

12 1 0 0 1 and, for (bαβ, ϕα) ∈ Cˇ ({Uα}; A ) ⊕ Cˇ ({Uα}; A ),

D(bαβ, ϕα) = (δ(bαβ), −dbαβ + (ϕα − ϕβ), −dϕα) 2 0 1 1 0 2 ∈ Cˇ ({Uα}; A ) ⊕ Cˇ ({Uα}; A ) ⊕ Cˇ ({Uα}; A ).

We can then see directly that

1 · · 1 · 1 H (s(Cˇ ({Uα}; A ),D) = H (Γ(X; A ), d) = H (X; F )

1 0 as follows: if D(bαβ, ϕα) = 0, then δ(bαβ) = 0. Since Hˇ ({Uα}; A ) = 0, we 0 0 can write bαβ = δ(cα) for some (cα) ∈ Cˇ ({Uα}; A ), unique up to adding a 0 0 σ ∈ H (X; A ). Thus, after modifying (bαβ, ϕα) by D(cα), we can assume that bαβ = 0 for all α, β, and that ϕα satisfies: ϕα = ϕβ for all α, β, hence 1 ϕα is the restriction of ϕ ∈ Γ(X; A ) such that dϕ = 0. Hence there is a representative for the class (bαβ, ϕα) of the form (0, ϕ), and it is unique up to adding Dσ = (0, dσ) for an arbitrary σ ∈ Γ(X; A0). This identifies 1 · · 1 · H (s(Cˇ ({Uα}; A )),D) with H (Γ(X; A ), d). 1 Going back to the original construction, given a class ξ ∈ Hˇ ({Uα}; F ) 1 1 0 represented by hαβ ∈ Cˇ ({Uα}; F ), with image tαβ ∈ Cˇ ({Uα}; A ) and such 0 0 that tαβ = sα − sβ for some s = (sα) ∈ Cˇ ({Uα}; A ), we have

1 0 0 1 D(sα) = tαβ + dsα ∈ Cˇ ({Uα}; A ) ⊕ Cˇ ({Uα}; A ).

Also Dtαβ = dsα − dsβ = 0, and likewise D(dsα) = 0. Thus, tαβ and −dsα 1 · · both define classes in H (s(Cˇ ({Uα}; A ),D), and these classes are equal. Tracing through the various identifications shows that tαβ is the image of 1 1 1 ξ ∈ Hˇ ({Uα}; F ) under the map Hˇ ({Uα}; F ) → H (X; F ), and that it is equal 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 of complexes, where I· is injective. Then there exists an extension of g, i.e. a morphism g˜: L· → I· with g =g ˜ ◦ f, and any two such extensions of g are chain 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 φi ◦f = 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), where · · · · I 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 canonical isomorphism. 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 functorial connecting homomoorphisms. For example, if K· = K[0] is a single sheaf K in dimension 0, then · · H (X; K[0]) = H (X; K). As before, we would like to compute hypercohomology by using acyclic sheaves. 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 a quasi-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 is then easy to show: Corollary 30. If f : K· → L· is a quasi-isomorphism of complexes of sheaves, then f induces an isomorphism · · ∼ · · H (X; K ) = H (X; L ). We can also use the Godement resolutions of each Ki to construct an acyclic resolution of the complex K·. In fact, we get a double complex · · i · i+1 · (CGo(K ), dGo, d), where dGo is the differential CGo(K ) → CGo (K ) coming · · j · j+1 from the Godement resolution for K and d: CGo(K ) → CGo(K ) is the differential induced by the functoriality of the Godement resolution. Note that, in this case, dGo and d commute, so if we want to form the associated · · p single complex (s(CGo(K )), d), we have to use D = dGo + (−1) d.

14 · · Lemma 31. The associated single complex (s(CGo(K )),D) is an acyclic resolution of K·.

Proof. Quite generally, we have the following lemma (whose statement and proof 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,·, d00) 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 the ·,· p,q spectral sequence of the double complex s(A ) whose E1 term is E1 = q p,· 0 p,· p Hd00 (A ). By assumption, this is 0 for q > 0 and Hd00 (A ) = K . Viewing K· 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(CGo(K )) is a · · · resolution of K , and it is acyclic since the terms of CGo(K ) are flasque. · · ∼ · · · Corollary 33. H (X; K ) = H (Γ(X; s(CGo(K ))),D). · · Looking now at the double complex of global sections Γ(X; s(CGo(K ))), with its two differentials dGo and d, we have the following:

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

p,q q p E1 = H (X; K ),

p p+1 and the differential d1 is induced by d: K → K . The second spectral sequence has E2 term given by

p,q p q · E2 = H (X; H (K )). Proof. The first statement follows by looking at the spectral sequence which p,q q p p,q q p has E0 = Γ(X; CGo(K )) and d0 = dGo. Then E1 = H (X; K ). For the second spectral sequence, consider the spectral sequence which p,q p q has E0 = Γ(X; CGo(K )) and d0 = d (up to a sign). For the E1 term, we p · have to compute the cohomology of the complex (Γ(X; CGo(K )), d). An easy argument using Lemma 13 shows that

q p · q p · p q · H (Γ(X; CGo(K ), d)) = Γ(X; H (CGo(K ), d)) = Γ(X; CGo(H K )),

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

p,q p q · E2 = H (X; H (K )) as claimed.

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

There is also a Cechˇ version of hypercohomology: given a complex K· ˇ · · of sheaves and an open cover {Uα} of X, we can define H ({Uα}; K ) by · · taking the single complex associated to the double complex Cˇ ({Uα}; K ) with the two commuting differentials δ and d (δ is the Cechˇ differential and d is the differential induced by the differential on K·) and total differential D = δ + (−1)pd. Note that, as in Remark 23, the inclusion Γ(X; K·) ⊆ 0 · Cˇ ({Uα}; K ) defines an inclusion of complexes

· p · · (Γ(X; K ), (−1) d) ⊆ (s(Cˇ ({Uα}; K )),D.

A straightforward spectral sequence argument shows:

p Proposition 36. If {Uα} is a Leray cover for K for every p, then

ˇ · · ∼ · · 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 filtration F ·K· by subcomplexes of of sheaves. Then there is an induced filtration of · · · the Godement resolution, F CGo(K ), by defining

· · · · · · · · F CGo(K ) = Im{CGo(F K ) → CGo(K )}.

· Note that CGo preserves injections. Also, by the exactness of the functor · CGo, p · · · p · grF CGo(K ) = CGo(grF K ). · · There is then an induced filtration on Γ(X; s(CGo(K ))) and hence a spectral sequence with

p,q p+q p · p+q · E1 = H (X; grF K ) =⇒ H (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 bˆete): ( 0; if q < p; σ≥pKq = Kq; if q ≥ p.

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

2. The canonical filtration, which is an increasing filtration defined by:  0, if q > p;  q p p+1 τ≤pK = Ker{d: K → K }, if q = p; Kq; if q < p.

τ · Thus the associated graded complex grp K 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 is the kernel of d: Kp → Kp+1, and the two terms are located in degrees p − 1 and p respectively. Note that the map Ker{d: Kp → Kp+1} → p · τ · (H K )[−p] defines a quasi-isomorphism from grp K 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, q q τ · then f induces a morphism τ≤pK → τ≤pL and hence grp K → τ · grp K , which is a quasi-isomorphism for every p because it induces the isomorphism HpK·[−p] → HpL·[−p]. By definition, the morphism of · · filtered 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

p,q p+q p q p E1 = H (X; K [−p]) = H (X; K ).

In fact, it is easy using the definitions to identify this spectral sequence with the 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 with E1 term

p,q p+q τ · p+q −p · 2p+q −p · E1 = H (X; gr−p K ) = H (X;(H )K [p]) = H (X; H K ).

Up to a shift in indices, this is the same as the E2 term of the second p,q p+1,q hypercohomology spectral sequence. Note d1 : E1 → E1 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 spectral sequence. And in fact one can show that the two spectral sequences agree after an appropriate shift.

5.5 Applications to complex and schemes Let M be a complex manifold. By the holomorphic Poincar´elemma, the · complex of sheaves (ΩM , d) is a resolution of C. Thus: k ∼ k · Theorem 38. For all k, H (M; C) = H (M;ΩM ). p,q q p By Theorem 34, there is a spectral sequence with E1 = H (M;ΩM ) k · k abutting to H (M;ΩM ) = H (M; C). Of course, we already know of such a spectral sequence, the Hodge spectral sequence. The following proposition states, somewhat informally, that these are the same spectral sequence:

Proposition 39. There is a natural identification of the hypercohomology k · spectral sequence for H (M;ΩM ) with the Hodge spectral sequence. Proof. We have seen that we can identify the hypercohomology spectral ≥p · sequence with the spectral sequence arising from the filtration σ ΩM . On the other hand, by Lemma 32, we have a quasi-isomorphism of complexes

· ·,· ΩM → s(AM ), and the ∂¯-Poincare lemma implies that this induces a quasi-isomorphism of filtered complexes · ≥p ·,· 0 (ΩM , σ ) → (s(AM ), F ), 0 p ·,· L r,· where F s(AM ) = r≥p A . Hence we get a morphism of spectral se- quences which is an isomorphism on the E1 terms, hence an isomorphism on Er for all r.

18 In particular, for M a smooth projective variety, or more generally the compact complex manifold associated to a smooth scheme proper over Spec C, the above spectral sequence degenerates at E1. Let X be a smooth scheme over Spec C, not necessarily proper, and let Xan be the corresponding complex manifold. We can define the coherent algebraic sheaves Ωi in the Zariski topology, as well as the de Rham dif- X/C ferential d. Note however that d is not O -linear, for the sheaves Ωi or X X/C i the analytic sheaves ΩXan . We can then consider the “algebraic” hyperco- , i.e. the hypercohomology · (X;Ω· ) computed with respect HZar X/C to the Zariski topology. For formal reasons, we have a homomorphism of hypercohomology groups

· · · an · (X;Ω ) → (X ;Ω an ). HZar X/C H X · · In fact, there is a morphism of filtered complexes Ω → π Ω an , where X/C ∗ X 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 morphism · · · an · HZar(X; π∗ΩXan ) → H (X ;ΩXan ), which we shall describe in more detail in remarks on the . The filtrations on all complexes are the trivial filtration σ≥p. Thus there is a morphism of the corresponding spectral sequences. For the E1 term, this morphism is the corresponding morphism q p q an p H (X;Ω ) → H (X ;Ω an ). Zar X/C X As a consequence, we obtain:

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

· · · an · (X;Ω ) → (X ;Ω an ) HZar X/C H X is an isomorphism.

Proof. In this case, by GAGA for X projective, and by Grothendieck’s gen- q p q an p eralization to the proper case, H (X;Ω ) → H (X ;Ω an ) is an iso- Zar X/C X 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 the abutments.

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

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

Hk(Xan; ) ∼ Hk(Γ(X;Ω· ), d). C = X/C Note that Theorem 40 implies in particular that, in case X is proper over Spec , the hypercohomology spectral sequence for · (X;Ω· ) de- C HZar X/C generates at E1. One way to interpret Theorem 40 is that can be defined purely algebraically. In particular, suppose that k is a subfield of C, for example a number field, and that X is a scheme defined over k, and for simplicity proper over Spec k. Let XC = X ×Spec k Spec C. Then we can · · consider the finite dimensional k- HZar(X;ΩX/k), and general results imply that

· · ∼ · · ∼ · an · ∼ · HZar(X;ΩX/k) ⊗k C = HZar(XC;ΩX/ ) = H (X ;ΩXan ) = H (XC; C). C C C

k In other words, H (XC; C) is the complexification of a k-vector space. But k even if k = Q, the rational structure induced on H (XC; C) via this isomor- k ∼ phism is very different from the rational structure given by H (XC; C) = k H √(XC; Q) ⊗Q C. This difference accounts for the mysterious powers of 2π −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 divisor D, defined over k. Then the integral cohomology class of D, or equivalently c1(L), is given as follows: take the transition functions hαβ of L with respect to an open cover {U } of Xan, where we can assume that the U ∩ U are α C α β simply connected. Then 1 c1(L) = √ δ log hαβ, 2π −1

ˇ 2 where δ is the Cech coboundary, a class in H ({Uα}; Z) which maps to an element of H2(Xan; ). However, using instead a cover by Zariski open C Z subsets Vij such that L|Vij is (algebraically) trivial, with transition func- ∗ 1 1 tions gij ∈ OX , then clearly dgij/gij ∈ H (X;ΩX/k). In fact, there is an 2 · analogue c(L) of the first Chern class c1(L) with values in HZar(X;ΩX/k), 1 2 · more precisely in F HZar(X;ΩX/k), which we can see as follows: the map ∗ 1 OX → ΩX/k defined by h 7→ dh/h gives rise to a morphism of complexes

∗ ≥1 · OX [−1] → σ ΩX/k.

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

2 ∗ 1 ∗ 2 ≥1 · 1 2 · c : HZar(X; OX [−1]) = HZar(X; OX ) → HZar(X; σ ΩX/k) → F HZar(X;ΩX/k),

1 2 · 1 1 which is compatible with the natural map F HZar(X;ΩX/k) → H (X;ΩX/k). 1 2 · ∼ 2 However, the image of c(L) in F HZar(XC;ΩX / ) = H (X; C) only agrees C C √ 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 (or rational) structure on H2(Xan; ). C C

5.6 The Leray spectral sequence In this section, we describe the Leray spectral sequence, one of the most important examples of the Grothendieck spectral sequence for the derived functors of a composition of two functors. Let f : X → Y be a continuous map of topological spaces. If F is a sheaf on X, we have the direct image sheaf f∗F . The functor f∗ from sheaves on X to sheaves on Y is left exact, and we define the higher direct image i · i sheaves R f∗ by: choose an injective resolution F → I . Then R f∗F is th · the i cohomology sheaf of the complex f∗I . As before, this definition is i independent of the choice of an injective resolution, and the R f∗ are the derived functors of f∗. A straightforward argument (cf. Hartshorne) shows i that R f∗F is the sheaf associated to the presheaf on Y defined by:

U 7→ Hi(f −1(U); F ).

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

i Definition 42. A sheaf K is f∗-acyclic if R f∗K = 0 for all i > 0. For · example, an injective sheaf is f∗-acyclic. An f∗-acyclic resolution F → K i of F is a resolution such that K 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, i i · R f∗F = H f∗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 a flasque 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 on Y . Theorem 46 (Leray spectral sequence). Let F be a sheaf on X. There is · a spectral sequence abutting to H (X; F ) with E2 term given by p,q p q E2 = H (Y ; R f∗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 image · complex f∗C . The first hypercohomology spectral sequence has E1 term p,q q p E1 = H (Y ; f∗C ). p q p By Lemma 45, f∗C is flasque for all p. Thus H (Y ; f∗C ) = 0 for q > 0. So p,q p,0 0 p 0 p E1 = 0 if q > 0, and E1 = H (Y ; f∗C ) = H (X; C ). The differential p,0 p+1,0 p p+1 d1 : E1 → E1 is the differential induced by d: C → C . Thus, p,0 p p,0 E2 = H (X; F ) = E∞ , the spectral sequence degenerates at E2, and the p · p · ∼ p filtration on H (Y ; f∗C ) is trivial. Hence H (Y ; f∗C ) = H (X; F ). · · The second spectral sequence for H (Y ; f∗C ) has E2 term p,q p q p E2 = H (Y ; H f∗C ). q p ∼ q By Lemmas 43 and 44, H f∗C = R f∗F . Putting the results about the two spectral sequences together gives the statement of the theorem. p,0 Remark 47. The edge homomorphisms induce homomorphisms E2 = p p p 0,p 0 p H (Y ; f∗F ) → H (X; F ) and H (X; F ) → E2 = H (Y ; R f∗F ). In fact, p p we can see the homomorphism H (Y ; f∗F ) → H (X; F ) quite directly. More generally, 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 easy −1 · ∼ · −1 −1 to check that f CGo(G) = CGo(f G). Moreover, f is an , −1 · ∼ · −1 −1 so we get: f CGo(G) = CGo(f G) is a flasque resolution of f G. The homomorphism Γ(Y ; G) → Γ(X; f −1G) extends to a morphism · −1 · ∼ · −1 Γ(Y ; CGo(G)) → Γ(X; f CGo(G)) = Γ(X; CGo(f G)). Taking cohomology yields Hp(Y ; G) → Hp(X; f −1G). Finally, setting G = −1 f∗F and using the canonical homomorphism f f∗F → F gives the desired p p homomorphism H (Y ; f∗F ) → H (X; F ). Moreover, if F is f∗-acyclic, then p p H (Y ; f∗F ) → H (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 page q p q p p is clearly H (Y ; f∗K ) → H (X; K ). By the above remark, since K is q p q p f∗-acyclic, H (Y ; f∗K ) → H (X; K ) is an isomorphism. Hence the map · · · · H (Y ; f∗K ) → H (X; K ) is an isomorphism as well. We can also generalize the above to hyperdirect images: Let K· be a complex of sheaves on X. Choosing an injective resolution K· → I·, we define i · i · R f∗K = H f∗I . Arguments similar to those given for the proof of Theorem 34 show:

k · Theorem 49. There are two spectral sequences abutting to R f∗K . The first has E1 term given by

p,q q p E1 = R f∗K ,

p p+1 and the differential d1 is induced by d: K → K . The second spectral sequence has E2 term given by

p,q p q · E2 = R f∗H K .

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

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

p,q p q E2 = H (Y ; R f∗K).

5.7 Products We give a very brief treatment of products in Cechˇ cohomology and Cechˇ hypercohomology.

23 Definition 51. Suppose that F1 and F2 are two sheaves on F and that we have an R-bilinear map from F1×F2 to a third sheaf G, i.e. a homomorphism F1 ⊗ F2 → G. We denote the value on sections s1 of F1 and s2 of F2 by s1 · s2. Let {Uα} be an open cover of X. Define the cup product ˇn ˇm ˇn+m `: C ({Uα}; F1) × C ({Uα}; F2) → C ({Uα}; G) by the formula

(s ` t)α0,α1,...,αk+1 = sα0,α1,...,αn · tαn,α1,...,αn+m . A straightforward calculation shows that

n δ(s ` t) = δ(s) ` t + (−1) s ` δ(t).

Hence ` induces an R-bilinear map, also denoted `: ˇ n ˇ m ˇ n+m `: H ({Uα}; F1) × H ({Uα}; F2) → H ({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 differential d(s⊗t) = ds⊗t+(−1)ps⊗dt. A basic example of this is as follows: Given a complex K·, define the complex Hom·(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 on Hom·(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 above sense. Returning to the general picture of a morphism ϕ: A· ⊗ K· → L· of complexes in the above sense, we would like to define analogous cup products

· · · · · · Cˇ ({Uα}; A ) ⊗ Cˇ ({Uα}; K ) → Cˇ ({Uα}; L ):

p q r s For s ∈ Cˇ ({Uα}; A ) and t ∈ Cˇ ({Uα}; K ), set

rq s ` t = (−1) s ` t. ˇp q ˇr s ˇp+q r+s Then ` : C ({Uα}; A )⊗C ({Uα}; K ) → C ({Uα}; L ) and one checks that n D(s ` t) = D(s) ` t + (−1) s ` D(t), in case s is homogeneous of degree n. Thus there is an induced cup product

ˇ · · ˇ · · ˇ · · 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 a general fact that we can also define pairings of complexes for the Godement resolution, or more generally for injective resolutions, giving a cup product

n m n+m H (X; F1) ⊗ H (X; F2) → H (X; G).

Moreover, if {Uα} is a Leray cover for F1, F2, and G, then these cup products agree under the isomorphisms. Finally, suppose that the cohomology of F1, · · F2, and G can be computed by complexes of sheaves A·, K , L which are acyclic with respect to the p-fold intersections of the open sets {Uα} and such that there is a pairing of complexes ϕ: A· ⊗ K· → L· as above which is · · compatible with the inclusions F1 → A·, F2 → K , G → L . Then the cup product ˇ · · ˇ · · ˇ · · 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 to check that it also agrees with the cup product

p q p+q Hˇ ({Uα}; F1) × Hˇ ({Uα}; F2) → Hˇ ({Uα}; G) as defined above. · For example, if M is a manifold, taking F1 = F2 = G = C and A = · · · K = L = AM with wedge product, we can identify wedge product on de Rham cohomology with usual cup product on Cechˇ cohomology (and on singular cohomology with a little more work). A similar argument, for a complex manifold M and a Stein cover {Uα}, identifies cup product

ˇ q p ˇ s r ˇ q+s p+r H ({Uα};ΩM ) × H ({Uα};ΩM ) → H ({Uα};ΩM ) 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 shall not write down the general statement here.

26