Chapter 8

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia

Last update: 11 June 2013

Sheaf Cohomology

We need some basic homological algebra. Let 𝒞 be an abelian category (for our purposes, 𝒞 will be the category of 𝒪-Modules, where 𝒪 is a sheaf of rings on a topological space X). An object I in 𝒞 is injective if the functor AI(A)=Hom(A,I) is exact and not merely left exact: that is to say, whenever AB is a monomorphism in 𝒞, the map I(B)(IA) is surjective.

The category 𝒞 has enough injectives if every object in 𝒞 can be embedded in an injective object. Suppose that 𝒞 has enough injectives, and let A be an object in 𝒞. Then there exists an injective I0 in 𝒞 and a monomorphrsm μ:AI0. Let A1=Coker(μ), then there exists an injective I1 in 𝒞 and a monomorphism μ1:A1I1. Let A2=Coker(μ1), and so on. The short exact sequences 0AI0A1 0, 0A1I1A2 0, etc., then stick together to form a long exact sequence:

() 0AI0 a0I1 a1I2 a2

called an injective resolution of A.

Now let F be a covariant additive left exact functor on 𝒞 with values in an abelian category 𝒞. If we operate on () with F, we get a complex, so we can form its cohomology:

Hp=KerF(ap) /ImF(ap-1) (p0;a-1=0) .

The central fact is that Hp depends (up to isomorphism) only on F and A and not on the injective resolution: it is denoted by RpF(A), and RpF is an additive functor, called-the pth right derived functor of F. Since F is left exact, we have R0F=F.

If A is injective then RpF(a)=0 for all p>0; for 0AA0 is an injective resolution of A.

Theorem (8.1). If 0AαB βC0 is an exact sequence in 𝒞, and if F is a covariant additive left exact functor on 𝒞 with values in an abelian category 𝒞, then there is an exact sequence in 𝒞:

0F(A) F(a)F(B) F(β)F(C) R1F(A) R1F(a) F1F(B) R1F(β) R1F(C) R2F(A)

For the definition of the 'coboundary morphisms' : Rp-1F(C) RpF(A) and the proof of (8.1) we refer to Godement's book (or any book on homological algebra).

Grothendieck cohomology

We shall apply this machinery to the following situation: (X,𝒪) is a ringed space and 𝒞 is the category of 𝒪-Modules. Then 𝒞 is abelian (as remarked in Chapter 7) and in fact 𝒞 has enough injectives (proof e.g. in Godement's book). By (7.2), the section functor Γ is a left exact functor on 𝒞 with values in the category of 𝒪(X)-modules. The cohomology groups (which are in fact 𝒪(X)-modules) of X with coefficients in the 𝒪-Module are then defined to be

Hp(X,)=Rp Γ() (p0).

In particular, H0(X,)=Γ(X,). From (8.1) we have an exact cohomology sequence: if 0𝒢0 is an exact sequence of 𝒪-Modules, then the sequence

0Γ(X,) Γ(X,𝒢) Γ(X,) H1(X,) H1(X,𝒢) H1(X,) H2(X,)

is exact.

This definition of the cohomology groups Hp(X,) is due to Grothendieck.

Čech cohomology

There is an earlier definition of sheaf cohomology, modelled on Čech theory, which goes as follows. Let 𝓊=(Ui)iI be any open covering of X. If σ=(i0,,ip) is any p-simplex, i.e, sequence of p+1 elements of the index set I, let Uσ denote the intersection Ui0Uip. An (alternating) p-cochain of the covering 𝓊 with coefficients in the sheaf is a function c which associates with each p-simplex σ an element cσ(Uσ) in such a way that cσ is alternating in the indices i0,,ip, and cσ=0 whenever any two of the indices are equal. The p-cochains form a group Cp(𝓊,), which has a natural 𝒪(X)-module structure: if a𝒪(X), then (ac)σ is defined to be (a|Uσ)·cσ. If we order the index set I linearly then we may write Cp(U,)= σ(Uσ), where in the product σ runs over all p-simplexes (i0,,ip) such that i0<i1<<ip.

Define a coboundary homomorphism

d:Cp(𝓊,) Cp+1(𝓊,)

as follows: if cCp(𝓊,), then

(dc) i0ip+1 =k=0p+1 (-1)k c i0i^k ip+1 | Ui0ip+1

One verifies that d2=0. Thus C(𝓊,)= p0Cp (𝓊,) is a complex of 𝒪(X)-modules, and we define the pth cohomology group of the covering 𝓊 with coefficients in to be

Hp(𝓊,)=Hp (C(𝓊,)) (p>0).

Next one shows that a refinement 𝓊 of 𝓊 gives rise to well-defined homomorphisms Hp(𝓊,) Hp(𝓊,) with the usual transitivity properties; these enable us to define the Čech cohomology groups of X with coefficients in :

Hˇp(X,)= lim𝓊Hp (𝓊,),

the direct limit being taken over arbitrarily fine open coverings 𝓊 of X.

The advantage of Čech cohomology is that one stands some chance of being able to compute it in given situations. The disadvantage, which is a serious one, is that the cohomology sequence in Čech cohomology is not necessarily exact. It is always the case that Hˇ0=H0 and Hˇ1=H1 (so that the Čech cohomology sequence is always exact as far as Hˇ1) but Hˇp(X,) and Hp(X,) are not necessarily the same for p>1. There is a spectral sequence relating the two cohomologies (details in Godement's book), from which one can assert that Hˇp=Hp for all p under suitable hypotheses on X or or both. Here are two such 'comparison theorems':

Theorem (8.2). Let 𝓊 be an open covering of X, let be a sheaf on X, and suppose that, for all simplexes σ=(i0,,ip), we have Hp(Uσ,|Uσ)=0 for all q>0. Then Hq(X,)= Hq(𝓊,) for all q0.

Theorem (8.3). (Cartan). Let 𝓊 be an open covering of X and a sheaf on X such that

(i) 𝓊 is closed under finite intersections;
(ii) the sets of 𝓊 form a basis of X;
(iii) Hˇq(U,|U)=0 for all U𝓊 and all q>0.

Then Hˇq(X,) Hq(X,) for all q0.

Theorems (8.2) and (8.3) are proved in Godement's book. We shall sketch a proof of (8.2) avoiding the use of spectral sequences, but not (8.3). There are other comparison theorems: thus the conclusion of (8.3) is valid if X is paracompact (and Hausdorff) and is any sheaf of abelian groups. This one is of use if X is a differentiable manifold or a complex manifold, but not in algebraic geometry.

The Čech resolution of a sheaf

Let (X,𝒪) be a ringed space and an 𝒪-Module, and let 𝓊=(Ui)iI be any open covering of X. For each open set V in X let V𝓊 denote the open covering (VUi)iI of V. Then we have 𝒪(V)-modules Cp(V𝓊,|V) for each open set V in X and each integer p0, hence presheaves VCp(V𝓊,|V) for each p0. These presheaves are easily verified to be sheaves; denote them by Cp(𝓊,). The coboundary operator d:CpCp+1 gives rise to sheaf homomorphisms

d:Cp(𝓊,) Cp+1(𝓊,);

also we have a sheaf homomorphism j:C0(𝓊,) defined as follows: if s is a section of over V, then

j(s)= (s|VUi)iI C0 (V𝓊,|V)= C0(𝓊,)(V).

Proposition (8.4). The sequence

0jC0 (𝓊,)d0 C1(𝓊,) d1

is exact.


(i) j is a monomorphism. For if j(s)=0 then s|VVi=0 for all iI, hence s=0 (since the VUi cover V).

(ii) Im(j)=Ker(d0). Let s=(si)C0(V). If ds=0 then (ds)ij=0 for all pairs (i,j) in I, i.e. si=sj in VUiUj; hence the si fit together to give a section s of over V such that s|VUi=si for each i; i.e. s=j(s). Conversely, if s=j(s) for some s(V), then si=s|VUi, hence (ds)ij= sj|VUiUj -si|VUiUj =0.

(iii) Im(dp-1)=Ker(dp). We have dpdp-1=0, hence Im(dp-1)Ker(dp). Conversely, let u(Cp(𝓊,))x be such that du=0: say xUi. Then there exists an open neighbourhood V of x contained in Ui and an element sCp(𝓊,)(V) such that sx=u. If σ=(i0,,ip-1) is a (p-1)-simplex, iσ let iσ denote the p-simplex (i,i0,,ip-1). We have Cp(𝓊,)(V)=Cp(V𝓊,|V), hence s is a family (sτ) where τ runs through the p-simplexes and sτ(VUτ). Define tCp-1(𝓊,)(V) by the rule tσ=siσ (VUiUσ) =(VUσ); then

(dt)τ = k=0p (-1)ktτk| VUτ (τk=kth 'face' ofτ) = k(-1)k siτk| VUτ = sτ-(ds)iτ = sτsinceds=0.

Hence dt=s and therefore Ker(dp) Im(dp-1).

Proof of (8.2).

(i) Any product of injectives is injective (this is true in any abelian category).

(ii) If is an injective 𝒪-Module and U is open in X, then |U is an injective 𝒪|U-Module. For we have Hom𝒪|U(𝒢,|U)= Hom𝒪(𝒢X,) for any 𝒪|U-Module 𝒢, where 𝒢X denotes the sheaf on X obtained by extending by zero outside U.

(iii) If 𝒥 is an injective 𝒪|U-Module and i:UX is the embedding of the open set U in X, then i𝒥 is an injective 𝒪-Module. For we have Hom𝒪(,i𝒥) Hom𝒪|U(|U,𝒥) for any 𝒪-Module .

(iv) With the notation of (8.4), we have Cq(𝓊,)(V)= σ(VUσ)= σiσ*(|Uσ)(V) where σ runs through all q-simplexes (i0,,iq) such that i0<<iq (with respect to some linear ordering of the index set I) and iσ is the embedding of Uσ in X. Hence if is injective, then Cq(𝓊,) is injective by (i), (ii) and (iii).

(v) Let 001 be an injective resolution of . Then for each simplex σ the sequence 0|Uσ0|Uσ1|Uσ is an injective resolution of |Uσ, by (iii) and the fact that restriction to an open set preserves exactness. Hence this sequence can be used to calculate the cohomology of |Uσ. But by hypothesis Hq(Uσ,|Uσ)=0 for all q>0. Hence the sequence

0(Uσ) 0(Uσ) 1(Uσ)

is exact. Hence, taking the product of these exact sequences for, all q-simplexes σ, the sequence

0Cq(𝓊,) Cq(𝓊,0) Cq(𝓊,1)

is exact.

(vi) Consider next the Čech resolution of p:

0pC0 (𝓊,p) C1(𝓊,p)

By (8.4) this is an exact sequence. By (iv) above, each Cq(𝓊,p) is injective, hence this is an injective resolution of p; but p has zero cohomology in dimensions >0, hence the sequence

0p(X) C0(𝓊,p)(X) C1(𝓊,p) (X),

that is to say the sequence

0p(X) C0(𝓊,p) C1(𝓊,p) ,

is exact.

(vii) We now have a double complex, in which all rows except for the top one, and all columns except for the left-hand one, are exact sequences (by (v) and (vi)):

0 0 0 0 (X) C0(𝓊,) C1(𝓊,) 0 0(X) C0(𝓊,0) C1(𝓊,0) 0 1(X) C0(𝓊,1) C1(𝓊,1)

In such a situation the cohomology of the top row is isomorphic to the cohomology of the left-hand column. But in the present case the cohomology of the top row is the Čech cohomology Hp(𝓊,), and the cohomology of the left-hand column is Grothendieck cohomology Hp(X,).

Notes and References

This is a typed excerpt of the book "Algebraic Geometry: Introduction to Schemes - I.G. Macdonald".

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