Sheaf cohomology

To continue, I am now going to have to use the language of sheaves. For it, and for all details I will omit here, I refer the reader to Charles Siegel’s post at Rigorous Trivialties and Hartshorne’s Algebraic Geometry. When I talk about sheaf cohomology, it will always be the derived functor cohomology. I will briefly review some of these ideas.

Sheaf cohomology

The basic properties of this are as follows.

First, if ${X}$ is a topological space and ${i \in \mathbb{Z}_{\geq 0}}$ , then ${H^i(X, \cdot)}$ is a covariant additive functor from sheaves on ${X}$ to the category of abelian groups. We have

$\displaystyle H^0(X,\mathcal{F}) = \Gamma(X,\mathcal{F}),$

that is to say, the global sections. Also, if

$\displaystyle 0 \rightarrow \mathcal{F} \rightarrow \mathcal{G} \rightarrow \mathcal{H} \rightarrow 0$

is a short exact sequence of sheaves, there is a long exact sequence

$\displaystyle H^i(X,\mathcal{F}) \rightarrow H^i(X, \mathcal{G})\rightarrow H^i(X, \mathcal{H}) \rightarrow H^{i+1}(X,\mathcal{F}) \rightarrow \dots .$

Finally, sheaf cohomology (except at 0) vanishes on injectives in the category of sheaves.

In other words, sheaf cohomology consists of the derived functors of the (left-exact) global section functor.

When ${\mathcal{F}}$ is flasque (i.e., for any ${U,V \subset X}$ open with ${U \subset V}$ , restriction ${\mathcal{F}(V) \rightarrow \mathcal{F}(U))}$ is surjective), we have

$\displaystyle H^i(\mathcal{F}, X) = 0 \ \mathrm{for} \ i \geq 1.$

(This is basically because of a well-known fact about sheaves: if ${0 \rightarrow \mathcal{F} \rightarrow \mathcal{G} \rightarrow \mathcal{H} \rightarrow 0 }$ is exact and ${\mathcal{F}}$ is flasque, then the sequence of global sections is also exact.)

Moreover, this gives a way to compute the cohomology of a sheaf ${\mathcal{F}}$ . If we have a flasque resolution ${0 \rightarrow \mathcal{F} \rightarrow \mathcal{G}_1 \rightarrow \mathcal{G}_2 \rightarrow \dots}$ , then consider the complex ${\Gamma(\mathcal{G})}$ of the global sections of the sheaves ${\mathcal{G}_i}$ (and ${0}$ ). The key formula is then:

$\displaystyle H^i(X, \mathcal{F}) = H^i(\Gamma(\mathcal{G})) .$

There are many interesting results about the cohomology of a coherent sheaf over schemes, though they are not really relevant to us now. We are interested in the analytic applications.

Cech cohomology

Cech cohomology gives a reasonable way to actually compute these cohomology groups. For instance, Hartshorne uses it to calculate the cohomology of line bundles on projective space. In general, Cech cohomology does not equal derived functor cohomology, though it does in certain cases (in algebraic geometry, if the scheme is separated, then it is ok).

In the analytic case, we are working with paracompact spaces, so Cech cohomology will always be acceptable, as we will see.

Let ${\mathfrak{U} = \{U_i, i \in I\}}$ be a covering of ${X}$ , and let ${\mathcal{F}}$ be a sheaf on ${X}$ . Consider the chain complex

$\displaystyle C^k(\mathfrak{U}, \mathcal{F}) := \prod_{i_1, \dots, i_k \in I} \mathcal{F}(U_{i_1} \cap \dots \cap U_{i_k} )$

with boundary maps defined as follows. Let ${c \in C^k(\mathfrak{U}, \mathcal{F})}$ with corresponding

$\displaystyle c_{i_1 \dots i_k} \in \mathcal{F}(U_{i_1} \cap \dots \cap U_{i_k} )$

for every ${k}$ -tuple ${i_1, \dots, i_k \in I}$ . Define

$\displaystyle (\delta c)_{i_1 \dots i_{k+1}} = \sum_{j=1}^{k+1} (-1)^j c_{i_1 \dots \hat{i_j} \dots i_{k+1} }| ( U_{i_1} \cap \dots \cap U_{i_{k+1}}),$

where the hat denotes omission, as usual. (Also, ${C^k(\mathfrak{U}, \mathcal{F})=0}$ when ${k \leq 0}$ .) This can be checked to be a complex. Then we define the Cech cohomology groups

$\displaystyle H^i( \mathfrak{U}, \mathcal{F}) := H^i( C^k(\mathfrak{U}, \mathcal{F}) )$

as the cohomology of this (co)chain complex.

There is another way to interpret the Cech cohomology groups that better illustrates their connection with regular cohomology. Given ${\mathfrak{U}, \mathcal{F}}$ , consider the sheaves ${\mathcal{C}^k := \mathcal{C}^k( \mathfrak{U}, \mathcal{F})}$ defined by

$\displaystyle \mathcal{C}^k(V) = \prod_{i_1, \dots, i_k \in I} \mathcal{F}(V \cap U_{i_1} \cap \dots \cap U_{i_k} ).$

The boundary map ${\delta}$ is defined the same as before, so we have a complex of sheaves

$\displaystyle 0 \rightarrow \mathcal{F} \rightarrow \mathcal{C}^1 \rightarrow \mathcal{C}^2 \rightarrow \dots.$

Proposition 1 The above complex is exact.

Exactness at the first step is basically the definition of a sheaf: cocyles in ${\mathcal{C}^1(V)}$ represents collections of elements of ${\mathcal{F}(V \cap U_i)}$ that agree on the common intersections, so piece together to a section of ${\mathcal{F}(V)}$ .

Now consider the other more general case. We need to check exactness on the stalks, say at ${x}$ . Choose ${i'}$ with ${x \in U_{i'}}$ . Let ${c \in \mathcal{C}^k(V)}$ be such that ${\delta c = 0}$ ; we can assume ${V \subset U_{i'}}$ . We need to lift ${c}$ on some neighborhood containing ${x}$ to get some ${d}$ that goes to ${c}$ via ${\delta}$ .

Thus, for all ${k-1}$ -tuples ${i_1, \dots, i_{k-1}}$ , define

$\displaystyle d_{i_1, \dots, i_{k-1}} = c_{i' , i_1 , \dots, i_{k-1}} | (V \cap U_1 \dots U_{k-1}).$

Then ${d}$ is a section of ${\mathcal{C}^{k-1}}$ on ${V}$ . In this way, we get a map from ${\mathcal{C}^k_x \rightarrow \mathcal{C}^{k-1}_x}$ , and it is easy to check that it is a chain homotopy, which implies exactness.

Cech cohomology versus derived functor cohomology

I will now discuss some examples of Cech cohomology. The notation remains the same.

Proposition 2 ${H^0(\mathfrak{U}, \mathcal{F}) = \Gamma(X,F)}$ .

This is really just the sheaf axiom; see the beginning of the proof of exactness of the Cech complex of sheaves.

The next proposition gives us some reason to suspect a relation between tehse two types of cohomology.

Proposition 3

There is a natural transformation of ${\delta}$ -functors $\displaystyle H^i(\frak{U},\cdot) \rightarrow H^i(X, \cdot)$

if ${\frak{U} = \{U_i\}}$ .

Let

$\displaystyle 0 \rightarrow \mathcal{F} \rightarrow \mathcal{G}^1 \rightarrow \mathcal{G}^2 \rightarrow \dots$

be an injective resolution of ${\mathcal{F}}$ . Then by a basic result in homological algebra, there is a unique commutative diagram of resolutions

which induces a corresponding commutative diagram on the global sections. Taking the cohomology of the complex then gives the result.

Next up: the Leray theorem.

Incidentally, I’m not sure how I should categorize this post. I’m using “algebraic geometry” because that subject seems to depend the most heavily on sheaves.

hilbertthm90 Says:

January 7, 2010 at 11:20 pm

It is interesting that you say “This is basically because of a well-known fact about sheaves: if ${0 \rightarrow \mathcal{F} \rightarrow \mathcal{G} \rightarrow \mathcal{H} \rightarrow 0 }$ is exact and ${\mathcal{F}}$ is flasque, then the sequence of global sections is also exact.”

I’d think to prove this by saying that the flasque resolution of $\mathcal{F}$ is $0\to \mathcal{F} \to \mathcal{F}\to 0$ . Thus $R^1\Gamma (X, \mathcal{F})=0$ .

Thus the induced long exact sequence from the short exact one is $0\to\Gamma (X, \mathcal{F})\to\Gamma (X, \mathcal{G})\to \Gamma(X, \mathcal{H})\to R^1\Gamma(X, \mathcal{F})=0$ .

I don’t see an obvious “low level” or “element-wise” proof of this.

Akhil Mathew Says:

January 8, 2010 at 7:35 pm
Well, the definition here uses injective resolutions to compute sheaf cohomology. If you use flasque resolutions, I don’t see how easy it would be to prove that the value of the sheaf cohomology is independent of the flasque resolution taken.

The claim using the derived functor definition can be proved in detail as follows: let $0 \to \mathcal{F} \to \mathcal{G}^1 \to \mathcal{G}^2 \dots$ be an injective resolution of $\mathcal{F}$ . Let $\mathcal{H}^i = \mathcal{F}^i \to \mathcal{F}^{i+1}$ ; this is also a cokernel, so thus flasque.

Now there is an exact sequence
$0 \to \mathcal{F} \to \mathcal{G}^1 \to \mathcal{H}^1 \to 0$ . There is an exact sequence
$0 \to \Gamma( X, \mathcal{F}) \to \Gamma(X, \mathcal{G}^1) \to \Gamma(X, \mathcal{H}^1) \to H^1( X, \mathcal{F}) ) \to 0$ , by the injectivity hypothesis. Now the map from $\Gamma(\mathcal{G}^1) \to \Gamma(\mathcal{H}^1)$ is surjective by flasqueness, which implies $H^1( X, \mathcal{F}) ) = 0$ . So since $\mathcal{F}$ was arbitrary and flasque, $H^1( X, \mathcal{H}^j) ) = 0$ . Then a dimension-shifting argument implies the cohomology of $\mathcal{F}$ vanishes.