Cohomology of Sheaves

In order to get at surfaces, we don’t just need the scheme theory of the last two posts, but we also need to understand the cohomology of sheaves. We will not be proving all of the details, we will also not be doing things in full generality with derived functors, at least not until we absolutely must. Today, we’re going to talk about Čech Cohomology.

To begin, we take $X$ to be any topological space and $\mathscr{F}$ to be a sheaf of abelian groups on $X$. We can then pick an open cover of $X$ consisting of $\{U_\alpha\}$ and call it $\mathcal{U}$. We will use the convention that $U_{i_0}\cap\ldots\cap U_{i_p}$ will be written as $U_{i_0,\ldots,i_p}$.

Now, for each $p\geq 0$, we define $C^p(\mathcal{U},\mathscr{F})=\prod_{i_0<\ldots. That is, we order the elements of the cover and then look at the $p+1$ fold intersections, and take a section of the sheaf over each one to give an element of $C^p(\mathcal{U},\mathscr{F})$.

Now we define the coboundary map $d:C^p\to C^{p+1}$ by, for $\alpha\in C^p$, $(d\alpha)_{i_0,\ldots,i_{p+1}}=\sum_{k=0}^{p+1} (-1)^k \alpha_{i_0,\ldots,\hat{i}_k,\ldots,i_{p+1}}|_{U_{i_0,\ldots,i_{p+1}}}$. That is, on each $p+2$ tuple, we take $\alpha$ on each $p+1$ subtuple, restrict to the smaller open set, and then add them together with alternating signs. It’s a nasty calculation to check that $d^2=0$, but it is, so the image of $d:C^p\to C^{p+1}$ is in the kernel of $d:C^{p+1}\to C^{p+2}$.

So we define $Z_p(\mathcal{U},\mathscr{F})$ to be the kernel of $d:C^p\to C^{p+1}$, and $B_p(\mathcal{U},\mathscr{F})$ to be the image of $d:C^{p-1}\to C^p$. We call the elements of $Z_p$ cocycles and the elements of $B_p$ coboundaries, mostly because when homology comes up first it’s in a topological context, and there is real meaning to boundary and cycles in the geometry/topology of the situation. Here it’s an algebraic condition on cocycles, and a very useful one.

So finally, we define the Čech Cohomology groups of the sheaf $\mathscr{F}$ with the cover $\mathcal{U}$ to be $H^p(\mathcal{U},\mathscr{F})=Z_p/B_p$, and those of just the sheaf to be the limit taken over all refinements of the cover. However, we won’t care about this for now, we’re going to jump to a special case.

We’ve previously discussed quasicoherent sheaves of modules over a locally ringed space. We’re now going to stick to quasicoherent sheaves on a scheme, but not just any scheme: a noetherian scheme. We say that a scheme is noetherian if it can be covered by a finite number of open affines $\mathrm{Spec}(A_i)$ with the $A_i$ all noetherian rings. This automatically forces the underlying topological space to be noetherian, which means that descending closed subsets stabilize after only finitely many.

The noetherian hypothesis is extremely valuable, because it gives us the Serre Vanishing Theorem, which states that if $X$ is a noetherian scheme, then $X$ is affine if and only if for all $\mathscr{F}$ quasicoherent, the nonzero cohomology all vanishes. The key point being that noetherian affine schemes have no cohomology to worry about.

So with this fact backing us up, until we look at derived functors, we will only take cohomology of quasicoherent sheaves on a noetherian separated scheme (all varieties are such, after all) and we will require that our covers $\mathcal{U}$ are open affine covers. With these hypotheses, we don’t lose anything in not doing the massive derived functor formalism, because the cohomologies turn out to be the same. However, we have restricted to noetherian schemes, and we have restricted to quasi-coherent sheaves, and we need to do more work to deal with things outside of this restricted world. In this context, the cohomology doesn’t depend on the cover, and we will write $H^p(X,\mathscr{F})$.

So before moving on, we should discuss some of the things that we can use cohomology for and a few theorems that we’ll just use. One big one is to compute the global sections of a sheaf, because $H^0(X,\mathscr{F})\cong \Gamma(X,\mathscr{F})$. This can, of course, be of use when looking at the line bundles associated to divisors, and their linear systems.

A nice theorem is the fact that if $X$ has dimension $n$, then for $i>n$ and $i<0$ and any quasicoherent sheaf, we have $H^i(X,\mathscr{F})=0$. This way we don’t have to compute all the possible cohomology groups in order to know if we’ve got all the information cohomology can give us.

And last, a classification theorem that does generalize we take $\mathscr{O}_X^*$ to be the sheaf whose sections are the invertible sections of $\mathscr{O}_X$. This sheaf has a remarkable property: $H^1(X,\mathscr{O}_X^1)\cong \mathrm{Pic}(X)$. This is, modulo some technical work, because the cocycle condition for elements of $H^1$ is that $g_{\alpha\beta}g_{\beta\gamma}g_{\gamma\alpha}=1$ where $g_{\alpha\beta}$ are invertible functions on $U_{\alpha\beta}$. That is, transition functions for line bundles. So this is saying that if you go in a circle on the base of a line bundle, you get back where you started, and any way you have of doing it gives some line bundle.

We’ll close on that, next time, a theorem of Serre’s about the cohomology of projective schemes.