I am now going to discuss Kempf’s proof of the theorem of Serre. Note that this is lifted verbatim of some notes I have been taking, so apologies if the style seems out of place as a result.  I use the (highly nonstandard) notation $\mathcal{G}m$ for global sections of a sheaf for entirely logistical (and typo-errorgraphical) reasons.  Since this is really better suited to a PDF, I’ll also post that.

(Note: You really should read the PDF, since some diagrams are missing from this post.)

Theorem 1 (Serre) Let ${X = \mathrm{Spec} A}$ be an affine scheme, ${\mathcal{F}}$ a quasi-coherent sheaf. Then ${H^i(X, \mathcal{F}) = 0}$ for ${i \geq 1}$.

We shall prove this result following Kempf. The idea is that ${X}$ has a very nice basis: namely, the family of all sets ${D(f), f \in A}$. These are themselves affine, and moreover the intersection of any two elements in this basis is still in this basis. For ${D(fg) = D(f) \cap D(g)}$.

0.1. A lemma of Kempf (more…)

There is a theorem of Serre that the higher cohomology groups of quasi-coherent sheaves on an affine scheme all vanish.  This is proved in Hartshorne for noetherian rings by showing that the sheaves associated to injective modules are flasque, so can be used to compute cohomology; this proof makes the annoying noetherian hypothesis though. There is a paper of Kempf where he explains how to avoid this, and in fact use pretty much nothing more about sheaf cohomology than its trivalty on flasque sheaves. I’ve been reading it as of late, and I recommend it. Perhaps it will become a blog post shortly.

I’ve been away from this blog for too long–partially it’s because most of my expository energy has gone into preparing a collection of notes on algebraic geometry (to help me learn the subject). Someday I’ll post them.

Today, however, I’d like to talk about a clever proof I learned recently.

The following result is neat:

Theorem: Let ${M}$ be a compact smooth manifold. Then the de Rham cohomology groups ${H^i_{DR}(X, \mathbb{R})}$ are finite-dimensional.

I’m pretty sure it follows from Hodge theory and the finite-dimensionality of the harmonic forms. However, I learned a neat elementary proof that I’d like to discuss.

By de Rham’s theorem, we can compute the de Rham cohomology groups as the sheaf cohomology groups ${H^i(X, \mathbb{R})}$ for ${\mathbb{R}}$ denoting the constant sheaf associated to the group ${\mathbb{R}}$. Now, pick a Riemannian metric on ${M}$. Each point has a neighborhood ${U}$ such that any two points in ${U}$ are joined by a unique geodesic contained in ${U}$—such a neighborhood is called geodesically convex. It is clear that a geodesically convex neighborhood is homeomorphic (via the exponential map) to a convex set in ${\mathbb{R}^n}$, which has trivial de Rham cohomology, and also that the intersection of two geodesically convex sets is geodesically convex.

So pick a finite cover of ${M}$ by geodesically convex sets ${U_1, \dots, U_k}$. Then on every intersection ${U_{i_1} \cap \dots \cap U_{i_n}}$, the sheaf ${\mathbb{R}}$ has trivial cohomology because this intersection is geodesically convex, hence diffeomorphic to a convex set in ${\mathbb{R}^n}$. In particular, the cover ${\{U_i\}}$ satisfies the hypotheses of Leray’s theorem. We can apply Cech cohomology with this cover to compute ${H^i(X, \mathbb{R})}$, or equivalently the de Rham cohomology.

But there are finitely many sets in this cover, and the sections of the sheaf ${\mathbb{R}}$ over each of these sets is just the abelian group ${\mathbb{R}}$ by connectedness of anything geodesically convex. So the Cech complex consists of finite-dimensional vector spaces; its cohomology thus consists of finite-dimensional vector spaces. $\Box$

I learned this from Bott and Tu’s Differential Forms in Algebraic Topology, which appears to be a really fun read.

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. (more…)