Earlier, our proof of the vanishing of higher quasi-coherent cohomology on an affine was actually very incomplete. We actually computed only Cech cohomology, and waved our hands while pointing to a fancy sheaf-theoretic result of Cartan. I would like to prove this result today, following Godement’s Theorie des faisceaux.

Cech cohomology is (comparatively) easy to compute, for instance via the Koszul complex. But the problem is that we don’t a priori know if coincides with derived functor cohomology. We have a natural map between Cech and derived functor cohomology in any case, but in general it won’t be an isomorphism. Leray’s theorem is a sufficient condition for this, but its expression is fundamentally in terms of derived functor cohomology: you have to have an acyclic covering–a covering on which the derived functor cohomology is trivial. But a priori, how can we tell that an open set is acyclic? What if we only know Cech cohomology? The point of today’s post is to use the heavy machinery of the Cech-to-derived functor spectral sequence to get such a purely Cech-theoretical criterion.

Cartan’s theorem gives a sufficient criterion for this to be the case. The result is:

Theorem 42 Let ${X}$ be a space, ${\mathcal{F}}$ a sheaf on ${X}$. Suppose there is a basis ${\mathfrak{A} }$ of open sets on ${X}$, closed under finite intersections, satisfying the following condition. If ${\mathfrak{B} \subset \mathfrak{A}}$ is a finite open covering of ${U \in \mathfrak{A}}$, then the Cech cohomology in positive dimension vanishes,$\displaystyle H^k(\mathfrak{B}, \mathcal{F})=0.$

Then the natural map:

$\displaystyle H^k(\mathfrak{A}, \mathcal{F}) \rightarrow H^k(X, \mathcal{F})$

is an isomorphism, for any ${k \in \mathbb{Z}_{\geq 0}.}$

I confess to having stated the result earlier incorrectly, when I claimed that the conclusion was ${H^k(X, \mathcal{F})=0}$ for ${k \geq 1}$.

But in any case, this will finally(!) complete the proof of the vanishing of the higher quasi-coherent cohomology of an affine. For then we just take ${\mathfrak{A}}$ to be the collection of basic open affines. We have shown that the Cech cohomology with respect to this family covers vanishes (on the whole space and on any basic open set, which is also affine!).

We shall now approach the proof of the Cartan vanishing theorem. First, however, it will be necessary to describe a spectral sequence between Cech cohomology and derived functor cohomology. For now, the reason is that there isn’t any obvious way for us to compute derived functor cohomology, because injective sheaves are big and scary, while Cech cohomology is nice and concrete. And indeed, all we’ve done so far is compute various Cech cohomologies.

I should mention that I don’t know a standard reference for the material in this post. I didn’t find Godement’s treatment in Theorie des faisceaux to be terribly enlightening, but after a fair bit of googling I found a sketch in James Milne’s online notes on étale cohomology. Fortunately, enough details are given to enable one to work it out more fully for oneself.

Let ${X}$ be a topological space covered by an open cover ${\mathfrak{A} = \left\{U_i\right\}_{i \in I}}$, and consider the category ${\mathfrak{C}}$ of presheaves of abelian groups on ${X}$. Let ${\mathfrak{C}'}$ be the subcategory of sheaves. The spectral sequence will be the Grothendieck spectral sequence of the composite of functors

$\displaystyle \mathfrak{C}' \stackrel{F}{\rightarrow} \mathfrak{C} \stackrel{G}{\rightarrow} \mathbf{Ab}.$

It’s funny that the topics I planned to discuss on this blog in the next month have turned out to largely match the topics covered in my commutative algebra class. As a result, I think I will move the focus closer to algebraic geometry. Today, I shall explain how the connection explained yesterday between Koszul and Cech cohomology lets us compute the cohomology of an affine space. This is the proof that Grothendieck gives in EGA III, and it has the advantage (unlike the argument in Hartshorne, though I’ll probably later end up explaining that too) of applying to non-noetherian rings. It is a nice application of the basic properties of the Koszul complex, though it has the disadvantage of requiring a rather non-elementary result in sheaf theory (which I’ll discuss next).

0.10. The cohomology of affine space

We are now going to prove the first fundamental theorem on the cohomology of quasi-coherent sheaves:

Theorem 35 (Cohomology of an affine) Let ${R}$ be a ring, and let ${\mathcal{F}}$ be a quasi-coherent sheaf on ${X=\mathrm{Spec} R}$. Then$\displaystyle H^k( X, \mathcal{F})=0, \quad k \geq 1.$

I have earlier discussed a proof due to Kempf. What we will now sketch is a much less elementary and significantly more complicated argument. Nonetheless, it has the virtue of being general, and telling us something about projective space too, as we shall see eventually.

Proof: This proof proceeds first by analyzing the Cech cohomology. We will show that this is zero. Then, we shall appeal to some general sheaf-theoretic business to prove the result for standard cohomology. In particular, we are going to prove:

Theorem 36 Let ${\mathcal{F}}$ be a quasi-coherent sheaf on ${X=\mathrm{Spec} R}$. Let ${\left\{f_i\right\} \subset R}$ be a finite set of elements generating the unit ideal. Then the higher Cech cohomology of ${\mathcal{F}}$ with respect to the open cover ${D(f_i)}$ vanishes.

Today, I will discuss the proof of a technical lemma in Hartshorne on sheaf cohomology, which he uses to prove Grothendieck’s vanishing theorem by reducing to finitely generated sheaves.

Prerequisite: Some familiarity with the Grothendieck spectral sequence. I’m probably going to blog about this (and spectral sequences more generally) soon, though I need to polish what I have written so far. Note that this has already been blogged about over at A Mind for Madness.

Proposition 1 Let ${X}$ be a noetherian topological space, and ${\mathcal{F}_\alpha, \alpha \in A}$ an inductive system of abelian groups on ${X}$. Then ${H^i( X, \varinjlim \mathcal{F}_\alpha) \simeq\varinjlim H^i(X, \mathcal{F}_\alpha)}$ for each ${i}$.

The idea is to use the Grothendieck spectral sequence. First, consider the abelian category ${\mathfrak{C}}$ of ${A}$-indexed inductive systems of sheaves on ${X}$. This is a functor category of ${A}$ (a poset is a category!) in an abelian category (namely, the category of sheaves) so ${\mathfrak{C}}$ is an abelian category. Then the maps ${\left\{\mathcal{F}_\alpha\right\} \rightarrow H^i( X, \varinjlim \mathcal{F}_\alpha) ,\varinjlim H^i(X, \mathcal{F}_\alpha)}$ are functors on this category. We first study the first functor. (more…)

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…)