So last time, we introduced the first form of the formal function theorem. We said that if ${X }$ was a proper scheme over ${\mathrm{Spec} A}$ with structure morphism ${f}$, and ${\mathcal{I} = f^*(I)}$ for some ideal ${I \subset A}$, then there were two constructions one could do on a coherent sheaf ${\mathcal{F}}$ on ${X}$ that were in fact the same. Namely, we could complete the cohomology ${H^n(X, \mathcal{F})}$ with respect to ${I}$, and we could take the inverse limit ${ \varprojlim H^n(X, \mathcal{F}/\mathcal{I}^k \mathcal{F})}$. The claim was that the natural map

$\displaystyle \widehat{H^n(X, \mathcal{F})} \rightarrow \varprojlim H^n(X, \mathcal{F}/\mathcal{I}^k \mathcal{F})$

was in fact an isomorphism. This is a very nontrivial statement, but in fact we saw yesterday that a reasonably straightforward proof could be given via diagram-chasing if one appeals to a strong form of the proper mapping theorem.

1. Formal functions, jazzed up

Now, however, we want to jazz this up a little. I won’t do this as much as possible because I don’t want to talk too much about formal schemes yet. On the other hand, I want to replace cohomology groups with higher direct images. (more…)

(Well, it looks like I should stop making promises on this blog. There hasn’t been a single post about spectra yet. I hope that will change before next semester.)

So, today I am going to talk about the formal function theorem. This is more or less a statement that the properties of taking completions and taking cohomologies are isomorphic for proper schemes. As we will see, it is the basic ingredient in the proof of the baby form of Zariski’s main theorem. In fact, this is a very important point: the formal function theorem allows one to make a comparison with the cohomology of a given sheaf over the entire space and its cohomology over an “infinitesimal neighborhood” of a given closed subset. Now localization always commutes with cohomology on non-pathological schemes. However, taking such “infinitesimal neighborhoods” is generally too fine a job for localization. This is why the formal function theorem is such a big deal.

I will give the argument following EGA III here, which is more general than that of Hartshorne (who only handles the case of a projective scheme). The form that I will state today is actually rather plain and down-to-earth. In fact, one can jazz it up a little by introducing formal schemes; perhaps this is worth discussion next time. (more…)

All right. I am now inclined to switch topics a little (I am looking forward to saying a few words about local cohomology), so I will sketch a few details in the present post. The goal is to compute the sheaf cohomology groups of the canonical line bundles on projective space. The argument will follow EGA III.2; Hartshorne does essentially the same thing (namely, analysis of the Cech complex) but without the Koszul machinery, so his approach seems more opaque to me.

Now, let us compute the cohomology of projective space ${X = \mathop{\mathbb P}^n_A}$ over a ring ${A}$. Note that ${X}$ is quasi-compact and separated, so we can compute the Cech cohomology by the above machinery. That is, we will use the Koszul-Cech connection discussed two days ago. In particular, we will consider the quasi-coherent sheaf

$\displaystyle \mathcal{H}=\bigoplus_{m \in \mathbb{Z}} \mathcal{O}(m)$

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…)