The paper of Kempf

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

First, we set up some notation, following Kempf. Given ${V \subset X}$ open and a sheaf ${\mathcal{F}}$ on ${X}$ , define ${{}_V \mathcal{F}}$ to be the sheaf ${i_*(i^{-1}\mathcal{F})}$ on ${X}$ , where ${i: V \rightarrow X}$ is the inclusion. This is equivalently the sheaf ${U \rightarrow \mathcal{F}(V \cap U)}$ . There is a natural map ${\mathcal{F} \rightarrow {}_V \mathcal{F}}$ , which of course induces maps on cohomology.

The elementary result that Kempf proves is:

Lemma 2 (Kempf) Let ${X}$ be a topological space and let ${\mathfrak{A}}$ be a basis for ${X}$ which is closed under finite intersections. Suppose ${n \in \mathbb{N}}$ is fixed. Suppose ${\mathcal{F} \in \mathrm{Sh}(X)}$ for ${X}$ a topological space is such that ${H^i(U, \mathcal{F}|_U) = 0}$ for all ${0<i<n}$ and ${U \in \mathfrak{A}}$ . Suppose ${\alpha \in H^n(X, \mathcal{F})}$ . Then there is a covering of ${X}$ by open sets ${V \in \mathfrak{A}}$ such that the image of ${\alpha}$ in ${H^n(X, {}_V \mathcal{F})}$ is zero for each ${V}$ .

Proof: We will prove this result by induction on ${n}$ . First, suppose ${n>1}$ , and that the result is valid for ${n-1}$ . The base case will be handled next. We can embed ${\mathcal{F}}$ in a flabby sheaf ${\mathcal{G}}$ , and let ${\mathcal{H}}$ be the cokernel. There is an exact sequence

and by the long exact sequence for cohomology (and since ${n>1}$ ), it follows that

is exact for every ${U}$ in the open basis ${\mathfrak{A}}$ . Now fix ${V \in \mathfrak{A}}$ and consider the complex of sheaves

In general, we know that ${i^{-1}}$ is exact, by looking at the stalks, but only that ${i_*}$ is left-exact. From this alone we get that (3) is exact except perhaps at the last step. But we also know that for any ${U \in \mathfrak{A}}$ , we have that the sequence of sections

is exact in view of the definition of ${{}_V}$ and exactness of (2). Consequently, since ${\mathfrak{A}}$ is a basis, we can pass to the direct limit to the stalks, and we see that (3) must be exact at the last step too.

But ${{}_V \mathcal{G}}$ is also flabby and consequently has trivial cohomology. As a result, we find that for any ${V \in \mathfrak{A}}$ , there is an isomorphism

$\displaystyle H^{n-1}(X, {}_V \mathcal{H}) \simeq H^{n}(X, {}_V\mathcal{F}).$

Moreover, since ${\mathcal{G}}$ is flabby and thus has trivial cohomology on ${X}$ , we get isomorphisms from the long exact sequence of (1)

$\displaystyle H^{n-1}(X, \mathcal{H}) \simeq H^{n}(X, \mathcal{F}).$

This means that ${\mathcal{H}}$ satisfies the conditions of the proposition with ${n-1}$ , and we have assumed inductively that the result is valid for ${n-1}$ . So ${\alpha}$ maps to some ${\beta \in H^{n-1}(X, \mathcal{H})}$ ; this means there is an open cover of ${X}$ by various ${V \in \mathfrak{A}}$ such that ${\beta}$ maps to zero in ${H^{n-1}(X, {}_V \mathcal{H})}$ . This means that ${\alpha}$ maps to zero in these ${H^{n}(X, {}_V\mathcal{F})}$ by naturality. This completes the proof of the inductive step.

The base case remains, i.e. ${n=1}$ . Fix ${\alpha \in H^1(X, \mathcal{F})}$ . We can still embed ${\mathcal{F}}$ in a flabby sheaf and obtain an exact sequence as in (1). So we get an exact sequence:

$\displaystyle 0 \rightarrow \mathcal{G}m(X, \mathcal{F}) \rightarrow \mathcal{G}m(X, \mathcal{G}) \rightarrow \mathcal{G}m(X, \mathcal{H}) \rightarrow H^{1}(X, \mathcal{F}) \rightarrow 0.$

However, exactness of (4) is now no longer valid, so we cannot conclude that (3) is exact. We do, however, have an exact sequence ${0 \rightarrow {}_V\mathcal{F} \rightarrow {}_V\mathcal{G} \rightarrow \mathcal{K}^{(V)} \rightarrow 0}$ exact for some cokernel ${\mathcal{K}^{(V)}}$ , and we can fit these into an exact commutative diagram

Let ${\alpha \in H^1(X, \mathcal{F})}$ . Then ${\alpha}$ lifts to some ${\beta \in \mathcal{G}m(X, \mathcal{H})}$ . We have a commutative diagram of exact sequences

So to say that ${\alpha}$ is annihilated by the map to ${H^1(X, {}_V \mathcal{F})}$ is the same as saying that the image of ${\beta}$ in ${\mathcal{K}^{(V)}}$ lifts to something in ${{}_V \mathcal{G}(X)}$ . But if ${V}$ is small, surjectivity of ${\mathcal{G} \rightarrow \mathcal{H}}$ implies that we can lift ${\beta}$ to something in ${{}_V \mathcal{G}(X)}$ . So we can cover ${X}$ by such sets ${V}$ in ${\mathfrak{A}}$ , completing the proof. $\Box$

0.2. Proof of the vanishing theorem

We now apply the lemma. Proof: Induction on ${n}$ .

Let ${X = \mathrm{Spec} A}$ be an affine scheme. Consider the basis ${\mathfrak{A}}$ of open sets ${D(f) = \mathrm{Spec} A_f}$ ; this is obviously closed under intersection, as ${D(fg) = D(f) \cap D(g)}$ . Now if ${\widetilde{M}}$ is the (quasi-coherent) sheaf on ${X}$ associated to an ${A}$ -module ${M}$ , then the pull-back to ${D(f)}$ is the sheaf associated to ${M \otimes_A A_f = M_f}$ . So the direct image ${{}_{D(f)} \widetilde{M}}$ to ${A}$ is just ${\widetilde{M_f}}$ . In particular, these are quasi-coherent on ${X}$ .

We will now apply the previous lemma. Suppose that ${n}$ is fixed and ${H^i(X, \mathcal{F}) = 0}$ for any quasi-coherent sheaf on ${X}$ and ${0<i<n}$ . Then, this is true for ${\widetilde{M}}$ , so the previous lemma says that given ${\alpha \in H^n(X, \widetilde{M})}$ , there is an open cover ${\{D(f_i)\}}$ of ${X}$ such the cohomology class of ${\alpha}$ in ${H^n(X, {}_{D(f_i)}\widetilde{M})}$ is zero. Now there is a map ${M \rightarrow \bigoplus M_{f_i}}$ which is injective, since the ${f_i}$ generate the unit ideal; this induces a map of sheaves

$\displaystyle 0 \rightarrow \widetilde{M} \rightarrow \bigoplus {}_{D(f_i)} \widetilde{M} \rightarrow \mathcal{K} \rightarrow 0$

where ${\mathcal{K}}$ is also quasi-coherent. There is thus a long exact sequence, of which we write a piece:

$\displaystyle H^{n-1}(X, \mathcal{K}) \rightarrow H^{n}(X, \widetilde{M}) \rightarrow H^n( X, \bigoplus {}_{D(f_i)} \widetilde{M}) .$

Since ${\alpha}$ is in the kernel of the second map, it is in the image of ${H^{n-1}(X, \mathcal{K})}$ .

But, if ${n>1}$ , inductively we assumed ${H^{n-1}(X, \mathcal{K})}$ was zero. This means ${\alpha = 0}$ . If ${n=1}$ , then we write out a bit more of the exact sequence:

$\displaystyle \mathcal{G}m(X, \bigoplus {}_{D(f_i)} \widetilde{M}) \rightarrow \mathcal{G}m(X, \mathcal{H}) \rightarrow H^1(X, \mathcal{F}) \rightarrow H^1(X, \bigoplus {}_{D(f_i)} \widetilde{M} )$

As before, we find that ${\alpha}$ is in the image of ${\mathcal{G}m(X, \mathcal{H})}$ . But since ${\mathcal{G}m}$ is exact on quasi-coherent sheaves, it follows that the first map is surjective, and the map out of ${\mathcal{G}m(X, \mathcal{H})}$ is zero. So again we find that ${\alpha = 0}$ . This proves Serre’s vanishing theorem. $\Box$

Jerome Says:

August 11, 2010 at 2:00 pm

Thansk for pointing out this article by Kempf. I’ve started to read it as well.

In this post, some of the diagrams seems to be missing. For example, there should be something after “There is an exact sequence”; which is at the begining of the proof of Lemma 2.

Akhil Mathew Says:

August 12, 2010 at 12:47 am
Ack, you’re right. I really can’t fix it right now but will take a look at it shortly.

henry Says:

August 12, 2010 at 12:49 am

Akhil, how do you prepare these posts? (It seems you prepare nearly one per day which is pretty remarkable; I’d like to know how one approaches it.) I mean, do you just copy the theorems and proofs out of a textbook? Or is it something deeper than that? I’d like to start a blog like this one day on mathematics and I’d like to know if there is a way to efficiently prepare these mathematic sblogs.

Akhil Mathew Says:

August 12, 2010 at 12:51 am
I just read or otherwise learn material and post what I find interesting or would like to understand better, more or less. Sometimes I get into a scheme of repeated posting on the same topic, as well. It’s nothing deeper than that; I’d impute the frequency simply to the fact that it’s fun and that I had lots of free time before (something which seems to be less true now, potentially contributing to my recent silence…).

Climbing Mount Bourbaki