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 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. 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. $\Box$