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!).

The proof is an inductive argument and a reduction to the theorem of Leray. Namely, we are going to show that the (derived functor) cohomology of {\mathcal{F}} on any element in this basis {\mathfrak{A}} vanishes. This means that the theorem of Leray goes into effect and gives us the conclusion, since you can always compute derived functor cohomology by Cech cohomology on an acyclic covering. Consider the following statement, which I’ll call {S_k}. For any {1<i \leq k}, and {U \in \mathfrak{A}},

\displaystyle H^i(U, \mathcal{F}) = 0 .

{S_1} is trivial. If we prove all the {S_k}, then we will see that {\mathfrak{A}} is an acyclic cover of {X}, and we can use it to compute sheaf cohomology for {\mathcal{F}}.

Suppose {S_{k-1}} is true. So {\mathcal{H}^i, i < k} is identically zero on the basis {\mathfrak{A}}. Fix {U \in \mathfrak{A}}. Then in the spectral sequence for Cech-to-derived-functor cohomology, things degenerate at {E_2}, and we find that

\displaystyle \boxed{H^k(U, \mathcal{F}) = H^k(\mathfrak{A}, U, \mathcal{F}).}

(Here {H^k(\mathfrak{A}, U, \mathcal{F})} denotes Cech cohomology of {\mathcal{F}} over the open set {U}—this can be computed using sets in the basis {\mathfrak{A}}.)

More precisely, on and below the {a+b = k} diagonal of the {E_2} page, everything in the spectral sequence is zero except on the horizontal row. This is because {\mathcal{H}^i} is zero on open sets in {\mathfrak{A}} and the spectral sequence looks like

\displaystyle H^a(\mathfrak{A}, U, \mathcal{H}^b) \rightarrow H^{a+b}(U, \mathcal{F})

Anyway, from the boxed equation, we find from the hypotheses about {\mathcal{F}} that {H^k(U, \mathcal{F})=0}. Since {U \in \mathfrak{A}} was arbitrary, this proves {S_k}. We can now climb up the natural numbers and see that the Leray theorem applies. [Edit: As pointed out below in the comments, there’s an extra step here to show that the Cech H^0 of the presheaves \mathcal{H}^k vanish, which follows from the fact that any class in the (derived functor) cohomology of a sheaf is always locally trivial.]

This completes the proof. And, finally, since we have spent an obscene amount of time on this, we can use to our heart’s content that quasi-coherent cohomology is trivial on an affine, even if you’re working with non-noetherian schemes!

There is now the question of where to go next. I would like to spend a little time on basic algebro-geometric results about cohomology. In particular, I want to compute the cohomology of line bundles on projective space, since that will be an application of the preceding result as well as another Koszul complex argument. I don’t want to spend too much time on that though—the big highlights of Zariski’s main theorem and the formal function theorem will come only much later, after I’ve discussed the foundational proper mapping theorem—since the theme for November is Koszul-type stuff. So instead, I plan to talk about local cohomology.

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