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 42Let be a space, a sheaf on . Suppose there is a basis of open sets on , closed under finite intersections, satisfying the following condition. If is a finite open covering of , then the Cech cohomology in positive dimension vanishes,Then the natural map:

is an isomorphism, for any

I confess to having stated the result earlier incorrectly, when I claimed that the conclusion was for .

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 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!).