# Alternating Cech cohomology

So Bhargav and I were, just earlier today, thinking about the alternating Cech complex in the setting of etale cohomology and this is what we came up with. Caveat: This may be wrong in which case it is my fault (I’m a little worried because the final result of this blog post seems to contradict a throwaway comment in some preprint). Also: it may be in the literature; if you know a reference for this construction please email, thanks.

Let X be an algebraic space. Let U be a separated scheme and let f : U —> X be a surjective etale morphism. Assume that there exists an integer d such that every geometric fibre of f has at most d points. (This is true if U is quasi-compact and X is quasi-separated.) Consider the trace map

f_!Z —> Z

and consider the Koszul complex on this

… —> ∧^2 f_!Z —> f_!Z —> Z

Looking at stalks we see that this is exact. Thus we obtain a quasi-isomorphism K^* —> Z[0] where the complex K^* has as terms K^i = ∧^{i + 1} f_!Z. Moreover, K^i = 0 for i ≥ d. Thus for any abelian sheaf F on X_{etale} we obtain a spectral sequence with E_1-page

E_1^{p, q} = Ext^q(K^p, F)

converging to H^{p + q}(X_{etale}, F). The complex E_1^{*, 0} is our alternating Cech complex.

Now, we want to make explicit the groups Ext^q(K^p, F). These are the right derived functors of Hom(K^p, F). To describe Hom(K^p, F) we introduce some notation. Namely, let W_p be the complement of ALL diagonals in U^{p + 1} = U ×_X … ×_X U. Since f is separated and etale W_p is both open and closed in U^{p + 1}. Moreover, the group S_{p + 1} has a free action on W_p. We claim that

Hom(K^p, F) = S_{p + 1}-anti-invariants in F(W_p)

To see this look at (W_p —> X)_!Z. The stalk of this sheaf at a geometric point x of X is the free Z-module with basis the set of injective maps {0, …, p} —> U_x. Hence ∧^{p + 1}f_!Z is the maximal S_{p + 1}-anti-invariant quotient of (W_p —> X)_!Z. This proves the displayed formula. Since S_{p + 1} acts freely on W_p over X the quotient U_p = W_p/S_{p + 1} is an algebraic space etale over X. There is a way to “twist” F|_{U_p} by the sign character S_{p + 1} —> {+1, -1} giving a sheaf F_p on U_p. With a little bit of work we obtain

Ext^q(K^p, F) = H^q(U_p, F_p).

Why is this useful? Suppose that F is a quasi-coherent O_X-module, X is quasi-compact, X is separated, and U is affine. Then each W_p is affine too, and so is U_p. Moreover, the sheaves F_p are still quasi-coherent. Thus we see that the E_1^{p, q} are nonzero only when q = 0 and we obtain vanishing of H^n(X, F) for all n >= d! This is exactly the vanishing you traditionally obtain from the alternating Cech complex associated to a finite affine open covering of a scheme.

For a quasi-compact, quasi-separated algebraic space X we can redo the argument with U an affine scheme. We find (because we can apply the previous result to the separated quasi-compact algebraic spaces U_p) that X has finite cohomological dimension for quasi-coherent sheaves. And that’s the thing I was stuck on in the stacks project yesterday…

[Edit Aug 19, 2011: This material is now in the stacks project. The spectral sequence is Lemma Tag 0728. The application to algebraic spaces is Proposition Tag 072B and Lemma Tag 072C. Note that the first vanishing result is interesting for schemes also.]