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.<\/p>\n
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<\/p>\n
f_!Z<\/strong> —> Z<\/strong><\/p><\/blockquote>\n
and consider the Koszul complex on this<\/p>\n
… —> \u2227^2 f_!Z<\/strong> —> f_!Z<\/strong> —> Z<\/strong><\/p><\/blockquote>\n
Looking at stalks we see that this is exact. Thus we obtain a quasi-isomorphism K^* —> Z<\/strong>[0] where the complex K^* has as terms K^i = \u2227^{i + 1} f_!Z<\/strong>. Moreover, K^i = 0 for i \u2265 d. Thus for any abelian sheaf F on X_{etale} we obtain a spectral sequence with E_1-page<\/p>\n
E_1^{p, q} = Ext^q(K^p, F)<\/p><\/blockquote>\n
converging to H^{p + q}(X_{etale}, F). The complex E_1^{*, 0} is our alternating Cech complex<\/em>.<\/p>\n
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 \u00d7_X … \u00d7_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<\/p>\n
Hom(K^p, F) = S_{p + 1}-anti-invariants in F(W_p)<\/p><\/blockquote>\n
To see this look at (W_p —> X)_!Z<\/strong>. The stalk of this sheaf at a geometric point x of X is the free Z<\/strong>-module with basis the set of injective maps {0, …, p} —> U_x. Hence \u2227^{p + 1}f_!Z is the maximal S_{p + 1}-anti-invariant quotient of (W_p —> X)_!Z<\/strong>. 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<\/p>\n
Ext^q(K^p, F) = H^q(U_p, F_p).<\/p><\/blockquote>\n
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.<\/p>\n
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<\/em>. And that’s the thing I was stuck on in the stacks project yesterday…<\/p>\n
[Edit Aug 19, 2011: This material is now in the stacks project. The spectral sequence is Lemma Tag 0728<\/a>. The application to algebraic spaces is Proposition Tag 072B<\/a> and Lemma Tag 072C<\/a>. Note that the first vanishing result is interesting for schemes also.]<\/p>\n","protected":false},"excerpt":{"rendered":"
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 … Continue reading