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Let X be a topological space, and T its category of open sets with the usual Grothendieck topology. Let T' be any sieve of T (a subcategory of T such that if U is in T' then any subset of U is also in T'). For example, T' might be the collection of open subsets subordinate to the open subsets in a cover \mathcal{U} http://latex.mathoverflow.net/png?%5Cmathcal%7BU%7D. Any sheaf on T induces a functor on T' which can be viewed as a sheaf on T' if T' is given the minimal topology (the only covers are the identity maps). This determines a morphism of topoi f : T \rightarrow T' http://latex.mathoverflow.net/png?f%20%3A%20T%20%5Crightarrow%20T%27, hence a spectral sequence

H^p(T', R^q f_\ast F) \Rightarrow H^{p+q}(T, F) http://latex.mathoverflow.net/png?H%5Ep%28T%27%2C%20R%5Eq%20f%5F%5Cast%20F%29%20%5CRightarrow%20H%5E%7Bp%2Bq%7D%28T%2C%20F%29 .

(One could surely also convince oneself that such a spectral sequence exists without any reference to topoi.)

The Cech cohomology of F with respect to some covering family \mathcal{U} http://latex.mathoverflow.net/png?%5Cmathcal%7BU%7D is

H^p(\mathcal{U}, F) = H^p(T', f_\ast F) http://latex.mathoverflow.net/png?H%5Ep%28%5Cmathcal%7BU%7D%2C%20F%29%20%3D%20H%5Ep%28T%27%2C%20f%5F%5Cast%20F%29

where T' = T'(U) http://latex.mathoverflow.net/png?T%27%20%3D%20T%27%28U%29 is the sieve associated to the cover \mathcal{U} http://latex.mathoverflow.net/png?%5Cmathcal%7BU%7D. The Cech cohomology is then the filtered colimit

\check{H}^p(T, F) = \varinjlim_{(T',f)} H^p(T', f_\ast F) http://latex.mathoverflow.net/png?%5Ccheck%7BH%7D%5Ep%28T%2C%20F%29%20%3D%20%5Cvarinjlim%5F%7B%28T%27%2Cf%29%7D%20H%5Ep%28T%27%2C%20f%5F%5Cast%20F%29

taken over the projections f : T \rightarrow T' http://latex.mathoverflow.net/png?f%20%3A%20T%20%5Crightarrow%20T%27 associated as above to covering families \mathcal{U} http://latex.mathoverflow.net/png?%5Cmathcal%7BU%7D.

One evidently has edge homomorphisms

\check{H}^p(T, F) \rightarrow H^p(T, F) http://latex.mathoverflow.net/png?%5Ccheck%7BH%7D%5Ep%28T%2C%20F%29%20%5Crightarrow%20H%5Ep%28T%2C%20F%29

from the spectral sequence, and the question is when these induce an isomorphism. If we could somehow eliminate the R^p f_\ast F http://latex.mathoverflow.net/png?R%5Ep%20f%5F%5Cast%20F, p > 0, by passing to a "small enough" cover we would have equality. This condition already holds in many cases; the following condition is more general (but I haven't checked carefully that it actually works!):

For every cover \mathcal{U} http://latex.mathoverflow.net/png?%5Cmathcal%7BU%7D of X, every U_1, \ldots, U_n \in \mathcal{U} http://latex.mathoverflow.net/png?U%5F1%2C%20%5Cldots%2C%20U%5Fn%20%5Cin%20%5Cmathcal%7BU%7D, and every class in \alpha \in H^p(U_1 \mathop{\times}_X \cdots \mathop{\times}_X U_n, F) http://latex.mathoverflow.net/png?%5Calpha%20%5Cin%20H%5Ep%28U%5F1%20%5Cmathop%7B%5Ctimes%7D%5FX%20%5Ccdots%20%5Cmathop%7B%5Ctimes%7D%5FX%20U%5Fn%2C%20F%29, p > 0, there exists a refinement \mathcal{U}' http://latex.mathoverflow.net/png?%5Cmathcal%7BU%7D%27 of \mathcal{U} http://latex.mathoverflow.net/png?%5Cmathcal%7BU%7D such that the restriction of \alpha http://latex.mathoverflow.net/png?%5Calpha under the map

H^p(U_1 \mathop{\times}_X \cdots \mathop{\times}_X U_n, F) \rightarrow H^p(U'_1 \mathop{\times}_X \cdots \mathop{\times}_X U'_n, F) http://latex.mathoverflow.net/png?H%5Ep%28U%5F1%20%5Cmathop%7B%5Ctimes%7D%5FX%20%5Ccdots%20%5Cmathop%7B%5Ctimes%7D%5FX%20U%5Fn%2C%20F%29%20%5Crightarrow%20H%5Ep%28U%27%5F1%20%5Cmathop%7B%5Ctimes%7D%5FX%20%5Ccdots%20%5Cmathop%7B%5Ctimes%7D%5FX%20U%27%5Fn%2C%20F%29

is zero.

To make sense of this, one must use some convention for the covers \mathcal{U} http://latex.mathoverflow.net/png?%5Cmathcal%7BU%7D and \mathcal{U}' http://latex.mathoverflow.net/png?%5Cmathcal%7BU%7D%27 to ensure there is a map as above. For example, one could work only with covers indexed by the points of X (a cover is then a collection of neighborhoods of each point of X).

A more refined version of the above condition would say that Cech cohomology equals cohomology in degrees at most q if the above condition holds for p \leq q http://latex.mathoverflow.net/png?p%20%5Cleq%20q. Since it always holds for p = 0,1 this implies that

\check{H}^1(T, F) = H^1(T, F) http://latex.mathoverflow.net/png?%5Ccheck%7BH%7D%5E1%28T%2C%20F%29%20%3D%20H%5E1%28T%2C%20F%29

in general.

Edit in response to David's comment:

The Cech complex always computes cohomology correctly in a presheaf category (i.e., when the topology is "chaotic": an object has no covers by anything except itself). Trying to compute cohomology in an arbitrary site using the Cech complex is (heuristically) something like trying to approximate the site by a presheaf category.

Here is how Cech cohomology computes cohomology of presheaves. Consider any category T'. If F is a presheaf of groups on T' then the sheaf cohomology groups of F are the derived functors of the inverse limit for diagrams of shape T'. They are also computed as

$Ext(\mathbf{Z}, F)$

where $\mathbf{Z}$ is the constant sheaf associated to the integers. Remarkably, in a presheaf category, $\mathbf{Z}$ has a canonical projective resolution associated to any cover of the final presheaf. A cover of the final presheaf is a collection of objects U of T' such that every object of T' has a map to at least one object of U. The i-th term of this complex is the direct sum, over all choices of i elements U_1, ..., U_i of U, of the groups $\mathbf{Z}_{U_1 \times \cdots \times U_i}$. (You can check this is projective by noting it is the extension by 0 of $\mathbf{Z}$ from the slice category $T' / U_1 \times \cdots \times U_i$ and extension by 0 preserves projectives (since it has an exact right adjoint) and $\mathbf{Z}$ is projective on the slice category since all higher cohomology of all sheaves vanishes (since it has a final object). It's also easy to check by a direct calculation.)

Denote this complex by K. Since this is a projective resolution of $\mathbf{Z}$, $\mathrm{Hom}(K, F)$ computes the cohomology of $F$. But it is also easy to see that this is just the Cech complex of F.