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Given a topological space X and a finite cover X = $\cup X_i$, one can define Cech cohomology of a sheaf of abelian groups F with respect to the cover $\{X_i\}$ in two different ways:

  1. (Ordered): The kth term of the Cech complex is $\bigoplus_{i_1 < \ldots < i_k} \Gamma(X_{i_1} \cap \ldots \cap X_{i_k}, F)$.
  2. (Unordered): The kth term of the Cech complex is $\bigoplus_{i_1, \ldots , i_k} \Gamma(X_{i_1} \cap \ldots \cap X_{i_k}, F)$.

In particular, the second description involves repetition and is non-zero in every degree. These two descriptions give isomorphic cohomology (the first maps you try to write down will likely be homotopy equivalences).


Question: Is there a canonical reference for this fact?

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4 Answers

up vote 10 down vote accepted

I wrote it up for my algebraic geometry course as a 2-page handout, inspired by EGA $0_{\rm{III}}$, 11.8.7 (which isn't to say this is a canonical reference; just some written reference...).

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I'd say that a canonical reference is Roger Godement's Topologie algébrique et théorie des faisceaux, §3.8, chapter I.

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I don't know if this is in SGA IV.5, but that's a good place to look for questions about Cech cohomology.

As I described here, the Cech cohomology with respect to a cover is the same as the sheaf cohomology in the sieve associated to that cover. If $\mathcal{U}$ is a cover of $X$, let $R$ be the category whose objects are maps $V \rightarrow X$ that factor through some object in $\mathcal{U}$. Then

$\check{H}^p(\mathcal{U}, F) = \varprojlim^{(p)}_{R} F = Ext^p(\mathbf{Z}_R, F)$

where $\varprojlim^{(p)}$ is the $p$-th derived functor $\varprojlim$. This can be calculated by taking a projective resolution of $\mathbf{Z}_R$. Here are two ways to do it:

$\displaystyle K_p = \sum_{i_1 < i_2 < \cdots < i_p} \mathbf{Z}_{U_{i_1} \cap \cdots \cap U_{i_p}}$

$\displaystyle L_p = \sum_{i_1, \ldots, i_p} \mathbf{Z}_{U_{i_1} \cap \cdots \cap U_{i_p}}$.`

One must check, of course, that these are indeed resolutions. (I don't have a slick explanation of why they are resolutions. The best I can do is to say that these complexes are associated via the Dold--Kan correspondence to simplicial resolutions of the final presheaf on $R$.) Taking $Hom$ into $F$ yields the two Cech complexes in question.

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A recent reference is Corollary 5.2.4 in Liu's "Algebraic geometry and arithmetic curves."

However, for the proof of the main step (reducing from cochains to alternating cochains, as in Brian Conrad's writeup) it refers to Serre's "Faisceaux Algébriques Cohérents‎", no. 20, Proposition 2.

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