Equivalence of ordered and unordered cech cohomology.

Question

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).

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Question: Is there a canonical reference for this fact?

Answer 1

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...).

Answer 2

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

Answer 3

I'll make use the category $\mathcal{P}=\mathrm{Fun}(\mathrm{Open}_X^{\mathrm{op}}, \mathrm{Ab})$ of presheaves of abelian groups on $X$. This is an abelian category with enough projectives: for any open set $U$, the free presheaf $\mathbb{Z}[U]$ defined by $\mathbb{Z}[U](V)=\mathbb{Z}$ if $V\subseteq U$ and $=0$ if not, is a projective object in presheaves.

We can define two chain complexes in $\mathcal{P}$: $$ A_k = \bigoplus_{i_0<\cdots<i_k} \mathbb{Z}[X_{i_0}\cap \cdots \cap X_{i_k}], \qquad B_k = \bigoplus_{i_0,\dots,i_k} \mathbb{Z}[X_{i_0}\cap \cdots \cap X_{i_k}], $$ with an evident chain map $\gamma\colon A_\bullet\to B_\bullet$. It is clear that for a sheaf $F$, the complexes $\mathrm{Hom}_{\mathcal{P}}(A_\bullet, F)$ and $\mathrm{Hom}_{\mathcal{P}}(B_\bullet, F)$ are respectively the ordered and unordered Cech complexes of $F$. So it suffices to show that $\gamma$ is a chain homotopy equivalence in $\mathrm{Ch}(\mathcal{P})$, as such data will thereby induce a chain homotopy equivalence between the two versions of the Cech complex.

Both $A_\bullet$ and $B_\bullet$ are bounded-below complexes of projectives in $\mathcal{P}$, so it suffices to show that $\gamma$ induces an isomorphism on homology. That is, we need to show that $H_k A_\bullet(U)\to H_k B_\bullet(U)$ is an isomorphism for all open sets $U$.

If $U$ is not contained in any $X_i$, then it is immediate that $A_\bullet(U)\equiv 0\equiv B_\bullet(U)$. If $U$ is contained in some $X_i$, then you can show by explicit calculation (e.g., an explicit chain homotopy, see below) that $$ H_0A_\bullet(U)\approx \mathbb{Z}\approx H_0B_\bullet(U),\qquad H_kA_\bullet(U)\approx 0 \approx H_kB_\bullet(U),\quad k>0. $$

To see how this works, write $I_U=\{i\in I\;\mid\; U\subseteq X_i\}$ where $I$ is the (well-ordered) indexing set of the cover. We see that each degree of $A_\bullet(U)$ and $B_\bullet(U)$ are free abelian groups: $$ A_k(U)=\mathbb{Z}\bigl\{ (i_0<\cdots<i_k),\; i_j\in I_U\bigr\},\qquad B_k(U)=\mathbb{Z}\bigl\{ (i_0,\dots,i_k),\; i_j\in I_U\bigr\}. $$ In fact, $A_\bullet(U)$ is the complex of normalized chains on the nerve of the poset $(I_U, \leq)$ with ordering inherited from $I$, which can be contracted to a point corresponding to the minimal element of $I_U$. Likewise, $B_\bullet(U)$ is the complex of unnormalized chains on the nerve of the category $(I_U, I_U\times I_U)$ with objects $I_U$ and a unique morphism between any pair of objects, and this nerve can also be contracted to a point.

(Explicit contractions for the augmented chain complexes $A_\bullet(U)\to \mathbb{Z}$ and $B_\bullet(U)\to \mathbb{Z}$ are given by $$ h_A(i_0<\cdots<i_k)=\text{$(m<i_0<\cdots<i_k)$ if $m\neq i_0$, or $0$ if $m=i_0$}, $$ and $$ h_B(i_0,\dots,i_k)=(m,i_0,\dots,i_k), $$ where $m\in I_U$ is the minimal element.)

Answer 4

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.

Answer 5

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.