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20.24 Alternative view of the Čech complex

In this section we discuss an alternative way to establish the relationship between the Čech complex and cohomology.

Lemma 20.24.1. Let $X$ be a ringed space. Let $\mathcal{U} : X = \bigcup _{i \in I} U_ i$ be an open covering of $X$. Let $\mathcal{F}$ be an $\mathcal{O}_ X$-module. Denote $\mathcal{F}_{i_0 \ldots i_ p}$ the restriction of $\mathcal{F}$ to $U_{i_0 \ldots i_ p}$. There exists a complex ${\mathfrak C}^\bullet (\mathcal{U}, \mathcal{F})$ of $\mathcal{O}_ X$-modules with

\[ {\mathfrak C}^ p(\mathcal{U}, \mathcal{F}) = \prod \nolimits _{i_0 \ldots i_ p} (j_{i_0 \ldots i_ p})_* \mathcal{F}_{i_0 \ldots i_ p} \]

and differential $d : {\mathfrak C}^ p(\mathcal{U}, \mathcal{F}) \to {\mathfrak C}^{p + 1}(\mathcal{U}, \mathcal{F})$ as in Equation ( Moreover, there exists a canonical map

\[ \mathcal{F} \to {\mathfrak C}^\bullet (\mathcal{U}, \mathcal{F}) \]

which is a quasi-isomorphism, i.e., ${\mathfrak C}^\bullet (\mathcal{U}, \mathcal{F})$ is a resolution of $\mathcal{F}$.

Proof. We check

\[ 0 \to \mathcal{F} \to \mathfrak {C}^0(\mathcal{U}, \mathcal{F}) \to \mathfrak {C}^1(\mathcal{U}, \mathcal{F}) \to \ldots \]

is exact on stalks. Let $x \in X$ and choose $i_{\text{fix}} \in I$ such that $x \in U_{i_{\text{fix}}}$. Then define

\[ h : \mathfrak {C}^ p(\mathcal{U}, \mathcal{F})_ x \to \mathfrak {C}^{p - 1}(\mathcal{U}, \mathcal{F})_ x \]

as follows: If $s \in \mathfrak {C}^ p(\mathcal{U}, \mathcal{F})_ x$, take a representative

\[ \widetilde{s} \in \mathfrak {C}^ p(\mathcal{U}, \mathcal{F})(V) = \prod \nolimits _{i_0 \ldots i_ p} \mathcal{F}(V \cap U_{i_0} \cap \ldots \cap U_{i_ p}) \]

defined on some neighborhood $V$ of $x$, and set

\[ h(s)_{i_0 \ldots i_{p - 1}} = \widetilde{s}_{i_{\text{fix}} i_0 \ldots i_{p - 1}, x}. \]

By the same formula (for $p = 0$) we get a map $\mathfrak {C}^{0}(\mathcal{U},\mathcal{F})_ x \to \mathcal{F}_ x$. We compute formally as follows:

\begin{align*} (dh + hd)(s)_{i_0 \ldots i_ p} & = \sum \nolimits _{j = 0}^ p (-1)^ j h(s)_{i_0 \ldots \hat i_ j \ldots i_ p} + d(s)_{i_{\text{fix}} i_0 \ldots i_ p}\\ & = \sum \nolimits _{j = 0}^ p (-1)^ j s_{i_{\text{fix}} i_0 \ldots \hat i_ j \ldots i_ p} + s_{i_0 \ldots i_ p} + \sum \nolimits _{j = 0}^ p (-1)^{j + 1} s_{i_{\text{fix}} i_0 \ldots \hat i_ j \ldots i_ p} \\ & = s_{i_0 \ldots i_ p} \end{align*}

This shows $h$ is a homotopy from the identity map of the extended complex

\[ 0 \to \mathcal{F}_ x \to \mathfrak {C}^0(\mathcal{U}, \mathcal{F})_ x \to \mathfrak {C}^1(\mathcal{U}, \mathcal{F})_ x \to \ldots \]

to zero and we conclude. $\square$

With this lemma it is easy to reprove the Čech to cohomology spectral sequence of Lemma 20.11.5. Namely, let $X$, $\mathcal{U}$, $\mathcal{F}$ as in Lemma 20.24.1 and let $\mathcal{F} \to \mathcal{I}^\bullet $ be an injective resolution. Then we may consider the double complex

\[ A^{\bullet , \bullet } = \Gamma (X, {\mathfrak C}^\bullet (\mathcal{U}, \mathcal{I}^\bullet )). \]

By construction we have

\[ A^{p, q} = \prod \nolimits _{i_0 \ldots i_ p} \mathcal{I}^ q(U_{i_0 \ldots i_ p}) \]

Consider the two spectral sequences of Homology, Section 12.25 associated to this double complex, see especially Homology, Lemma 12.25.1. For the spectral sequence $({}'E_ r, {}'d_ r)_{r \geq 0}$ we get ${}'E_2^{p, q} = \check{H}^ p(\mathcal{U}, \underline{H}^ q(\mathcal{F}))$ because taking products is exact (Homology, Lemma 12.32.1). For the spectral sequence $({}''E_ r, {}''d_ r)_{r \geq 0}$ we get ${}''E_2^{p, q} = 0$ if $p > 0$ and ${}''E_2^{0, q} = H^ q(X, \mathcal{F})$. Namely, for fixed $q$ the complex of sheaves ${\mathfrak C}^\bullet (\mathcal{U}, \mathcal{I}^ q)$ is a resolution (Lemma 20.24.1) of the injective sheaf $\mathcal{I}^ q$ by injective sheaves (by Lemmas 20.7.1 and 20.11.11 and Homology, Lemma 12.27.3). Hence the cohomology of $\Gamma (X, {\mathfrak C}^\bullet (\mathcal{U}, \mathcal{I}^ q))$ is zero in positive degrees and equal to $\Gamma (X, \mathcal{I}^ q)$ in degree $0$. Taking cohomology of the next differential we get our claim about the spectral sequence $({}''E_ r, {}''d_ r)_{r \geq 0}$. Whence the result since both spectral sequences converge to the cohomology of the associated total complex of $A^{\bullet , \bullet }$.

Definition 20.24.2. Let $X$ be a topological space. An open covering $X = \bigcup _{i \in I} U_ i$ is said to be locally finite if for every $x \in X$ there exists an open neighbourhood $W$ of $x$ such that $\{ i \in I \mid W \cap U_ i \not= \emptyset \} $ is finite.

Remark 20.24.3. Let $X = \bigcup _{i \in I} U_ i$ be a locally finite open covering. Denote $j_ i : U_ i \to X$ the inclusion map. Suppose that for each $i$ we are given an abelian sheaf $\mathcal{F}_ i$ on $U_ i$. Consider the abelian sheaf $\mathcal{G} = \bigoplus _{i \in I} (j_ i)_*\mathcal{F}_ i$. Then for $V \subset X$ open we actually have

\[ \Gamma (V, \mathcal{G}) = \prod \nolimits _{i \in I} \mathcal{F}_ i(V \cap U_ i). \]

In other words we have

\[ \bigoplus \nolimits _{i \in I} (j_ i)_*\mathcal{F}_ i = \prod \nolimits _{i \in I} (j_ i)_*\mathcal{F}_ i \]

This seems strange until you realize that the direct sum of a collection of sheaves is the sheafification of what you think it should be. See discussion in Modules, Section 17.3. Thus we conclude that in this case the complex of Lemma 20.24.1 has terms

\[ {\mathfrak C}^ p(\mathcal{U}, \mathcal{F}) = \bigoplus \nolimits _{i_0 \ldots i_ p} (j_{i_0 \ldots i_ p})_* \mathcal{F}_{i_0 \ldots i_ p} \]

which is sometimes useful.

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