## 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 (20.9.0.1). 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.

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