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.
$\square$

## Comments (2)

Comment #937 by correction_bot on

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