The Stacks project

Lemma 20.36.1. Let $(X, \mathcal{O}_ X)$ be a ringed space. Let $\mathcal{U} : X = \bigcup _{i \in I} U_ i$ be a finite open covering. For a complex $\mathcal{F}^\bullet $ of $\mathcal{O}_ X$-modules there is a canonical map

\[ \text{Tot}(\check{\mathcal{C}}^\bullet _{alt}(\mathcal{U}, \mathcal{F}^\bullet )) \longrightarrow R\Gamma (X, \mathcal{F}^\bullet ) \]

functorial in $\mathcal{F}^\bullet $ and compatible with (20.36.0.1).

Proof. Let ${\mathcal I}^\bullet $ be a K-injective complex whose terms are injective $\mathcal{O}_ X$-modules. The map (20.36.0.1) for $\mathcal{I}^\bullet $ is a map $\Gamma (X, {\mathcal I}^\bullet ) \to \text{Tot}(\check{\mathcal{C}}^\bullet _{alt}({\mathcal U}, {\mathcal I}^\bullet ))$. This is a quasi-isomorphism of complexes of abelian groups as follows from Homology, Lemma 12.25.4 applied to the double complex $\check{\mathcal{C}}^\bullet _{alt}({\mathcal U}, {\mathcal I}^\bullet )$ using Lemmas 20.11.1 and 20.23.6. Suppose ${\mathcal F}^\bullet \to {\mathcal I}^\bullet $ is a quasi-isomorphism of ${\mathcal F}^\bullet $ into a K-injective complex whose terms are injectives (Injectives, Theorem 19.12.6). Since $R\Gamma (X, {\mathcal F}^\bullet )$ is represented by the complex $\Gamma (X, {\mathcal I}^\bullet )$ we obtain the map of the lemma using

\[ \text{Tot}(\check{\mathcal{C}}^\bullet _{alt}({\mathcal U}, {\mathcal F}^\bullet )) \longrightarrow \text{Tot}(\check{\mathcal{C}}^\bullet _{alt}({\mathcal U}, {\mathcal I}^\bullet )). \]

We omit the verification of functoriality and compatibilities. $\square$


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