Lemma 36.9.4 (08D2)—The Stacks project

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Table of contents
Part 2: Schemes
Chapter 36: Derived Categories of Schemes
Section 36.9: Koszul complexes
Lemma 36.9.4 (cite)

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Lemma 36.9.4. In Situation 36.9.1. Let $\mathcal{F}^\bullet$ be a complex of quasi-coherent $\mathcal{O}_ X$ -modules. Then there is a canonical isomorphism

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

in $D(A)$ functorial in $\mathcal{F}^\bullet$ .

Proof. Let $\mathcal{B}$ be the set of affine opens of $U$ . Since the higher cohomology groups of a quasi-coherent module on an affine scheme are zero (Cohomology of Schemes, Lemma 30.2.2) this is a special case of Cohomology, Lemma 20.40.2. $\square$

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