Lemma 85.22.3 (0DAD)—The Stacks project

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Table of contents
Part 5: Topics in Geometry
Chapter 85: Simplicial Spaces
Section 85.22: Hypercovering by a simplicial object of the site: modules
Lemma 85.22.3 (cite)

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Lemma 85.22.3. Let $\mathcal{C}$ be a site with fibre products and $X \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$ . Let $\mathcal{O}_\mathcal {C}$ be a sheaf of rings. Let $U$ be a hypercovering of $X$ in $\mathcal{C}$ . Then we have a canonical isomorphism

$R\Gamma (X, E) = R\Gamma ((\mathcal{C}/U)_{total}, La^*E)$

for $E \in D(\mathcal{O}_\mathcal {C})$ .

Proof. This is a special case of Lemma 85.20.3. $\square$

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