Lemma 85.20.3 (0DA6)—The Stacks project

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Table of contents
Part 5: Topics in Geometry
Chapter 85: Simplicial Spaces
Section 85.20: Cohomological descent for hypercoverings of an object: modules
Lemma 85.20.3 (cite)

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Lemma 85.20.3. 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 $K$ be a hypercovering of $X$ . Then we have a canonical isomorphism

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

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

Proof. Via Remarks 85.15.7 and 85.16.6 and the discussion in the introduction to this section this follows from Lemma 85.18.3. $\square$

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