Lemma 85.36.6 (0DHP)—The Stacks project

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Table of contents
Part 5: Topics in Geometry
Chapter 85: Simplicial Spaces
Section 85.36: Proper hypercoverings of algebraic spaces
Lemma 85.36.6 (cite)

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Lemma 85.36.6. Let $S$ be a scheme. Let $X$ be an algebraic space over $S$ . Let $U$ be a simplicial algebraic space over $S$ . Let $a : U \to X$ be an augmentation. Let $\mathcal{F}$ be an abelian sheaf on $X_{\acute{e}tale}$ . Let $\mathcal{F}_ n$ be the pullback to $U_{n, {\acute{e}tale}}$ . If $U$ is a ph hypercovering of $X$ , then there exists a canonical spectral sequence

$E_1^{p, q} = H^ q_{\acute{e}tale}(U_ p, \mathcal{F}_ p)$

converging to $H^{p + q}_{\acute{e}tale}(X, \mathcal{F})$ .

Proof. Immediate consequence of Lemmas 85.36.4 and 85.8.3. $\square$

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