Lemma 25.4.5 (01GF)—The Stacks project

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Table of contents
Part 1: Preliminaries
Chapter 25: Hypercoverings
Section 25.4: Acyclicity
Lemma 25.4.5 (cite)

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Lemma 25.4.5. Let $\mathcal{C}$ be a site with fibre products. Let $X$ be an object of $\mathcal{C}$ . Let $K$ be a hypercovering of $X$ . The homology of the simplicial presheaf $F(K)$ is $0$ in degrees $> 0$ and equal to $\mathbf{Z}_ X^\#$ in degree $0$ .

Proof. Combine Lemmas 25.4.4 and 25.3.9. $\square$

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