Lemma 85.22.1 (0DAB)—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 85.22.1 (cite)

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Lemma 85.22.1. 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 $\mathcal{C}$ . With notation as above

$a^* : \textit{Mod}(\mathcal{O}_ X) \to \textit{Mod}(\mathcal{O})$

is fully faithful with essential image the cartesian $\mathcal{O}$ -modules. The functor $a_*$ provides the quasi-inverse.

Proof. This is a special case of Lemma 85.20.1. $\square$

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