Lemma 18.19.8 (0F6Z)—The Stacks project

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Table of contents
Part 1: Preliminaries
Chapter 18: Modules on Sites
Section 18.19: Localization of ringed sites
Lemma 18.19.8 (cite)

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Lemma 18.19.8. Let $\mathcal{C}$ be a site. Let $U \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$ . Assume that every $X$ in $\mathcal{C}$ has at most one morphism to $U$ . Let $\mathcal{F}$ be an abelian sheaf on $\mathcal{C}/U$ . The canonical maps $\mathcal{F} \to j_ U^{-1}j_{U!}\mathcal{F}$ and $j_ U^{-1}j_{U*}\mathcal{F} \to \mathcal{F}$ are isomorphisms.

Proof. This is a special case of Lemma 18.16.4 because the assumption on $U$ is equivalent to the fully faithfulness of the localization functor $\mathcal{C}/U \to \mathcal{C}$ . $\square$

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