Lemma 6.31.10 (00A9)—The Stacks project

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Part 1: Preliminaries
Chapter 6: Sheaves on Spaces
Section 6.31: Open immersions and (pre)sheaves
Lemma 6.31.10 (cite)

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Lemma 6.31.10. Let $X$ be a topological space. Let $j : U \to X$ be the inclusion of an open subset. The functor

$j_! : \textit{Ab}(U) \longrightarrow \textit{Ab}(X)$

is fully faithful. Its essential image consists exactly of those sheaves $\mathcal{G}$ such that $\mathcal{G}_ x = 0$ for all $x \in X \setminus U$ .

Proof. Omitted. $\square$

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