The Stacks project

Lemma 6.31.12. Let $(X, \mathcal{O})$ be a ringed space. Let $j : (U, \mathcal{O}|_ U) \to (X, \mathcal{O})$ be an open subspace. The functor

\[ j_! : \textit{Mod}(\mathcal{O}|_ U) \longrightarrow \textit{Mod}(\mathcal{O}) \]

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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