# The Stacks Project

## Tag 00AB

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. $\square$

The code snippet corresponding to this tag is a part of the file sheaves.tex and is located in lines 4813–4825 (see updates for more information).

\begin{lemma}
\label{lemma-equivalence-categories-open-modules}
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 $x \in X \setminus U$.
\end{lemma}

\begin{proof}
Omitted.
\end{proof}

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