The Stacks project

Lemma 6.32.2. Let $X$ be a topological space. Let $i : Z \to X$ be the inclusion of a closed subset. The functor

\[ i_* : \mathop{\mathit{Sh}}\nolimits (Z) \longrightarrow \mathop{\mathit{Sh}}\nolimits (X) \]

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

Proof. Fully faithfulness follows formally from $i^{-1} i_* = \text{id}$. We have seen that any sheaf in the image of the functor has the property on the stalks mentioned in the lemma. Conversely, suppose that $\mathcal{G}$ has the indicated property. Then it is easy to check that

\[ \mathcal{G} \to i_* i^{-1} \mathcal{G} \]

is an isomorphism on all stalks and hence an isomorphism. $\square$

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