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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