Lemma 67.13.5. Let $S$ be a scheme. Let $i : Z \to X$ be a closed immersion of algebraic spaces over $S$.
The functor
\[ i_{small, *} : \mathop{\mathit{Sh}}\nolimits (Z_{\acute{e}tale}) \longrightarrow \mathop{\mathit{Sh}}\nolimits (X_{\acute{e}tale}) \]
is fully faithful and its essential image is those sheaves of sets $\mathcal{F}$ on $X_{\acute{e}tale}$ whose restriction to $X \setminus Z$ is isomorphic to $*$, and
the functor
\[ i_{small, *} : \textit{Ab}(Z_{\acute{e}tale}) \longrightarrow \textit{Ab}(X_{\acute{e}tale}) \]
is fully faithful and its essential image is those abelian sheaves on $X_{\acute{e}tale}$ whose support is contained in $|Z|$.
In both cases $i_{small}^{-1}$ is a left inverse to the functor $i_{small, *}$.
Proof.
Let $U$ be a scheme and let $U \to X$ be surjective étale. Set $V = Z \times _ X U$. Then $V$ is a scheme and $i' : V \to U$ is a closed immersion of schemes. By Properties of Spaces, Lemma 66.18.12 for any sheaf $\mathcal{G}$ on $Z$ we have
\[ (i_{small}^{-1}i_{small, *}\mathcal{G})|_ V = (i')_{small}^{-1}i'_{small, *}(\mathcal{G}|_ V) \]
By Étale Cohomology, Proposition 59.46.4 the map $(i')_{small}^{-1}i'_{small, *}(\mathcal{G}|_ V) \to \mathcal{G}|_ V$ is an isomorphism. Since $V \to Z$ is surjective and étale this implies that $i_{small}^{-1}i_{small, *}\mathcal{G} \to \mathcal{G}$ is an isomorphism. This clearly implies that $i_{small, *}$ is fully faithful, see Sites, Lemma 7.41.1. To prove the statement on the essential image, consider a sheaf of sets $\mathcal{F}$ on $X_{\acute{e}tale}$ whose restriction to $X \setminus Z$ is isomorphic to $*$. As in the proof of Étale Cohomology, Proposition 59.46.4 we consider the adjunction mapping
\[ \mathcal{F} \longrightarrow i_{small, *}i_{small}^{-1}\mathcal{F}. \]
As in the first part we see that the restriction of this map to $U$ is an isomorphism by the corresponding result for the case of schemes. Since $U$ is an étale covering of $X$ we conclude it is an isomorphism.
$\square$
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