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The Stacks project

Lemma 67.13.5. Let S be a scheme. Let i : Z \to X be a closed immersion of algebraic spaces over S.

  1. 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

  2. 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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