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. $\square$


Comments (0)


Post a comment

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.




In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 04E5. Beware of the difference between the letter 'O' and the digit '0'.