The Stacks project

Proposition 59.46.4. Let $i : Z \to X$ be a closed immersion of schemes.

  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's discuss the case of sheaves of sets. For any sheaf $\mathcal{G}$ on $Z$ the morphism $i_{small}^{-1}i_{small, *}\mathcal{G} \to \mathcal{G}$ is an isomorphism by Lemma 59.46.3 (and Theorem 59.29.10). This implies formally that $i_{small, *}$ is fully faithful, see Sites, Lemma 7.41.1. It is clear that $i_{small, *}\mathcal{G}|_{U_{\acute{e}tale}} \cong *$ where $U = X \setminus Z$. Conversely, suppose that $\mathcal{F}$ is a sheaf of sets on $X$ such that $\mathcal{F}|_{U_{\acute{e}tale}} \cong *$. Consider the adjunction mapping

\[ \mathcal{F} \longrightarrow i_{small, *}i_{small}^{-1}\mathcal{F} \]

Combining Lemmas 59.46.3 and 59.36.2 we see that it is an isomorphism. This finishes the proof of (1). The proof of (2) is identical. $\square$


Comments (0)

There are also:

  • 2 comment(s) on Section 59.46: Closed immersions and pushforward

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Your email address will not be published.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 04CA. Beware of the difference between the letter 'O' and the digit '0'.