The Stacks project

Definition 59.70.1. Let $j : U \to X$ be an étale morphism of schemes.

  1. The restriction functor $j^{-1} : \mathop{\mathit{Sh}}\nolimits (X_{\acute{e}tale}) \to \mathop{\mathit{Sh}}\nolimits (U_{\acute{e}tale})$ has a left adjoint $j_!^{Sh} : \mathop{\mathit{Sh}}\nolimits (U_{\acute{e}tale}) \to \mathop{\mathit{Sh}}\nolimits (X_{\acute{e}tale})$.

  2. The restriction functor $j^{-1} : \textit{Ab}(X_{\acute{e}tale}) \to \textit{Ab}(U_{\acute{e}tale})$ has a left adjoint which is denoted $j_! : \textit{Ab}(U_{\acute{e}tale}) \to \textit{Ab}(X_{\acute{e}tale})$ and called extension by zero.

  3. Let $\Lambda $ be a ring. The restriction functor $j^{-1} : \textit{Mod}(X_{\acute{e}tale}, \Lambda ) \to \textit{Mod}(U_{\acute{e}tale}, \Lambda )$ has a left adjoint which is denoted $j_! : \textit{Mod}(U_{\acute{e}tale}, \Lambda ) \to \textit{Mod}(X_{\acute{e}tale}, \Lambda )$ and called extension by zero.


Comments (4)

Comment #74 by Keenan Kidwell on

Is "...functor is right exact, so it has a left adjoint..." what is intended? Is there some result elsewhere which states that, in this context, having a left adjoint is implied by being right exact?

Comment #81 by on

Fixed. There is such a result (maybe with more hypotheses), but since we have an explicit description of the extension by zero functor it is better not to appeal to it here. Thanks!

Comment #3538 by Timo Keller on

In (1), the functor goes in the wrong direction.


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Your email address will not be published.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 03S3. Beware of the difference between the letter 'O' and the digit '0'.