The Stacks project

Lemma 62.6.2. Let $j : U \to X$ be an étale morphism. Then $j^! = j^{-1}$.

Proof. This is true because $j_!$ as defined in Section 62.4 agrees with $j_!$ as defined in Étale Cohomology, Section 59.70, see Lemma 62.4.3. Finally, in Étale Cohomology, Section 59.70 the functor $j_!$ is defined as the left adjoint of $j^{-1}$ and hence we conclude by uniqueness of adjoint functors. $\square$


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Your email address will not be published.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 0F5A. Beware of the difference between the letter 'O' and the digit '0'.