The Stacks project

Lemma 84.6.6. In Lemma 84.6.4 if $f$ is finite, then $a_ Y^{-1}(Rf_{small, *}K) = Rf_{big, fppf, *}(a_ X^{-1}K)$ for $K$ in $D^+(X_{\acute{e}tale})$.

Proof. Let $V \to Y$ be a surjective étale morphism where $V$ is a scheme. It suffices to prove the base change map is an isomorphism after restricting to $V$. Hence we may assume that $Y$ is a scheme. As the morphism is finite, hence representable, we conclude that we may assume both $X$ and $Y$ are schemes. In this case the result follows from the case of schemes (Étale Cohomology, Lemma 59.100.6 part (2)) using the comparison of topoi discussed in Section 84.3 and in particular given in Lemma 84.3.1. Some details omitted. $\square$

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Your email address will not be published.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