The Stacks project

Lemma 59.63.5. Let $X$ be a proper scheme over an algebraically closed field $k$ of characteristic $p > 0$. Then

  1. $H_{\acute{e}tale}^ q(X, \underline{\mathbf{Z}/p\mathbf{Z}})$ is a finite $\mathbf{Z}/p\mathbf{Z}$-module for all $q$, and

  2. $H^ q_{\acute{e}tale}(X, \underline{\mathbf{Z}/p\mathbf{Z}}) \to H^ q_{\acute{e}tale}(X_{k'}, \underline{\mathbf{Z}/p\mathbf{Z}}))$ is an isomorphism if $k'/k$ is an extension of algebraically closed fields.

Proof. By Cohomology of Schemes, Lemma 30.19.2) and the comparison of cohomology of Theorem 59.22.4 the cohomology groups $H^ q_{\acute{e}tale}(X, \mathbf{G}_ a) = H^ q(X, \mathcal{O}_ X)$ are finite dimensional $k$-vector spaces. Hence by Lemma 59.63.2 the long exact cohomology sequence associated to the Artin-Schreier sequence, splits into short exact sequences

\[ 0 \to H_{\acute{e}tale}^ q(X, \underline{\mathbf{Z}/p\mathbf{Z}}) \to H^ q(X, \mathcal{O}_ X) \xrightarrow {F - 1} H^ q(X, \mathcal{O}_ X) \to 0 \]

and moreover the $\mathbf{F}_ p$-dimension of the cohomology groups $H_{\acute{e}tale}^ q(X, \underline{\mathbf{Z}/p\mathbf{Z}})$ is equal to the $k$-dimension of the vector space $H^ q(X, \mathcal{O}_ X)$. This proves the first statement. The second statement follows as $H^ q(X, \mathcal{O}_ X) \otimes _ k k' \to H^ q(X_{k'}, \mathcal{O}_{X_{k'}})$ is an isomorphism by flat base change (Cohomology of Schemes, Lemma 30.5.2). $\square$

Comments (0)

There are also:

  • 4 comment(s) on Section 59.63: The Artin-Schreier sequence

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Your email address will not be published.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 0A3P. Beware of the difference between the letter 'O' and the digit '0'.