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). $\square$

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