Lemma 59.83.12. Let k'/k be an extension of separably closed fields. Let X be a proper scheme over k of dimension \leq 1. Let \mathcal{F} be a torsion abelian sheaf on X. Then the map H^ q_{\acute{e}tale}(X, \mathcal{F}) \to H^ q_{\acute{e}tale}(X_{k'}, \mathcal{F}|_{X_{k'}}) is an isomorphism for q \geq 0.
Proof. We have seen this for algebraically closed fields in Theorem 59.83.10. Given k \subset k' as in the statement of the lemma we can choose a diagram
\xymatrix{ k' \ar[r] & \overline{k}' \\ k \ar[u] \ar[r] & \overline{k} \ar[u] }
where k \subset \overline{k} and k' \subset \overline{k}' are the algebraic closures. Since k and k' are separably closed the field extensions \overline{k}/k and \overline{k}'/k' are algebraic and purely inseparable. In this case the morphisms X_{\overline{k}} \to X and X_{\overline{k}'} \to X_{k'} are universal homeomorphisms. Thus the cohomology of \mathcal{F} may be computed on X_{\overline{k}} and the cohomology of \mathcal{F}|_{X_{k'}} may be computed on X_{\overline{k}'}, see Proposition 59.45.4. Hence we deduce the general case from the case of algebraically closed fields. \square
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