The Stacks project

Lemma 57.80.11. Let $k \subset k'$ 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 57.80.9. 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 $k \subset \overline{k}$ and $k' \subset \overline{k}'$ 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 57.45.4. Hence we deduce the general case from the case of algebraically closed fields. $\square$


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