The Stacks project

Lemma 59.90.2. Let $K/k$ be an extension of separably closed fields. Let $X$ be a quasi-compact and quasi-separated scheme over $k$. Let $E \in D^+(X_{\acute{e}tale})$ have cohomology sheaves whose stalks are torsion of orders invertible in $k$. Then

  1. the maps $H^ q_{\acute{e}tale}(X, E) \to H^ q_{\acute{e}tale}(X_ K, E|_{X_ K})$ are isomorphisms, and

  2. $E \to R(X_ K \to X)_*E|_{X_ K}$ is an isomorphism.

Proof. Proof of (1). First let $\overline{k}$ and $\overline{K}$ be the algebraic closures of $k$ and $K$. The morphisms $\mathop{\mathrm{Spec}}(\overline{k}) \to \mathop{\mathrm{Spec}}(k)$ and $\mathop{\mathrm{Spec}}(\overline{K}) \to \mathop{\mathrm{Spec}}(K)$ are universal homeomorphisms as $\overline{k}/k$ and $\overline{K}/K$ are purely inseparable (see Algebra, Lemma 10.46.7). Thus $H^ q_{\acute{e}tale}(X, \mathcal{F}) = H^ q_{\acute{e}tale}(X_{\overline{k}}, \mathcal{F}_{X_{\overline{k}}})$ by the topological invariance of étale cohomology, see Proposition 59.45.4. Similarly for $X_ K$ and $X_{\overline{K}}$. Thus we may assume $k$ and $K$ are algebraically closed. In this case $K$ is a limit of smooth $k$-algebras, see Algebra, Lemma 10.158.11. We conclude our lemma is a special case of Theorem 59.89.2 as reformulated in Lemma 59.89.3.

Proof of (2). For any quasi-compact and quasi-separated $U$ in $X_{\acute{e}tale}$ the above shows that the restriction of the map $E \to R(X_ K \to X)_*E|_{X_ K}$ determines an isomorphism on cohomology. Since every object of $X_{\acute{e}tale}$ has an étale covering by such $U$ this proves the desired statement. $\square$

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