The Stacks project

Lemma 59.39.5. Let $K/k$ be an extension of fields with $k$ separably algebraically closed. Let $S$ be a scheme over $k$. Denote $p : S_ K = S \times _{\mathop{\mathrm{Spec}}(k)} \mathop{\mathrm{Spec}}(K) \to S$ the projection. Let $\mathcal{F}$ be a sheaf on $S_{\acute{e}tale}$. Then $\Gamma (S, \mathcal{F}) = \Gamma (S_ K, p^{-1}_{small}\mathcal{F})$.

Proof. Follows from Lemma 59.39.3. Namely, it is clear that $p$ is flat and quasi-compact as the base change of $\mathop{\mathrm{Spec}}(K) \to \mathop{\mathrm{Spec}}(k)$. On the other hand, if $\overline{s} : \mathop{\mathrm{Spec}}(L) \to S$ is a geometric point, then the fibre of $p$ over $\overline{s}$ is the spectrum of $K \otimes _ k L$ which is irreducible hence connected by Algebra, Lemma 10.47.2. $\square$

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