Lemma 72.12.6 (0A15)—The Stacks project

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Table of contents
Part 4: Algebraic Spaces
Chapter 72: Algebraic Spaces over Fields
Section 72.12: Geometrically connected algebraic spaces
Lemma 72.12.6 (cite)

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Lemma 72.12.6. Let $k$ be a field. Let $X$ be an algebraic space over $k$ . Let $\overline{k}$ be a (possibly infinite) Galois extension of $k$ . Let $V \subset X_{\overline{k}}$ be a quasi-compact open. Then

there exists a finite subextension $\overline{k}/k'/k$ and a quasi-compact open $V' \subset X_{k'}$ such that $V = (V')_{\overline{k}}$ ,
there exists an open subgroup $H \subset \text{Gal}(\overline{k}/k)$ such that $\sigma (V) = V$ for all $\sigma \in H$ .

Proof. Choose a scheme $U$ and a surjective étale morphism $U \to X$ . Choose a quasi-compact open $W \subset U_{\overline{k}}$ whose image in $X_{\overline{k}}$ is $V$ . This is possible because $|U_{\overline{k}}| \to |X_{\overline{k}}|$ is continuous and because $|U_{\overline{k}}|$ has a basis of quasi-compact opens. We can apply Varieties, Lemma 33.7.9 to $W \subset U_{\overline{k}}$ to obtain the lemma. $\square$

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