The Stacks project

Lemma 70.12.6. 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

  1. there exists a finite subextension $k \subset k' \subset \overline{k}$ and a quasi-compact open $V' \subset X_{k'}$ such that $V = (V')_{\overline{k}}$,

  2. 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 ├ętale 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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