The Stacks project

Lemma 33.7.9. Let $k$ be a field. Let $X$ be a scheme 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 $\overline{k}/k'/k$ 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. By Lemma 33.7.8 there exists a finite subextension $k'/k \subset \overline{k}$ and an open $V' \subset X_{k'}$ which pulls back to $V$. This proves (1). Since $\text{Gal}(\overline{k}/k')$ is open in $\text{Gal}(\overline{k}/k)$ part (2) is clear as well. $\square$

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