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The Stacks project

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

  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. 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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