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.
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