Lemma 70.10.3. Let $k \subset k'$ be a finite Galois extension with Galois group $G$. Let $X$ be an algebraic space over $k$. Then $G$ acts freely on the algebraic space $X_{k'}$ and $X = X_{k'}/G$ in the sense of Properties of Spaces, Lemma 64.34.1.

Proof. Omitted. Hints: First show that $\mathop{\mathrm{Spec}}(k) = \mathop{\mathrm{Spec}}(k')/G$. Then use compatibility of taking quotients with base change. $\square$

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