Lemma 72.10.4 (0B87)—The Stacks project

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Table of contents
Part 4: Algebraic Spaces
Chapter 72: Algebraic Spaces over Fields
Section 72.10: Schematic locus and field extension
Lemma 72.10.4 (cite)

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Lemma 72.10.4. Let $S$ be a scheme. Let $X$ be an algebraic space over $S$ and let $G$ be a finite group acting freely on $X$ . Set $Y = X/G$ as in Properties of Spaces, Lemma 66.34.1. For $y \in |Y|$ the following are equivalent

$y$ is in the schematic locus of $Y$ , and
there exists an affine open $U \subset X$ containing the preimage of $y$ .

Proof. It follows from the construction of $Y = X/G$ in Properties of Spaces, Lemma 66.34.1 that the morphism $X \to Y$ is surjective and étale. Of course we have $X \times _ Y X = X \times G$ hence the morphism $X \to Y$ is even finite étale. It is also surjective. Thus the lemma follows from Decent Spaces, Lemma 68.10.3. $\square$

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