Lemma 101.32.1. Let Y be an algebraic space. Let (U, R, s, t, c) be a groupoid in algebraic spaces over Y. Assume U \to Y is flat and locally of finite presentation and R \to U \times _ Y U an open immersion. Then X = [U/R] = U/R is an algebraic space and X \to Y is étale.
Proof. The quotient stack [U/R] is an algebraic stacks by Criteria for Representability, Theorem 97.17.2. On the other hand, since R \to U \times U is a monomorphism, it is an algebraic space (by our abuse of language and Algebraic Stacks, Proposition 94.13.3) and of course it is equal to the algebraic space U/R (of Bootstrap, Theorem 80.10.1). Since U \to X is surjective, flat, and locally of finite presentation (Bootstrap, Lemma 80.11.6) we conclude that X \to Y is flat and locally of finite presentation by Morphisms of Spaces, Lemma 67.31.5 and Descent on Spaces, Lemma 74.8.2. Finally, consider the cartesian diagram
\xymatrix{ R \ar[d] \ar[r] & U \times _ Y U \ar[d] \\ X \ar[r] & X \times _ Y X }
Since the right vertical arrow is surjective, flat, and locally of finite presentation (small detail omitted), we find that X \to X \times _ Y X is an open immersion as the top horizontal arrow has this property by assumption (use Properties of Stacks, Lemma 100.3.3). Thus X \to Y is unramified by Morphisms of Spaces, Lemma 67.38.9. We conclude by Morphisms of Spaces, Lemma 67.39.12. \square
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