Lemma 100.32.1. 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 96.17.2. 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 93.13.3) and of course it is equal to the algebraic space $U/R$ (of Bootstrap, Theorem 79.10.1). Since $U \to X$ is surjective, flat, and locally of finite presenation (Bootstrap, Lemma 79.11.6) we conclude that $X \to Y$ is flat and locally of finite presentation by Morphisms of Spaces, Lemma 66.31.5 and Descent on Spaces, Lemma 73.8.2. Finally, consider the cartesian diagram

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 horizonal arrow has this property by assumption (use Properties of Stacks, Lemma 99.3.3). Thus $X \to Y$ is unramified by Morphisms of Spaces, Lemma 66.38.9. We conclude by Morphisms of Spaces, Lemma 66.39.12. $\square$

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