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 presenation (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

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 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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