The Stacks project

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

\[ \xymatrix{ R \ar[d] \ar[r] & U \times _ Y U \ar[d] \\ X \ar[r] & X \times _ Y X } \]

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 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. $\square$


Comments (0)


Post a comment

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.




In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 0DU1. Beware of the difference between the letter 'O' and the digit '0'.