Lemma 79.11.6. Let $S$ be a scheme. Consider an algebraic space $F$ of the form $F = U/R$ where $(U, R, s, t, c)$ is a groupoid in algebraic spaces over $S$ such that $s, t$ are flat and locally of finite presentation, and $j = (t, s) : R \to U \times _ S U$ is an equivalence relation. Then $U \to F$ is surjective, flat, and locally of finite presentation.

Proof. This is almost but not quite a triviality. Namely, by Groupoids in Spaces, Lemma 77.19.5 and the fact that $j$ is a monomorphism we see that $R = U \times _ F U$. Choose a scheme $W$ and a surjective étale morphism $W \to F$. As $U \to F$ is a surjection of sheaves we can find an fppf covering $\{ W_ i \to W\}$ and maps $W_ i \to U$ lifting the morphisms $W_ i \to F$. Then we see that

$W_ i \times _ F U = W_ i \times _ U U \times _ F U = W_ i \times _{U, t} R$

and the projection $W_ i \times _ F U \to W_ i$ is the base change of $t : R \to U$ hence flat and locally of finite presentation, see Morphisms of Spaces, Lemmas 66.30.4 and 66.28.3. Hence by Descent on Spaces, Lemmas 73.11.13 and 73.11.10 we see that $U \to F$ is flat and locally of finite presentation. It is surjective by Spaces, Remark 64.5.2. $\square$

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.

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 06PG. Beware of the difference between the letter 'O' and the digit '0'.