The Stacks project

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 ├ętale 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. $\square$


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