The Stacks project

Theorem 97.17.2. Let $S$ be a scheme contained in $\mathit{Sch}_{fppf}$. Let $(U, R, s, t, c)$ be a groupoid in algebraic spaces over $S$. Assume $s, t$ are flat and locally of finite presentation. Then the quotient stack $[U/R]$ is an algebraic stack over $S$.

Proof. We check the two conditions of Theorem 97.16.1 for the morphism

\[ (\mathit{Sch}/U)_{fppf} \longrightarrow [U/R]. \]

The first is trivial (as $U$ is an algebraic space). The second is Lemma 97.17.1. $\square$

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