The Stacks project

Theorem 90.17.3. Let $S$ be a scheme contained in $\mathit{Sch}_{fppf}$. Let $(U, R, s, t, c)$ be a smooth groupoid in algebraic spaces over $S$. Then the quotient stack $[U/R]$ is an algebraic stack over $S$.

Proof. We check the three conditions of Definition 90.12.1. By construction we have that $[U/R]$ is a stack in groupoids which is the first condition.

The second condition follows from the stronger Lemma 90.17.1.

Finally, we have to show there exists a scheme $W$ over $S$ and a surjective smooth $1$-morphism $(\mathit{Sch}/W)_{fppf} \longrightarrow \mathcal{X}$. First choose $W \in \mathop{\mathrm{Ob}}\nolimits ((\mathit{Sch}/S)_{fppf})$ and a surjective ├ętale morphism $W \to U$. Note that this gives a surjective ├ętale morphism $\mathcal{S}_ W \to \mathcal{S}_ U$ of categories fibred in sets, see Lemma 90.10.3. Of course then $\mathcal{S}_ W \to \mathcal{S}_ U$ is also surjective and smooth, see Lemma 90.10.9. Hence $\mathcal{S}_ W \to \mathcal{S}_ U \to [U/R]$ is surjective and smooth by a combination of Lemmas 90.17.2 and 90.10.5. $\square$


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