The Stacks project

Lemma 101.32.2. Let $S$ be a scheme. Let $(U, R, s, t, c)$ be a groupoid in algebraic spaces over $S$. Assume $s, t$ are flat and locally of finite presentation. Let $P \subset R$ be an open subspace such that $(U, P, s|_ P, t|_ P, c|_{P \times _{s, U, t} P})$ is a groupoid in algebraic spaces over $S$. Then

\[ [U/P] \longrightarrow [U/R] \]

is a morphism of algebraic stacks which is representable by algebraic spaces, surjective, and étale.

Proof. Since $P \subset R$ is open, we see that $s|_ P$ and $t|_ P$ are flat and locally of finite presentation. Thus $[U/R]$ and $[U/P]$ are algebraic stacks by Criteria for Representability, Theorem 97.17.2. To see that the morphism is representable by algebraic spaces, it suffices to show that $[U/P] \to [U/R]$ is faithful on fibre categories, see Algebraic Stacks, Lemma 94.15.2. This follows immediately from the fact that $P \to R$ is a monomorphism and the explicit description of quotient stacks given in Groupoids in Spaces, Lemma 78.24.1. Having said this, we know what it means for $[U/P] \to [U/R]$ to be surjective and étale by Algebraic Stacks, Definition 94.10.1. Surjectivity follows for example from Criteria for Representability, Lemma 97.7.3 and the description of objects of quotient stacks (see lemma cited above) over spectra of fields. It remains to prove that our morphism is étale.

To do this it suffices to show that $U \times _{[U/R]} [U/P] \to U$ is étale, see Properties of Stacks, Lemma 100.3.3. By Groupoids in Spaces, Lemma 78.21.2 the fibre product is equal to $[R/P \times _{s, U, t} R]$ with morphism to $U$ induced by $s : R \to U$. The maps $s', t' : P \times _{s, U, t} R \to R$ are given by $s' : (p, r) \mapsto r$ and $t' : (p, r) \mapsto c(p, r)$. Since $P \subset R$ is open we conclude that $(t', s') : P \times _{s, U, t} R \to R \times _{s, U, s} R$ is an open immersion. Thus we may apply Lemma 101.32.1 to conclude. $\square$

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