The Stacks project

Lemma 94.18.1. Let $S$ be a scheme and let $B$ be an algebraic space over $S$. Let $(U, R, s, t, c)$ be a groupoid in algebraic spaces over $B$. The quotient stack $[U/R]$ is an algebraic stack if and only if there exists a morphism of algebraic spaces $g : U' \to U$ such that

  1. the composition $U' \times _{g, U, t} R \to R \xrightarrow {s} U$ is a surjection of sheaves, and

  2. the morphisms $s', t' : R' \to U'$ are flat and locally of finite presentation where $(U', R', s', t', c')$ is the restriction of $(U, R, s, t, c)$ via $g$.

Proof. First, assume that $g : U' \to U$ satisfies (1) and (2). Property (1) implies that $[U'/R'] \to [U/R]$ is an equivalence, see Groupoids in Spaces, Lemma 75.24.2. By Theorem 94.17.2 the quotient stack $[U'/R']$ is an algebraic stack. Hence $[U/R]$ is an algebraic stack too, see Algebraic Stacks, Lemma 91.12.4.

Conversely, assume that $[U/R]$ is an algebraic stack. We may choose a scheme $W$ and a surjective smooth $1$-morphism

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

By the $2$-Yoneda lemma (Algebraic Stacks, Section 91.5) this corresponds to an object $\xi $ of $[U/R]$ over $W$. By the description of $[U/R]$ in Groupoids in Spaces, Lemma 75.23.1 we can find a surjective, flat, locally finitely presented morphism $b : U' \to W$ of schemes such that $\xi ' = b^*\xi $ corresponds to a morphism $g : U' \to U$. Note that the $1$-morphism

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

corresponding to $\xi '$ is surjective, flat, and locally of finite presentation, see Algebraic Stacks, Lemma 91.10.5. Hence $(\mathit{Sch}/U')_{fppf} \times _{[U/R]} (\mathit{Sch}/U')_{fppf}$ which is represented by the algebraic space

\[ \mathit{Isom}_{[U/R]}(\text{pr}_0^*\xi ', \text{pr}_1^*\xi ') = (U' \times _ S U') \times _{(g \circ \text{pr}_0, g \circ \text{pr}_1), U \times _ S U} R = R' \]

(see Groupoids in Spaces, Lemma 75.21.1 for the first equality; the second is the definition of restriction) is flat and locally of finite presentation over $U'$ via both $s'$ and $t'$ (by base change, see Algebraic Stacks, Lemma 91.10.6). By this description of $R'$ and by Algebraic Stacks, Lemma 91.16.1 we obtain a canonical fully faithful $1$-morphism $[U'/R'] \to [U/R]$. This $1$-morphism is essentially surjective because $f'$ is flat, locally of finite presentation, and surjective (see Stacks, Lemma 8.4.8); another way to prove this is to use Algebraic Stacks, Remark 91.16.3. Finally, we can use Groupoids in Spaces, Lemma 75.24.2 to conclude that the composition $U' \times _{g, U, t} R \to R \xrightarrow {s} U$ is a surjection of sheaves. $\square$


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