The Stacks project

Lemma 94.18.2. Let $S$ be a scheme and let $B$ be an algebraic space over $S$. Let $G$ be a group algebraic space over $B$. Let $X$ be an algebraic space over $B$ and let $a : G \times _ B X \to X$ be an action of $G$ on $X$ over $B$. The quotient stack $[X/G]$ is an algebraic stack if and only if there exists a morphism of algebraic spaces $\varphi : X' \to X$ such that

  1. $G \times _ B X' \to X$, $(g, x') \mapsto a(g, \varphi (x'))$ is a surjection of sheaves, and

  2. the two projections $X'' \to X'$ of the algebraic space $X''$ given by the rule

    \[ T \longmapsto \{ (x'_1, g, x'_2) \in (X' \times _ B G \times _ B X')(T) \mid \varphi (x'_1) = a(g, \varphi (x'_2))\} \]

    are flat and locally of finite presentation.

Proof. This lemma is a special case of Lemma 94.18.1. Namely, the quotient stack $[X/G]$ is by Groupoids in Spaces, Definition 75.19.1 equal to the quotient stack $[X/G \times _ B X]$ of the groupoid in algebraic spaces $(X, G \times _ B X, s, t, c)$ associated to the group action in Groupoids in Spaces, Lemma 75.14.1. There is one small observation that is needed to get condition (1). Namely, the morphism $s : G \times _ B X \to X$ is the second projection and the morphism $t : G \times _ B X \to X$ is the action morphism $a$. Hence the morphism $h : U' \times _{g, U, t} R \to R \xrightarrow {s} U$ from Lemma 94.18.1 corresponds to the morphism

\[ X' \times _{\varphi , X, a} (G \times _ B X) \xrightarrow {\text{pr}_1} X \]

in the current setting. However, because of the symmetry given by the inverse of $G$ this morphism is isomorphic to the morphism

\[ (G \times _ B X) \times _{\text{pr}_1, X, \varphi } X' \xrightarrow {a} X \]

of the statement of the lemma. Details omitted. $\square$


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