Lemma 97.18.2. 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

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

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.

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