The Stacks project

Definition 78.20.1. Quotient stacks. Let $B \to S$ be as above.

  1. Let $(U, R, s, t, c)$ be a groupoid in algebraic spaces over $B$. The quotient stack

    \[ p : [U/R] \longrightarrow (\mathit{Sch}/S)_{fppf} \]

    of $(U, R, s, t, c)$ is the stackification (see Stacks, Lemma 8.9.1) of the category fibred in groupoids $[U/_{\! p}R]$ over $(\mathit{Sch}/S)_{fppf}$ associated to (78.20.0.1).

  2. Let $(G, m)$ be a group algebraic space over $B$. Let $a : G \times _ B X \to X$ be an action of $G$ on an algebraic space over $B$. The quotient stack

    \[ p : [X/G] \longrightarrow (\mathit{Sch}/S)_{fppf} \]

    is the quotient stack associated to the groupoid in algebraic spaces $(X, G \times _ B X, s, t, c)$ over $B$ of Lemma 78.15.1.

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