The Stacks project

94.13 Quotient stacks

Let $(U, R, s, t, c)$ be a groupoid in algebraic spaces over $S$. In this case the quotient stack

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

is a stack in groupoids by construction, see Groupoids in Spaces, Definition 77.20.1. It is even the case that the $\mathit{Isom}$-sheaves are representable by algebraic spaces, see Bootstrap, Lemma 79.11.5. These quotient stacks are of fundamental importance to the theory of algebraic stacks.

A special case of the construction above is the quotient stack

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

associated to a datum $(B, G/B, m, X/B, a)$. Here

  1. $B$ is an algebraic space over $S$,

  2. $(G, m)$ is a group algebraic space over $B$,

  3. $X$ is an algebraic space over $B$, and

  4. $a : G \times _ B X \to X$ is an action of $G$ on $X$ over $B$.

Namely, by Groupoids in Spaces, Definition 77.20.1 the stack in groupoids $[X/G]$ is the quotient stack $[X/G \times _ B X]$ given above. It behooves us to spell out what the category $[X/G]$ really looks like. We will do this in Section 94.15.

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