77.20 Quotient stacks

In this section and the next few sections we describe a kind of generalization of Section 77.19 above and Groupoids, Section 39.20. It is different in the following way: We are going to take quotient stacks instead of quotient sheaves.

Let us assume we have a scheme $S$, and algebraic space $B$ over $S$ and a groupoid in algebraic spaces $(U, R, s, t, c)$ over $B$. Given these data we consider the functor

77.20.0.1
$$\label{spaces-groupoids-equation-quotient-stack} \begin{matrix} (\mathit{Sch}/S)_{fppf}^{opp} & \longrightarrow & \textit{Groupoids} \\ S' & \longmapsto & (U(S'), R(S'), s, t, c) \end{matrix}$$

By Categories, Example 4.37.1 this “presheaf in groupoids” corresponds to a category fibred in groupoids over $(\mathit{Sch}/S)_{fppf}$. In this chapter we will denote this

$[U/_{\! p}R] \to (\mathit{Sch}/S)_{fppf}$

where the subscript ${}_ p$ is there to distinguish from the quotient stack.

Definition 77.20.1. 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 (77.20.0.1).

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 77.15.1.

Thus $[U/R]$ and $[X/G]$ are stacks in groupoids over $(\mathit{Sch}/S)_{fppf}$. These stacks will be very important later on and hence it makes sense to give a detailed description. Recall that given an algebraic space $X$ over $S$ we use the notation $\mathcal{S}_ X \to (\mathit{Sch}/S)_{fppf}$ to denote the stack in sets associated to the sheaf $X$, see Categories, Lemma 4.38.6 and Stacks, Lemma 8.6.2.

Lemma 77.20.2. Assume $B \to S$ and $(U, R, s, t, c)$ as in Definition 77.20.1 (1). There are canonical $1$-morphisms $\pi : \mathcal{S}_ U \to [U/R]$, and $[U/R] \to \mathcal{S}_ B$ of stacks in groupoids over $(\mathit{Sch}/S)_{fppf}$. The composition $\mathcal{S}_ U \to \mathcal{S}_ B$ is the $1$-morphism associated to the structure morphism $U \to B$.

Proof. During this proof let us denote $[U/_{\! p}R]$ the category fibred in groupoids associated to the presheaf in groupoids (77.20.0.1). By construction of the stackification there is a $1$-morphism $[U/_{\! p}R] \to [U/R]$. The $1$-morphism $\mathcal{S}_ U \to [U/R]$ is simply the composition $\mathcal{S}_ U \to [U/_{\! p}R] \to [U/R]$, where the first arrow associates to the scheme $S'/S$ and morphism $x : S' \to U$ over $S$ the object $x \in U(S')$ of the fibre category of $[U/_{\! p}R]$ over $S'$.

To construct the $1$-morphism $[U/R] \to \mathcal{S}_ B$ it is enough to construct the $1$-morphism $[U/_{\! p}R] \to \mathcal{S}_ B$, see Stacks, Lemma 8.9.2. On objects over $S'/S$ we just use the map

$U(S') \longrightarrow B(S')$

coming from the structure morphism $U \to B$. And clearly, if $a \in R(S')$ is an “arrow” with source $s(a) \in U(S')$ and target $t(a) \in U(S')$, then since $s$ and $t$ are morphisms over $B$ these both map to the same element $\overline{a}$ of $B(S')$. Hence we can map an arrow $a \in R(S')$ to the identity morphism of $\overline{a}$. (This is good because the fibre category $(\mathcal{S}_ B)_{S'}$ only contains identities.) We omit the verification that this rule is compatible with pullback on these split fibred categories, and hence defines a $1$-morphism $[U/_{\! p}R] \to \mathcal{S}_ B$ as desired.

We omit the verification of the last statement. $\square$

Lemma 77.20.3. Assumptions and notation as in Lemma 77.20.2. There exists a canonical $2$-morphism $\alpha : \pi \circ s \to \pi \circ t$ making the diagram

$\xymatrix{ \mathcal{S}_ R \ar[r]_ s \ar[d]_ t & \mathcal{S}_ U \ar[d]^\pi \\ \mathcal{S}_ U \ar[r]^-\pi & [U/R] }$

$2$-commutative.

Proof. Let $S'$ be a scheme over $S$. Let $r : S' \to R$ be a morphism over $S$. Then $r \in R(S')$ is an isomorphism between the objects $s \circ r, t \circ r \in U(S')$. Moreover, this construction is compatible with pullbacks. This gives a canonical $2$-morphism $\alpha _ p : \pi _ p \circ s \to \pi _ p \circ t$ where $\pi _ p : \mathcal{S}_ U \to [U/_{\! p}R]$ is as in the proof of Lemma 77.20.2. Thus even the diagram

$\xymatrix{ \mathcal{S}_ R \ar[r]_ s \ar[d]^ t & \mathcal{S}_ U \ar[d]^{\pi _ p} \\ \mathcal{S}_ U \ar[r]^-{\pi _ p} & [U/_{\! p}R] }$

is $2$-commutative. Thus a fortiori the diagram of the lemma is $2$-commutative. $\square$

Remark 77.20.4. In future chapters we will use the ambiguous notation where instead of writing $\mathcal{S}_ X$ for the stack in sets associated to $X$ we simply write $X$. Using this notation the diagram of Lemma 77.20.3 becomes the familiar diagram

$\xymatrix{ R \ar[r]_ s \ar[d]_ t & U \ar[d]^\pi \\ U \ar[r]^-\pi & [U/R] }$

In the following sections we will show that this diagram has many good properties. In particular we will show that it is a $2$-fibre product (Section 77.22) and that it is close to being a $2$-coequalizer of $s$ and $t$ (Section 77.23).

Comment #1812 by OS on

In \label{equation-quotient-stack} I guess it should not be $(\Sch/S)_{fppf}^{opp}$ but $(\Sch/B)_{fppf}^{opp}$. There are several other places below where the same change would be necessary.

Comment #1830 by on

Hmm.... no! The point is that we always want our categories fibred in groupoids to be on the category of schemes over $S$. What you are sayng is actually discussed in Lemma 77.20.2.

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