Quotient stack

In algebraic geometry, a quotient stack is a stack that parametrizes equivariant objects. Geometrically, it generalizes a quotient of a scheme or a variety by a group: a quotient variety, say, would be a coarse approximation of a quotient stack. The notion is of fundamental importance in the study of stacks: a stack that arises in nature is often either a quotient stack itself or admits a stratification by quotient stacks (e.g., a Deligne–Mumford stack.) A quotient stack is also used to construct other stacks like classifying stacks.

Definition

A quotient stack is defined as follows. Let G be an affine smooth group scheme over a scheme S and X an S-scheme on which G acts. Let the quotient stack [math]\displaystyle{ [X/G] }[/math] be the category over the category of S-schemes:

• an object over T is a principal G-bundle [math]\displaystyle{ P\to T }[/math] together with equivariant map [math]\displaystyle{ P\to X }[/math];
• an arrow from [math]\displaystyle{ P\to T }[/math] to [math]\displaystyle{ P'\to T' }[/math] is a bundle map (i.e., forms a commutative diagram) that is compatible with the equivariant maps [math]\displaystyle{ P\to X }[/math] and [math]\displaystyle{ P'\to X }[/math].

Suppose the quotient [math]\displaystyle{ X/G }[/math] exists as an algebraic space (for example, by the Keel–Mori theorem). The canonical map

[math]\displaystyle{ [X/G] \to X/G }[/math],

that sends a bundle P over T to a corresponding T-point, need not be an isomorphism of stacks; that is, the space "X/G" is usually coarser. The canonical map is an isomorphism if and only if the stabilizers are trivial (in which case [math]\displaystyle{ X/G }[/math] exists.)

In general, [math]\displaystyle{ [X/G] }[/math] is an Artin stack (also called algebraic stack). If the stabilizers of the geometric points are finite and reduced, then it is a Deligne–Mumford stack.

Burt Totaro (2004) has shown: let X be a normal Noetherian algebraic stack whose stabilizer groups at closed points are affine. Then X is a quotient stack if and only if it has the resolution property; i.e., every coherent sheaf is a quotient of a vector bundle. Earlier, Robert Wayne Thomason proved that a quotient stack has the resolution property.

Examples

An effective quotient orbifold, e.g., [math]\displaystyle{ [M/G] }[/math] where the [math]\displaystyle{ G }[/math] action has only finite stabilizers on the smooth space [math]\displaystyle{ M }[/math], is an example of a quotient stack.

If [math]\displaystyle{ X = S }[/math] with trivial action of [math]\displaystyle{ G }[/math] (often [math]\displaystyle{ S }[/math] is a point), then [math]\displaystyle{ [S/G] }[/math] is called the classifying stack of [math]\displaystyle{ G }[/math] (in analogy with the classifying space of [math]\displaystyle{ G }[/math]) and is usually denoted by [math]\displaystyle{ BG }[/math]. Borel's theorem describes the cohomology ring of the classifying stack.

Moduli of line bundles

One of the basic examples of quotient stacks comes from the moduli stack [math]\displaystyle{ B\mathbb{G}_m }[/math] of line bundles [math]\displaystyle{ [*/\mathbb{G}_m] }[/math] over [math]\displaystyle{ \text{Sch} }[/math], or [math]\displaystyle{ [S/\mathbb{G}_m] }[/math] over [math]\displaystyle{ \text{Sch}/S }[/math] for the trivial [math]\displaystyle{ \mathbb{G}_m }[/math]-action on [math]\displaystyle{ S }[/math]. For any scheme (or [math]\displaystyle{ S }[/math]-scheme) [math]\displaystyle{ X }[/math], the [math]\displaystyle{ X }[/math]-points of the moduli stack are the groupoid of principal [math]\displaystyle{ \mathbb{G}_m }[/math]-bundles [math]\displaystyle{ P \to X }[/math].

Moduli of line bundles with n-sections

There is another closely related moduli stack given by [math]\displaystyle{ [\mathbb{A}^n/\mathbb{G}_m] }[/math] which is the moduli stack of line bundles with [math]\displaystyle{ n }[/math]-sections. This follows directly from the definition of quotient stacks evaluated on points. For a scheme [math]\displaystyle{ X }[/math], the [math]\displaystyle{ X }[/math]-points are the groupoid whose objects are given by the set

[math]\displaystyle{ [\mathbb{A}^n/\mathbb{G}_m](X) = \left\{ \begin{matrix} P & \to & \mathbb{A}^n \\ \downarrow & & \\ X \end{matrix} : \begin{align} &P \to \mathbb{A}^n \text{ is }\mathbb{G}_m\text{ equivariant and} \\ &P \to X \text{ is a principal } \mathbb{G}_m\text{-bundle} \end{align} \right\} }[/math]

The morphism in the top row corresponds to the [math]\displaystyle{ n }[/math]-sections of the associated line bundle over [math]\displaystyle{ X }[/math]. This can be found by noting giving a [math]\displaystyle{ \mathbb{G}_m }[/math]-equivariant map [math]\displaystyle{ \phi: P \to \mathbb{A}^1 }[/math] and restricting it to the fiber [math]\displaystyle{ P|_x }[/math] gives the same data as a section [math]\displaystyle{ \sigma }[/math] of the bundle. This can be checked by looking at a chart and sending a point [math]\displaystyle{ x \in X }[/math] to the map [math]\displaystyle{ \phi_x }[/math], noting the set of [math]\displaystyle{ \mathbb{G}_m }[/math]-equivariant maps [math]\displaystyle{ P|_x \to \mathbb{A}^1 }[/math] is isomorphic to [math]\displaystyle{ \mathbb{G}_m }[/math]. This construction then globalizes by gluing affine charts together, giving a global section of the bundle. Since [math]\displaystyle{ \mathbb{G}_m }[/math]-equivariant maps to [math]\displaystyle{ \mathbb{A}^n }[/math] is equivalently an [math]\displaystyle{ n }[/math]-tuple of [math]\displaystyle{ \mathbb{G}_m }[/math]-equivariant maps to [math]\displaystyle{ \mathbb{A}^1 }[/math], the result holds.

Moduli of formal group laws

Example: Let L be the Lazard ring; i.e., [math]\displaystyle{ L = \pi_* \operatorname{MU} }[/math]. Then the quotient stack [math]\displaystyle{ [\operatorname{Spec}L/G] }[/math] by [math]\displaystyle{ G }[/math],

[math]\displaystyle{ G(R) = \{g \in R[\![t]\!] | g(t) = b_0 t + b_1t^2+ \cdots, b_0 \in R^\times \} }[/math],

is called the moduli stack of formal group laws, denoted by [math]\displaystyle{ \mathcal{M}_\text{FG} }[/math].

See also

• Homotopy quotient
• Moduli stack of principal bundles (which, roughly, is an infinite product of classifying stacks.)
• Group-scheme action
• Moduli of algebraic curves

References

• Deligne, Pierre; Mumford, David (1969), "The irreducibility of the space of curves of given genus", Publications Mathématiques de l'IHÉS 36 (36): 75–109, doi:10.1007/BF02684599, http://www.numdam.org/item?id=PMIHES_1969__36__75_0 
• Totaro, Burt (2004). "The resolution property for schemes and stacks". Journal für die reine und angewandte Mathematik 577: 1–22. doi:10.1515/crll.2004.2004.577.1. 

Some other references are

• Behrend, Kai (1991). The Lefschetz trace formula for the moduli stack of principal bundles (PDF) (Thesis). University of California, Berkeley.
• Edidin, Dan. "Notes on the construction of the moduli space of curves". http://www.math.missouri.edu/~edidin/Papers/mfile.pdf.