Quotient stack explained

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

[X/G] be the category over the category of S-schemes:P\to T together with equivariant map P\to X ;

• an arrow from

P\to T to P'\to T' is a bundle map (i.e., forms a commutative diagram) that is compatible with the equivariant maps P\to X and P'\to X .X/G exists as an algebraic space (for example, by the Keel–Mori theorem). The canonical map[X/G] \to X/G ,that sends a bundle P over T to a corresponding T-point,^[1] 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 X/G exists.)

In general,

[X/G] 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.

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.,

[M/G] where the G action has only finite stabilizers on the smooth space M , is an example of a quotient stack.^[2]

If

X = S with trivial action of G (often S is a point), then [S/G] is called the classifying stack of G (in analogy with the classifying space of G ) and is usually denoted by BG . 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

B G _{m} of line bundles [*/ G _{m]} over Sch , or [S/ G _{m]} over Sch /S for the trivial G _{m} -action on S . For any scheme (or S -scheme) X , the X -points of the moduli stack are the groupoid of principal G _{m} -bundles P \to X .

Moduli of line bundles with n-sections

There is another closely related moduli stack given by

[ A ^{n/ G} _{m]} which is the moduli stack of line bundles with n -sections. This follows directly from the definition of quotient stacks evaluated on points. For a scheme X , the X -points are the groupoid whose objects are given by the set

[ A ^{n/ G} _{m](X)} = \left\{ \begin{matrix} P & \to & A ^{n} \\ \downarrow & & \\ X \end{matrix} : \begin{align} &P \to A ^{n} is G _{mequivariantand} \\ &P \to X isaprincipal G _{m-bundle \end{align} \right\}}

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

Moduli of formal group laws

Example:^[3] Let L be the Lazard ring; i.e.,

L = \pi _{*} \operatorname{MU} . Then the quotient stack [\operatorname{Spec}L/G] by G ,G(R) = \{g \in R[[t]] | g(t) = b _{0} t + b ^{2+} _{1 t} … , b _{0} \in R ^{x} \} ,is called the moduli stack of formal group laws, denoted by l{M} _{FG} .

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

• • Burt. Totaro. Burt Totaro. The resolution property for schemes and stacks. Journal für die reine und angewandte Mathematik. 577 . 2004. 1–22. 2108211. 10.1515/crll.2004.2004.577.1. math/0207210.

Some other references are

• Behrend. Kai. Kai Behrend. The Lefschetz trace formula for the moduli stack of principal bundles . University of California, Berkeley. 1991.
• Web site: Dan. Edidin. Notes on the construction of the moduli space of curves.

Notes and References

1. The T-point is obtained by completing the diagram T \leftarrow P \to X \to X/G .
2. Book: Orbifolds and Stringy Topology. Cambridge Tracts in Mathematics . Definition 1.7 . 4.
3. Taken from http://www.math.harvard.edu/~lurie/252xnotes/Lecture11.pdf