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.

An orbifold is an example of a quotient stack.[citation needed]

Definition

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

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

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

${\displaystyle [X/G]\to X/G}$ ,

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 ${\displaystyle X/G}$  exists.)[citation needed]

In general, ${\displaystyle [X/G]}$  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.

(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, Thomason proved that a quotient stack has the resolution property.

Remark: It is possible to approach the construction from the point of view of simplicial sheaves.[2]

Examples

If ${\displaystyle X=S}$  with trivial action of G (often S is a point), then ${\displaystyle [S/G]}$  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.

Example:[3] Let L be the Lazard ring; i.e., ${\displaystyle L=\pi _{*}\operatorname {MU} }$ . Then the quotient stack ${\displaystyle [\operatorname {Spec} L/G]}$  by ${\displaystyle G}$ ,

${\displaystyle G(R)=\{g\in R[\![t]\!]|g(t)=b_{0}t+b_{1}t^{2}+\cdots ,b_{0}\in R^{\times }\}}$ ,

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

1. ^ The T-point is obtained by completing the diagram ${\displaystyle T\leftarrow P\to X\to X/G}$ .