# 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 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.)

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

An effective quotient orbifold, e.g. ${\displaystyle [M/G]}$ where the ${\displaystyle G}$ action has only finite stabilizers on the smooth space ${\displaystyle M}$, is an example of a quotient stack.[3]

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:[4] 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 ${\displaystyle {\mathcal {M}}_{\text{FG}}}$.

## References

1. The T-point is obtained by completing the diagram ${\displaystyle T\leftarrow P\to X\to X/G}$.
2. Jardine, John F. (2015). Local homotopy theory. Springer Monographs in Mathematics. New York: Springer-Verlag. section 9.2. doi:10.1007/978-1-4939-2300-7. MR 3309296.
3. Orbifolds and Stringy Topology. Definition 1.7: CAMBRIDGE TRACTS IN MATHEMATICS. p. 4.

Some other references are