# 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]}

## Contents

## DefinitionEdit

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 be the category over the category of *S*-schemes:

- an object over
*T*is a principal*G*-bundle together with equivariant map ; - an arrow from to is a bundle map (i.e., forms a cartesian diagram) that is compatible with the equivariant maps and .

Suppose the quotient exists as an algebraic space (for example, by the Keel–Mori theorem). The canonical map

- ,

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 exists.)^{[citation needed]}

In general, 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]}

## ExamplesEdit

If with trivial action of *G* (often *S* is a point), then 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., . Then the quotient stack by
,

- ,

is called the moduli stack of formal group laws, denoted by .

## See alsoEdit

- homotopy quotient
- moduli stack of principal bundles (which, roughly, is an infinite product of classifying stacks.)

## ReferencesEdit

**^**The*T*-point is obtained by completing the diagram .**^**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.**^**Taken from http://www.math.harvard.edu/~lurie/252xnotes/Lecture11.pdf

- Deligne, Pierre; Mumford, David (1969), "The irreducibility of the space of curves of given genus",
*Publications Mathématiques de l'IHÉS*,**36**(36): 75–109, CiteSeerX 10.1.1.589.288, doi:10.1007/BF02684599, MR 0262240 - Totaro, Burt (2004). "The resolution property for schemes and stacks".
*Journal für die reine und angewandte Mathematik*.**577**: 1–22. arXiv:math/0207210. doi:10.1515/crll.2004.2004.577.1. MR 2108211.

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" (PDF).