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 or toric stacks.

An orbifold is an example of a quotient stack.

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 $[X/G]$ be the category over the category of S-schemes: an object over T is a principal G-bundle P →T together with equivariant map P →X; an arrow from P →T to P' →T' is a bundle map (i.e., forms a cartesian diagram) that is compatible with the equivariant maps P →X and P' →X.

Suppose the quotient $X/G$ exists as, say, 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 $X/G$ usually 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.

(Totaro & 04) 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; cf. 9.2. of Jardine's "local homotopy theory".^[2]

Examples

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.

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

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

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

See also

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

References

1. The T-point is obtained by completing the diagram $T \leftarrow P \to X \to X/G$.
2. http://www.math.uwo.ca/~jardine/papers/preprints/book.pdf
3. 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, doi:10.1007/BF02684599, MR 0262240
• Burt Totaro, The resolution property for schemes and stacks, J. Reine Angew. Math. 577 (2004), 1–22. 25

Some other references are

• http://www.math.missouri.edu/~edidin/Papers/mfile.pdf
• http://www.math.ubc.ca/~behrend/thesis.pdf