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 $[X/G]$ be the category over the category of S-schemes:

an object over T is a principal G-bundle $P\to T$ 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 cartesian diagram) that is compatible with the equivariant maps $P\to X$ and $P'\to X$ .

Suppose the quotient $X/G$ 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.)^{[citation needed]}

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 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 $X=S$ 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}+\cdots ,b_{0}\in R^{\times }\}

,

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

References

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

[1] The T-point is obtained by completing the diagram $T\leftarrow P\to X\to X/G$ .

[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] Taken from http://www.math.harvard.edu/~lurie/252xnotes/Lecture11.pdf

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[2]

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Definition

Examples

See also

References