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 an S-scheme on which G acts. Let the quotient stack $[X/G]$ 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 commutative 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.)

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.

Burt 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, Robert Wayne Thomason proved that a quotient stack has the resolution property.

Examples

An effective quotient orbifold, e.g., $[M/G]$ where the $G$ action has only finite stabilizers on the smooth space $M$ , is an example of a quotient stack.[2]

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.

Moduli of line bundles

One of the basic examples of quotient stacks comes from the moduli stack $B\mathbb {G} _{m}$ of line bundles $[*/\mathbb {G} _{m}]$ over ${\text{Sch}}$ , or $[S/\mathbb {G} _{m}]$ over ${\text{Sch}}/S$ for the trivial $\mathbb {G} _{m}$ -action on $S$ . For any scheme (or $S$ -scheme) $X$ , the $X$ -points of the moduli stack are the groupoid of principal $\mathbb {G} _{m}$ -bundles $P\to X$ .

Moduli of line bundles with n-sections

There is another closely related moduli stack given by $[\mathbb {A} ^{n}/\mathbb {G} _{m}]$ which is the moduli stack of line bundles with $n$ -sections. This follows directly from the definition of quotient stacks evaluated on points. For a scheme $X$ , the $X$ -points are the groupoid whose objects are given by the set

$[\mathbb {A} ^{n}/\mathbb {G} _{m}](X)=\left\{{\begin{matrix}P&\to &\mathbb {A} ^{n}\\\downarrow &&\\X\end{matrix}}:{\begin{aligned}&P\to \mathbb {A} ^{n}{\text{ is }}\mathbb {G} _{m}{\text{ equivariant and}}\\&P\to X{\text{ is a principal }}\mathbb {G} _{m}{\text{-bundle}}\end{aligned}}\right\}$

The morphism in the top row corresponds to the $n$ -sections of the associated line bundle over $X$ . This can be found by noting giving a $\mathbb {G} _{m}$ -equivariant map $\phi :P\to \mathbb {A} ^{1}$ and restricting it to the fiber $P|_{x}$ gives the same data as a section $\sigma$ of the bundle. This can be checked by looking at a chart and sending a point $x\in X$ to the map $\phi _{x}$ , noting the set of $\mathbb {G} _{m}$ -equivariant maps $P|_{x}\to \mathbb {A} ^{1}$ is isomorphic to $\mathbb {G} _{m}$ . This construction then globalizes by gluing affine charts together, giving a global section of the bundle. Since $\mathbb {G} _{m}$ -equivariant maps to $\mathbb {A} ^{n}$ is equivalently an $n$ -tuple of $\mathbb {G} _{m}$ -equivariant maps to $\mathbb {A} ^{1}$ , the result holds.

Moduli of formal group laws

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 $T\leftarrow P\to X\to X/G$ .
Orbifolds and Stringy Topology. Definition 1.7: Cambridge Tracts in Mathematics. p. 4.{{cite book}}: CS1 maint: location (link)
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).

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

[2] Orbifolds and Stringy Topology. Definition 1.7: Cambridge Tracts in Mathematics. p. 4.{{cite book}}: CS1 maint: location (link)

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