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, where

• an object over T is a principal G-bundle $P\to T$ together with equivariant map $P\to X$;
• a morphism 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, 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.

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: 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 ${\mathcal {M}}_{\text{FG}}$.

See also

• Homotopy quotient
• Moduli stack of principal bundles (which, roughly, is an infinite product of classifying stacks.)
• Group-scheme action
• Moduli of algebraic curves

References

• 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).