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 ${\displaystyle [X/G]}$ be the category over the category of S-schemes:

• an object over T is a principal G-bundle ${\displaystyle P\to T}$ together with equivariant map ${\displaystyle P\to X}$;
• an arrow from ${\displaystyle P\to T}$ to ${\displaystyle P'\to T'}$ is a bundle map (i.e., forms a commutative diagram) that is compatible with the equivariant maps ${\displaystyle P\to X}$ and ${\displaystyle P'\to X}$.

Suppose the quotient ${\displaystyle X/G}$ exists as an algebraic space (for example, by the Keel–Mori theorem). The canonical map

${\displaystyle [X/G]\to X/G}$,

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 ${\displaystyle X/G}$ exists.)^{[citation needed]}

In general, ${\displaystyle [X/G]}$ 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., ${\displaystyle [M/G]}$ where the ${\displaystyle G}$ action has only finite stabilizers on the smooth space ${\displaystyle M}$, is an example of a quotient stack.^[2]

If ${\displaystyle X=S}$ with trivial action of ${\displaystyle G}$ (often ${\displaystyle S}$ is a point), then ${\displaystyle [S/G]}$ is called the classifying stack of ${\displaystyle G}$ (in analogy with the classifying space of ${\displaystyle G}$) and is usually denoted by ${\displaystyle BG}$. 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 ${\displaystyle B\mathbb {G} _{m}}$ of line bundles ${\displaystyle [*/\mathbb {G} _{m}]}$ over ${\displaystyle {\text{Sch}}}$, or ${\displaystyle [S/\mathbb {G} _{m}]}$ over ${\displaystyle {\text{Sch}}/S}$ for the trivial ${\displaystyle \mathbb {G} _{m}}$-action on ${\displaystyle S}$. For any scheme (or ${\displaystyle S}$-scheme) ${\displaystyle X}$, the ${\displaystyle X}$-points of the moduli stack are the groupoid of principal ${\displaystyle \mathbb {G} _{m}}$-bundles ${\displaystyle P\to X}$.

Moduli of line bundles with n-sections

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

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

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

Moduli of formal group laws

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

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

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

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).