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.

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

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

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

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

In general, ${\displaystyle [X/G]}${\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.

Remark: It is possible to approach the construction from the point of view of simplicial sheaves.^[2] See also: simplicial diagram.

An effective quotient orbifold, e.g., ${\displaystyle [M/G]}${\displaystyle [M/G]} where the ${\displaystyle G}${\displaystyle G} action has only finite stabilizers on the smooth space ${\displaystyle M}${\displaystyle M}, is an example of a quotient stack.^[3]

If ${\displaystyle X=S}${\displaystyle X=S} with trivial action of ${\displaystyle G}${\displaystyle G} (often ${\displaystyle S}${\displaystyle S} is a point), then ${\displaystyle [S/G]}${\displaystyle [S/G]} is called the classifying stack of ${\displaystyle G}${\displaystyle G} (in analogy with the classifying space of ${\displaystyle G}${\displaystyle G}) and is usually denoted by ${\displaystyle BG}${\displaystyle BG}. Borel's theorem describes the cohomology ring of the classifying stack.

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

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

${\displaystyle [\mathbb {A} ^{n}/\mathbb {G} _{m}](X)=\left\{{\begin{matrix}P&\to &\mathbb {A} ^{n}\\\downarrow &&\\X\end{matrix}}:{\begin{aligned}&P\to \mathbb {A} ^{n}{\text{ is }}\mathbb {G} _{m}{\text{ equivariant and}}\\&P\to X{\text{ is a principal }}\mathbb {G} _{m}{\text{-bundle}}\end{aligned}}\right\}}${\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 ${\displaystyle n}${\displaystyle n}-sections of the associated line bundle over ${\displaystyle X}${\displaystyle X}. This can be found by noting giving a ${\displaystyle \mathbb {G} _{m}}${\displaystyle \mathbb {G} _{m}}-equivariant map ${\displaystyle \phi :P\to \mathbb {A} ^{1}}${\displaystyle \phi :P\to \mathbb {A} ^{1}} and restricting it to the fiber ${\displaystyle P|_{x}}${\displaystyle P|_{x}} gives the same data as a section ${\displaystyle \sigma }${\displaystyle \sigma } of the bundle. This can be checked by looking at a chart and sending a point ${\displaystyle x\in X}${\displaystyle x\in X} to the map ${\displaystyle \phi _{x}}${\displaystyle \phi _{x}}, noting the set of ${\displaystyle \mathbb {G} _{m}}${\displaystyle \mathbb {G} _{m}}-equivariant maps ${\displaystyle P|_{x}\to \mathbb {A} ^{1}}${\displaystyle P|_{x}\to \mathbb {A} ^{1}} is isomorphic to ${\displaystyle \mathbb {G} _{m}}${\displaystyle \mathbb {G} _{m}}. This construction then globalizes by gluing affine charts together, giving a global section of the bundle. Since ${\displaystyle \mathbb {G} _{m}}${\displaystyle \mathbb {G} _{m}}-equivariant maps to ${\displaystyle \mathbb {A} ^{n}}${\displaystyle \mathbb {A} ^{n}} is equivalently an ${\displaystyle n}${\displaystyle n}-tuple of ${\displaystyle \mathbb {G} _{m}}${\displaystyle \mathbb {G} _{m}}-equivariant maps to ${\displaystyle \mathbb {A} ^{1}}${\displaystyle \mathbb {A} ^{1}}, the result holds.

Example:^[4] Let L be the Lazard ring; i.e., ${\displaystyle L=\pi _{*}\operatorname {MU} }${\displaystyle L=\pi _{*}\operatorname {MU} }. Then the quotient stack ${\displaystyle [\operatorname {Spec} L/G]}${\displaystyle [\operatorname {Spec} L/G]} by ${\displaystyle G}${\displaystyle G},

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

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

• 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). Archived from the original (PDF) on 2013年05月16日. Retrieved 2013年09月19日.

• Algebraic geometry

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