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

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

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

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

In general, [ 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.

BurtTotaro(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. See also: simplicial diagram.

Examples

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

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

Moduli of line bundles with n-sections

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

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

Moduli of formal group laws

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

G ( R ) = { g ∈ R [ [ t ] ] | g ( t ) = b 0 t + b 1 t 2 + ⋯ , b 0 ∈ R × } {\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 M FG {\displaystyle {\mathcal {M}}_{\text{FG}}}.