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.

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., $[M / G]$ 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 $B G$ . 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 $B G_{m}$ of line bundles $[* / G_{m}]$ over $Sch$ , or $[S / G_{m}]$ over $Sch / S$ for the trivial $G_{m}$ -action on $S$ . For any scheme (or $S$ -scheme) $X$ , the $X$ -points of the moduli stack are the groupoid of principal $G_{m}$ -bundles $P \to X$ .

Moduli of line bundles with n-sections

There is another closely related moduli stack given by $[A^{n} / G_{m}]$ 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

$[A^{n} / G_{m}] (X) = {\begin{matrix} P & \to & A^{n} \\ ↓ \\ X \end{matrix} : \begin{array}{l} P \to A^{n} is G_{m} equivariant and \\ P \to X is a principal G_{m} -bundle \end{array}}$

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

G_{m}

-equivariant map

ϕ : P \to A^{1}

and restricting it to the fiber

P |_{x}

gives the same data as a section

σ

of the bundle. This can be checked by looking at a chart and sending a point

x \in X

to the map

ϕ_{x}

, noting the set of

G_{m}

-equivariant maps

P |_{x} \to A^{1}

is isomorphic to

G_{m}

. This construction then globalizes by gluing affine charts together, giving a global section of the bundle. Since

G_{m}

-equivariant maps to

A^{n}

is equivalently an

n

-tuple of

G_{m}

-equivariant maps to

A^{1}

, the result holds.

Moduli of formal group laws

Example: For a S-scheme X, let ${Bun}_{G} (X) = Map (X, B G)$ , where $Map$ is a mapping stack. It is called the moduli stack of principal bundles on X.-->
Example: Let L be the Lazard ring; i.e., $L = π_{*} MU$ . Then the quotient stack $[Spec L / G]$ by

$G$ ,

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

is called the moduli stack of formal group laws, denoted by

M_{FG}

References

John F., Jardine, Rick Jardine, Local homotopy theory, Springer Monographs in Mathematics, Springer-Verlag, New York, 2015, 3309296, 10.1007/978-1-4939-2300-7, section 9.2
Orbifolds and Stringy Topology, Cambridge Tracts in Mathematics, Definition 1.7, 4

Some other references are

Category:Algebraic geometry

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