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 $P^{\prime }\to T^{\prime }$ is a bundle map (i.e., forms a commutative diagram) that is compatible with the equivariant maps $P\to X$ and $P^{\prime }\to X$.

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.

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 $BG$. 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 $B{𝔾}_m$ of line bundles $[*/{𝔾}_m]$ over $\text{Sch}$, or $[S/{𝔾}_m]$ over $\text{Sch}/S$ 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

$[{𝔸}^n/{𝔾}_m](X)=\{\begin{array}{ccc}P& \to & {𝔸}^n\\ \downarrow & & \\ X\end{array}:\begin{array}{cc}& P\to {𝔸}^n\text{ is }{𝔾}_m\text{ equivariant and}\\ & P\to X\text{ is a principal }{𝔾}_m\text{-bundle}\end{array}\}$

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

$\varphi :P\to {𝔸}^1$

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

${\varphi }_x$

, 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: Let L be the Lazard ring; i.e., $L={\pi }_{*}\mathrm{MU}$. Then the quotient stack $[\mathrm{Spec}L/G]$ 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 ${ℳ}_{\text{FG}}$.

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

Some other references are