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:
 * an object over T is a principal G-bundle $$P\to T$$ together with equivariant map $$P\to X$$;
 * an arrow 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.

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

Moduli of line bundles with n-sections
There is another closely related moduli stack given by $$[\mathbb{A}^n/\mathbb{G}_m]$$ 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 $$[\mathbb{A}^n/\mathbb{G}_m](X) = \left\{ \begin{matrix} P & \to & \mathbb{A}^n \\ \downarrow & & \\ X \end{matrix} : \begin{align} &P \to \mathbb{A}^n \text{ is }\mathbb{G}_m\text{ equivariant and} \\ &P \to X \text{ is a principal } \mathbb{G}_m\text{-bundle} \end{align} \right\}$$ 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 $$\mathbb{G}_m$$-equivariant map $$\phi: P \to \mathbb{A}^1$$ 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 $$\phi_x$$, noting the set of $$\mathbb{G}_m$$-equivariant maps $$P|_x \to \mathbb{A}^1$$ is isomorphic to $$\mathbb{G}_m$$. This construction then globalizes by gluing affine charts together, giving a global section of the bundle. Since $$\mathbb{G}_m$$-equivariant maps to $$\mathbb{A}^n$$ is equivalently an $$n$$-tuple of $$\mathbb{G}_m$$-equivariant maps to $$\mathbb{A}^1$$, the result holds.

Moduli of formal group laws
Example: Let L be the Lazard ring; i.e., $$L = \pi_* \operatorname{MU}$$. Then the quotient stack $$[\operatorname{Spec}L/G]$$ by $$G$$,
 * $$G(R) = \{g \in R[\![t]\!] | g(t) = b_0 t + b_1t^2+ \cdots, b_0 \in R^\times \}$$,

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