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 or toric stacks.

An orbifold is an example of a quotient stack.

Definition

A quotient stack is defined as follows. Let G be an affine smooth group scheme over a scheme S and X a S-scheme on which G acts. Let $[X/G]$ be the category over the category of S-schemes: an object over T is a principal G-bundle P →T together with equivariant map P →X; an arrow from P →T to P' →T' is a bundle map (i.e., forms a cartesian diagram) that is compatible with the equivariant maps P →X and P' →X.

Suppose the quotient $X/G$ exists as, say, 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 $X/G$ usually 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.

(Totaro 04) 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, 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; cf. 9.2. of Jardine's "local homotopy theory".

Examples

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.

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

$G(R) = \{g \in R[\![t]\!] | g(t) = b_0 t + b_1t^2 \dots, b_0 \in R^\times \}$,

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

See also

• homotopy quotient
• moduli stack of principal bundles (which, roughly, is an infinite product of classifying stacks.)

References

• Burt Totaro, The resolution property for schemes and stacks, J. Reine Angew. Math. 577 (2004), 1–22. 25

Some other references are

• http://www.math.missouri.edu/~edidin/Papers/mfile.pdf
• http://www.math.ubc.ca/~behrend/thesis.pdf