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

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 = π ∗ MU . Then the quotient stack [ Spec ⁡ L / G ] by G ,

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