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 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 a S-scheme on which G acts. Let be the category over the category of S-schemes:
Suppose the quotient exists as an algebraic space. 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 general, is an Artin 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., where the action has only finite stabilizers on the smooth space, is an example of a quotient stack.
If with trivial action of G, then is called the classifying stack 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.,. Then the quotient stack by
is called the moduli stack of formal group laws, denoted by.