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.
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.
exists as 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,[1] 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
)[citation needed] In general,
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.
An effective quotient orbifold, e.g.,
action has only finite stabilizers on the smooth space
, is an example of a quotient stack.
is called the classifying stack of
(in analogy with the classifying space of
Borel's theorem describes the cohomology ring of the classifying stack.
-points of the moduli stack are the groupoid of principal
There is another closely related moduli stack given by
which is the moduli stack of line bundles with
This follows directly from the definition of quotient stacks evaluated on points.
-points are the groupoid whose objects are given by the set
The morphism in the top row corresponds to the
This can be found by noting giving a
gives the same data as a section
This can be checked by looking at a chart and sending a point
This construction then globalizes by gluing affine charts together, giving a global section of the bundle.
Example:[3] Let L be the Lazard ring; i.e.,
, is called the moduli stack of formal group laws, denoted by