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

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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   be the category over the category of S-schemes, where

  • an object over T is a principal G-bundle   together with equivariant map  ;
  • a morphism from   to   is a bundle map (i.e., forms a commutative diagram) that is compatible with the equivariant maps   and  .

Suppose the quotient   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   exists.)[citation needed]

In general,   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.

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.

Examples

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An effective quotient orbifold, e.g.,   where the   action has only finite stabilizers on the smooth space  , is an example of a quotient stack.[2]

If   with trivial action of   (often   is a point), then   is called the classifying stack of   (in analogy with the classifying space of  ) and is usually denoted by  . Borel's theorem describes the cohomology ring of the classifying stack.

Moduli of line bundles

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One of the basic examples of quotient stacks comes from the moduli stack   of line bundles   over  , or   over   for the trivial  -action on  . For any scheme (or  -scheme)  , the  -points of the moduli stack are the groupoid of principal  -bundles  .

Moduli of line bundles with n-sections

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There is another closely related moduli stack given by   which is the moduli stack of line bundles with  -sections. This follows directly from the definition of quotient stacks evaluated on points. For a scheme  , the  -points are the groupoid whose objects are given by the set

 

The morphism in the top row corresponds to the  -sections of the associated line bundle over  . This can be found by noting giving a  -equivariant map   and restricting it to the fiber   gives the same data as a section   of the bundle. This can be checked by looking at a chart and sending a point   to the map  , noting the set of  -equivariant maps   is isomorphic to  . This construction then globalizes by gluing affine charts together, giving a global section of the bundle. Since  -equivariant maps to   is equivalently an  -tuple of  -equivariant maps to  , the result holds.

Moduli of formal group laws

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Example:[3] Let L be the Lazard ring; i.e.,  . Then the quotient stack   by  ,

 ,

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

See also

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References

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  1. ^ The T-point is obtained by completing the diagram  .
  2. ^ "Definition 1.7". Orbifolds and Stringy Topology. Cambridge Tracts in Mathematics. p. 4.
  3. ^ Taken from http://www.math.harvard.edu/~lurie/252xnotes/Lecture11.pdf

Some other references are