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Quotient stack

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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:

  • an object over T is a principal G-bundle together with equivariant map ;
  • an arrow 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