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.
- 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 cartesian diagram) that is compatible with the equivariant maps and .
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 exists.)
(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, 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.
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 (often S is a point), then 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., . Then the quotient stack by ,
is called the moduli stack of formal group laws, denoted by .
- The T-point is obtained by completing the diagram .
- Jardine, John F. (2015). Local homotopy theory. Springer Monographs in Mathematics. New York: Springer-Verlag. section 9.2. doi:10.1007/978-1-4939-2300-7. MR 3309296.
- Orbifolds and Stringy Topology. Definition 1.7: CAMBRIDGE TRACTS IN MATHEMATICS. p. 4.
- Taken from http://www.math.harvard.edu/~lurie/252xnotes/Lecture11.pdf
- Deligne, Pierre; Mumford, David (1969), "The irreducibility of the space of curves of given genus", Publications Mathématiques de l'IHÉS, 36 (36): 75–109, CiteSeerX 10.1.1.589.288, doi:10.1007/BF02684599, MR 0262240
- Totaro, Burt (2004). "The resolution property for schemes and stacks". Journal für die reine und angewandte Mathematik. 577: 1–22. arXiv:math/0207210. doi:10.1515/crll.2004.2004.577.1. MR 2108211.
Some other references are
- Behrend, Kai (1991). The Lefschetz trace formula for the moduli stack of principal bundles (PDF) (Thesis). University of California, Berkeley.
- Edidin, Dan. "Notes on the construction of the moduli space of curves" (PDF).