“Quotient stack”的意思、由来-开放百科全书

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Quotient stack

释义

Definition
Examples
See also
References

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 orbifold is an example of a quotient stack.{{fact|date=November 2017}}

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:

an object over T is a principal G-bundle P →T together with equivariant map P →X
an arrow from P →T to P' →T' is a bundle map (i.e., forms a cartesian diagram) that is compatible with the equivariant maps P →X and P' →X.

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.){{fact|date=April 2018}}

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.

{{harv|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; cf. 9.2. of Jardine's "local homotopy theory".^[2]

Examples

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:^[4] Let L be the Lazard ring; i.e., . Then the quotient stack by

,

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

References

1. ^The T-point is obtained by completing the diagram

.
2. ^{{Cite web | url=http://www.math.uwo.ca/~jardine/papers/preprints/book.pdf | title=The Department of Mathematics - Western University}}
3. ^This definition is given at http://www.math.harvard.edu/~gaitsgde/grad_2009/SeminarNotes/Sept17(Bun(G)).pdf
4. ^Taken from http://www.math.harvard.edu/~lurie/252xnotes/Lecture11.pdf

{{Citation | last1=Deligne | first1=Pierre | author1-link=Pierre Deligne | last2=Mumford | first2=David | author2-link=David Mumford | title=The irreducibility of the space of curves of given genus | url=http://www.numdam.org/item?id=PMIHES_1969__36__75_0 |mr=0262240 | year=1969 | journal=Publications Mathématiques de l'IHÉS | issue=36 | pages=75–109 | doi=10.1007/BF02684599 | volume=36| citeseerx=10.1.1.589.288 }}
Burt Totaro, The resolution property for schemes and stacks, J. Reine Angew. Math. 577 (2004), 1–22. 25

Some other references are

http://www.math.missouri.edu/~edidin/Papers/mfile.pdf
http://www.math.ubc.ca/~behrend/thesis.pdf

1 : Algebraic geometry

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Definition

Examples

See also

References