Quotient stack explained

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

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

[X/G]

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

P\toT

together with equivariant map

P\toX

;

a morphism from

P\toT

P'\toT'

is a bundle map (i.e., forms a commutative diagram) that is compatible with the equivariant maps

P\toX

and

P'\toX

X/G

exists as an algebraic space (for example, by the Keel–Mori theorem). The canonical map

[X/G]\toX/G

,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

X/G

exists.)

In general,

[X/G]

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.

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.

Remark: It is possible to approach the construction from the point of view of simplicial sheaves.^[2] See also: simplicial diagram.

Examples

An effective quotient orbifold, e.g.,

[M/G]

where the

action has only finite stabilizers on the smooth space

, is an example of a quotient stack.^[3]

X=S

with trivial action of

(often

is a point), then

[S/G]

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

One of the basic examples of quotient stacks comes from the moduli stack

BG_m

of line bundles

[*/G_m]

over

Sch

, or

[S/G_m]

over

Sch/S

for the trivial

G_m

-action on

. For any scheme (or

-scheme)

, the

-points of the moduli stack are the groupoid of principal

G_m

-bundles

P\toX

Moduli of line bundles with n-sections

There is another closely related moduli stack given by

	n/G
[A
	m]

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

n/G
[A
m](X)

=\left\{ \begin{matrix} P&\to&Aⁿ\\ \downarrow&&\\ X \end{matrix}:\begin{align} &P\toAⁿisG_{mequivariantand}\\ &P\toXisaprincipalG_{m-bundle
\end{align}
\right\}}

The morphism in the top row corresponds to the

-sections of the associated line bundle over

. This can be found by noting giving a

G_m

-equivariant map

\phi:P\toA¹

and restricting it to the fiber

P|_x

gives the same data as a section

\sigma

of the bundle. This can be checked by looking at a chart and sending a point

x\inX

to the map

\phi_x

, noting the set of

G_m

-equivariant maps

P|_x\toA¹

is isomorphic to

G_m

. This construction then globalizes by gluing affine charts together, giving a global section of the bundle. Since

G_m

-equivariant maps to

Aⁿ

is equivalently an

-tuple of

G_m

-equivariant maps to

A¹

, the result holds.

Moduli of formal group laws

Example:^[4] Let L be the Lazard ring; i.e.,

L=\pi_*\operatorname{MU}

. Then the quotient stack

[\operatorname{Spec}L/G]

G(R)=\{g\inR[[t]]|g(t)=b₀t+

	2+
b
	1t

… ,b₀\inR^x\}

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

l{M}_FG

References

- Burt. Totaro. Burt Totaro. The resolution property for schemes and stacks. Journal für die reine und angewandte Mathematik. 577 . 2004. 1–22. 2108211. 10.1515/crll.2004.2004.577.1. math/0207210.

Some other references are

Behrend. Kai. Kai Behrend. The Lefschetz trace formula for the moduli stack of principal bundles . University of California, Berkeley. 1991.
Web site: Dan. Edidin. Notes on the construction of the moduli space of curves. 2013-09-19. 2013-05-16. https://web.archive.org/web/20130516201721/http://www.math.missouri.edu/~edidin/Papers/mfile.pdf. dead.

Notes and References

The T-point is obtained by completing the diagram
T\leftarrowP\toX\toX/G

.
Book: Jardine, John F.. Rick Jardine
. Rick Jardine. Local homotopy theory. Springer Monographs in Mathematics. Springer-Verlag. New York. 2015. 3309296. 10.1007/978-1-4939-2300-7. section 9.2.
Book: Orbifolds and Stringy Topology. Cambridge Tracts in Mathematics . Definition 1.7 . 4.
Taken from http://www.math.harvard.edu/~lurie/252xnotes/Lecture11.pdf