quotient stack

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

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

Suppose the quotient $X/G$ exists as an algebraic space (for example, by the Keel–Mori theorem). The canonical map

: $[X/G] \to X/G$ ,

that sends a bundle P over T to a corresponding T-point,The T-point is obtained by completing the diagram $T \leftarrow P \to X \to X/G$ . 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.){{fact|date=April 2018}}

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.

{{harvs|txt|last=Totaro|first=Burt|authorlink=Burt Totaro|year=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.

Remark: It is possible to approach the construction from the point of view of simplicial sheaves.{{Cite book| first=John F.|last= Jardine|authorlink=Rick Jardine| title=Local homotopy theory|series= Springer Monographs in Mathematics| publisher=Springer-Verlag|location= New York|year=2015|mr=3309296|doi=10.1007/978-1-4939-2300-7|at=section 9.2}} See also: simplicial diagram.

Examples

An effective quotient orbifold, e.g., $[M/G]$ where the $G$ action has only finite stabilizers on the smooth space $M$ , is an example of a quotient stack.{{cite book|title=Orbifolds and Stringy Topology|publisher=Cambridge Tracts in Mathematics |chapter=Definition 1.7 |pages=4}}

If $X = S$ with trivial action of $G$ (often $S$ is a point), then $[S/G]$ 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.

= Moduli of line bundles =

One of the basic examples of quotient stacks comes from the moduli stack $B\mathbb{G}_m$ of line bundles $[*/\mathbb{G}_m]$ over $\text{Sch}$ , or $[S/\mathbb{G}_m]$ over $\text{Sch}/S$ for the trivial $\mathbb{G}_m$ -action on $S$ . For any scheme (or $S$ -scheme) $X$ , the $X$ -points of the moduli stack are the groupoid of principal $\mathbb{G}_m$ -bundles $P \to X$ .

= Moduli of line bundles with n-sections =

There is another closely related moduli stack given by $[\mathbb{A}^n/\mathbb{G}_m]$ which is the moduli stack of line bundles with $n$ -sections. This follows directly from the definition of quotient stacks evaluated on points. For a scheme $X$ , the $X$ -points are the groupoid whose objects are given by the set

$[\mathbb{A}^n/\mathbb{G}_m](X) = \left\{
\begin{matrix}
P & \to & \mathbb{A}^n \\
\downarrow & & \\
X
\end{matrix} : \begin{align}
&P \to \mathbb{A}^n \text{ is }\mathbb{G}_m\text{ equivariant and} \\
&P \to X \text{ is a principal } \mathbb{G}_m\text{-bundle}
\end{align}
\right\}$

The morphism in the top row corresponds to the

n

-sections of the associated line bundle over

X

. This can be found by noting giving a

\mathbb{G}_m

-equivariant map

\phi: P \to \mathbb{A}^1

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 \in X

to the map

\phi_x

, noting the set of

\mathbb{G}_m

-equivariant maps

P|_x \to \mathbb{A}^1

is isomorphic to

\mathbb{G}_m

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

\mathbb{G}_m

-equivariant maps to

\mathbb{A}^n

is equivalently an

n

-tuple of

\mathbb{G}_m

-equivariant maps to

\mathbb{A}^1

, the result holds.

=== Moduli of formal group laws ===

Example:Taken from http://www.math.harvard.edu/~lurie/252xnotes/Lecture11.pdf Let L be the Lazard ring; i.e., $L = \pi_* \operatorname{MU}$ . Then the quotient stack $[\operatorname{Spec}L/G]$ by

$G$ ,

: $G(R) = \{g \in R[\![t]\!] | g(t) = b_0 t + b_1t^2+ \cdots, b_0 \in R^\times \}$ ,

is called the moduli stack of formal group laws, denoted by $\mathcal{M}_\text{FG}$ .

References

{{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 }}
{{cite journal|first=Burt|last= Totaro|authorlink=Burt Totaro|title= The resolution property for schemes and stacks|journal=Journal für die reine und angewandte Mathematik|volume= 577 |year=2004|pages= 1–22|mr=2108211|doi=10.1515/crll.2004.2004.577.1|arxiv=math/0207210}}

Some other references are

{{cite thesis|last=Behrend|first=Kai|authorlink=Kai Behrend|title=The Lefschetz trace formula for the moduli stack of principal bundles |publisher=University of California, Berkeley| year=1991|url=http://www.math.ubc.ca/~behrend/thesis.pdf}}
{{cite web|first=Dan|last=Edidin|title=Notes on the construction of the moduli space of curves|url=http://www.math.missouri.edu/~edidin/Papers/mfile.pdf}}

Category:Algebraic geometry