Quotient stack

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 ${\displaystyle [X/G]}$ be the category over the category of S-schemes:

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

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

${\displaystyle [X/G]\to X/G}$,

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 ${\displaystyle X/G}$ exists.)

In general, ${\displaystyle [X/G]}$ 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.

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.

An effective quotient orbifold, e.g., ${\displaystyle [M/G]}$ where the ${\displaystyle G}$ action has only finite stabilizers on the smooth space ${\displaystyle M}$, is an example of a quotient stack.[2]

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

Example:[3] Let L be the Lazard ring; i.e., ${\displaystyle L=\pi _{*}\operatorname {MU} }$. Then the quotient stack ${\displaystyle [\operatorname {Spec} L/G]}$ by ${\displaystyle G}$,

${\displaystyle G(R)=\{g\in R[\![t]\!]|g(t)=b_{0}t+b_{1}t^{2}+\cdots ,b_{0}\in R^{\times }\}}$,

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

• Homotopy quotient
• Moduli stack of principal bundles (which, roughly, is an infinite product of classifying stacks.)
• Group-scheme action

• 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).