Classifying 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 /G/math> be the category over the category of ''S''-schemes:
*an object over ''T'' is a principal ''G''-bundle $P\to T$ together with equivariant map $P\to X$;
*an arrow 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
: /G\to X/G,
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 $X/G$ exists.)

In general, /G/math> 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.

Examples

An effective quotient orbifold, e.g., /G/math> where the $G$ action has only finite stabilizers on the smooth space $M$, is an example of a quotient stack.

If $X = S$ with trivial action of $G$ (often $S$ is a point), then /G/math> 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 $B\mathbb_m$ of line bundles /\mathbb_m/math> over $\text$, or /\mathbb_m/math> over $\text/S$ for the trivial $\mathbb_m$-action on $S$. For any scheme (or $S$-scheme) $X$, the $X$-points of the moduli stack are the groupoid of principal $\mathbb_m$-bundles $P \to X$.

Moduli of line bundles with n-sections

There is another closely related moduli stack given by mathbb^n/\mathbb_m/math> 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^n/\mathbb_mX) = \left\

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 $\mathbb_m$-equivariant map $\phi: P \to \mathbb^1$ 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 $\mathbb_m$-equivariant maps $P, _x \to \mathbb^1$ is isomorphic to $\mathbb_m$. This construction then globalizes by gluing affine charts together, giving a global section of the bundle. Since $\mathbb_m$-equivariant maps to $\mathbb^n$ is equivalently an $n$-tuple of $\mathbb_m$-equivariant maps to $\mathbb^1$, 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., $L = \pi_* \operatorname$. Then the quotient stack operatornameL/G/math> by
$G$,
:$G(R) = \$,
is called the moduli stack of formal group laws, denoted by $\mathcal_\text$.

See also

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

References

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Some other references are
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*{{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

Algebraic geometry