Quotient stack

 ( Algebraic Geometry ) 

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.

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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 fibered category the category of S-schemes, where

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

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

$X/G\; \backslash to\; X/G$,

that sends a bundle P over T to a corresponding T-point,The T-point is obtained by completing the diagram $T\; \backslash leftarrow\; P\; \backslash to\; X\; \backslash to\; X/G$. 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.)

In general, $X/G$ is an Artin stack (also called algebraic stack). If the stabilizers of the 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 sheaf. See also: simplicial diagram.

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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.

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.

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

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Moduli of line bundles with n-sections There is another closely related moduli stack given by $\backslash mathbb\{A\}^n/\backslash mathbb\{G\}\_m$ 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

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 $\backslash mathbb\{G\}\_m$-equivariant map $\backslash phi:\; P\; \backslash to\; \backslash mathbb\{A\}^1$ and restricting it to the fiber $P|\_x$ gives the same data as a section $\backslash sigma$ of the bundle. This can be checked by looking at a chart and sending a point $x\; \backslash in\; X$ to the map $\backslash phi\_x$, noting the set of $\backslash mathbb\{G\}\_m$-equivariant maps $P|\_x\; \backslash to\; \backslash mathbb\{A\}^1$ is isomorphic to $\backslash mathbb\{G\}\_m$. This construction then globalizes by gluing affine charts together, giving a global section of the bundle. Since $\backslash mathbb\{G\}\_m$-equivariant maps to $\backslash mathbb\{A\}^n$ is equivalently an $n$-tuple of $\backslash mathbb\{G\}\_m$-equivariant maps to $\backslash mathbb\{A\}^1$, the result holds.

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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., $L\; =\; \backslash pi\_*\; \backslash operatorname\{MU\}$. Then the quotient stack $\backslash operatorname\{Spec\}L/G$ by $G$,

$G(R)\; =\; \backslash \{g\; \backslash in\; R\backslash ![t\backslash !]\; |\; g(t)\; =\; b\_0\; t\; +\; b\_1t^2+\; \backslash cdots,\; b\_0\; \backslash in\; R^\backslash times\; \backslash \}$,

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

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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

Some other references are

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