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'''<!--boldface per WP:R#PLA-->.
== 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 <math>[X/G]</math> be the category over the category of ''S''-schemes, where *an object over ''T'' is a principal ''G''-bundle <math>P\to T</math> together with equivariant map <math>P\to X</math>; *a morphism from <math>P\to T</math> to <math>P'\to T'</math> is a bundle map (i.e., forms a commutative diagram) that is compatible with the equivariant maps <math>P\to X</math> and <math>P'\to X</math>.
Suppose the quotient <math>X/G</math> exists as an algebraic space (for example, by the Keel–Mori theorem). The canonical map :<math>[X/G] \to X/G</math>, that sends a bundle ''P'' over ''T'' to a corresponding ''T''-point,<ref>The ''T''-point is obtained by completing the diagram <math>T \leftarrow P \to X \to X/G</math>.</ref> 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 <math>X/G</math> exists).{{fact|date=April 2018}}
In general, <math>[X/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. <!--if ''X'' is a smooth variety over an algebraically closed field ''k'', then <math>[X/G] \to X /\!/ G</math> is a bijection from the set of isomorphism classes to the set of closed points in the GIT quotient of ''X'' by ''G''. reference?-->
{{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.<ref>{{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}}</ref> See also: simplicial diagram.
== Examples == An effective quotient orbifold, e.g., <math>[M/G]</math> where the <math>G</math> action has only finite stabilizers on the smooth space <math>M</math>, is an example of a quotient stack.<ref>{{cite book|title=Orbifolds and Stringy Topology|publisher=Cambridge Tracts in Mathematics |chapter=Definition 1.7 |pages=4}}</ref>
If <math>X = S</math> with trivial action of <math>G</math> (often <math>S</math> is a point), then <math>[S/G]</math> is called the '''classifying stack''' of <math>G</math> (in analogy with the classifying space of <math>G</math>) and is usually denoted by <math>BG</math>. 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 <math>B\mathbb{G}_m</math> of line bundles <math>[*/\mathbb{G}_m]</math> over <math>\text{Sch}</math>, or <math>[S/\mathbb{G}_m]</math> over <math>\text{Sch}/S</math> for the trivial <math>\mathbb{G}_m</math>-action on <math>S</math>. For any scheme (or <math>S</math>-scheme) <math>X</math>, the <math>X</math>-points of the moduli stack are the groupoid of principal <math>\mathbb{G}_m</math>-bundles <math>P \to X</math>.
=== Moduli of line bundles with n-sections === There is another closely related moduli stack given by <math>[\mathbb{A}^n/\mathbb{G}_m]</math> which is the moduli stack of line bundles with <math>n</math>-sections. This follows directly from the definition of quotient stacks evaluated on points. For a scheme <math>X</math>, the <math>X</math>-points are the groupoid whose objects are given by the set<blockquote><math>[\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\}</math></blockquote>The morphism in the top row corresponds to the <math>n</math>-sections of the associated line bundle over <math>X</math>. This can be found by noting giving a <math>\mathbb{G}_m</math>-equivariant map <math>\phi: P \to \mathbb{A}^1</math> and restricting it to the fiber <math>P|_x</math> gives the same data as a section <math>\sigma</math> of the bundle. This can be checked by looking at a chart and sending a point <math>x \in X</math> to the map <math>\phi_x</math>, noting the set of <math>\mathbb{G}_m</math>-equivariant maps <math>P|_x \to \mathbb{A}^1</math> is isomorphic to <math>\mathbb{G}_m</math>. This construction then globalizes by gluing affine charts together, giving a global section of the bundle. Since <math>\mathbb{G}_m</math>-equivariant maps to <math>\mathbb{A}^n</math> is equivalently an <math>n</math>-tuple of <math>\mathbb{G}_m</math>-equivariant maps to <math>\mathbb{A}^1</math>, the result holds.
=== Moduli of formal group laws ===<!-- Example: For a ''S''-scheme ''X'', let <math>\operatorname{Bun}_G (X) = \operatorname{Map}(X, BG)</math>, where <math>\operatorname{Map}</math> is a mapping stack.<ref>This definition is given at http://www.math.harvard.edu/~gaitsgde/grad_2009/SeminarNotes/Sept17(Bun(G)).pdf</ref> It is called the moduli stack of principal bundles on ''X''.-->
Example:<ref>Taken from http://www.math.harvard.edu/~lurie/252xnotes/Lecture11.pdf</ref> Let ''L'' be the Lazard ring; i.e., <math>L = \pi_* \operatorname{MU}</math>. Then the quotient stack <math>[\operatorname{Spec}L/G]</math> by <math>G</math>, :<math>G(R) = \{g \in R[\![t]\!] | g(t) = b_0 t + b_1t^2+ \cdots, b_0 \in R^\times \}</math>, is called the moduli stack of formal group laws, denoted by <math>\mathcal{M}_\text{FG}</math>.
== 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 == {{reflist}} *{{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|access-date=2013-09-19|archive-date=2013-05-16|archive-url=https://web.archive.org/web/20130516201721/http://www.math.missouri.edu/~edidin/Papers/mfile.pdf|url-status=dead}}
Category:Stacks (mathematics)