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Quotient stack

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<div id="mw-content-text" class="mw-body-content mw-content-ltr" lang="en" dir="ltr"><div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr">In <a href="Algebraic_geometry" title="Algebraic geometry">algebraic geometry</a>, a quotient stack is a <a href="Stack_(mathematics)" title="Stack (mathematics)">stack</a> that parametrizes equivariant objects. Geometrically, it generalizes a quotient of a <a href="Scheme_(mathematics)" title="Scheme (mathematics)">scheme</a> or a <a href="Algebraic_variety" title="Algebraic variety">variety</a> by a <a href="Group_(mathematics)" title="Group (mathematics)">group</a>: 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 <a href="Deligne%E2%80%93Mumford_stack" title="Deligne–Mumford stack">Deligne–Mumford stack</a>.) A quotient stack is also used to construct other stacks like classifying stacks.

<div class="mw-heading mw-heading2"><h2 id="Definition">Definition</h2></div>

A quotient stack is defined as follows. Let G be an affine smooth <a href="Group_scheme" title="Group scheme">group scheme</a> over a scheme S and X an S-scheme on which G <a href="Group-scheme_action" title="Group-scheme action">acts</a>. Let the quotient stack <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [X/G]}">

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<annotation encoding="application/x-tex">{\displaystyle [X/G]}</annotation>

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</math><img src="./6c10708eac95ed8fcce0dd132f18c6d844536ebf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.263ex; height:2.843ex;" alt="{\displaystyle [X/G]}" loading="lazy"> be the <a href="Fibered_category" class="mw-redirect" title="Fibered category">category over</a> the category of S-schemes, where

<ul><li>an object over T is a <a href="Torsor_(algebraic_geometry)" title="Torsor (algebraic geometry)">principal G-bundle</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P\to T}">

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<annotation encoding="application/x-tex">{\displaystyle P\to T}</annotation>

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</math><img src="./167b0dc298a4fb91fe08fbaacb03655c36bf4657.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.996ex; height:2.176ex;" alt="{\displaystyle P\to T}" loading="lazy"> together with equivariant map <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P\to X}">

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<annotation encoding="application/x-tex">{\displaystyle P\to X}</annotation>

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</math><img src="./706af59e3c62c6fa253d5f87c0810d24703b99ff.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.34ex; height:2.176ex;" alt="{\displaystyle P\to X}" loading="lazy">;</li>

<li>a morphism from <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P\to T}">

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<annotation encoding="application/x-tex">{\displaystyle P\to T}</annotation>

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</math><img src="./167b0dc298a4fb91fe08fbaacb03655c36bf4657.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.996ex; height:2.176ex;" alt="{\displaystyle P\to T}" loading="lazy"> to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P'\to T'}">

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<annotation encoding="application/x-tex">{\displaystyle P'\to T'}</annotation>

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</math><img src="./31adccc85a1a150cf69884829f589f7077d7124b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.525ex; height:2.509ex;" alt="{\displaystyle P'\to T'}" loading="lazy"> is a <a href="Bundle_map" title="Bundle map">bundle map</a> (i.e., forms a commutative diagram) that is compatible with the equivariant maps <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P\to X}">

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<annotation encoding="application/x-tex">{\displaystyle P\to X}</annotation>

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</math><img src="./706af59e3c62c6fa253d5f87c0810d24703b99ff.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.34ex; height:2.176ex;" alt="{\displaystyle P\to X}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P'\to X}">

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<annotation encoding="application/x-tex">{\displaystyle P'\to X}</annotation>

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</math><img src="./93dd2cc03a1bfc331857bde605e296b76f47bf28.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.1ex; height:2.509ex;" alt="{\displaystyle P'\to X}" loading="lazy">.</li></ul>

Suppose the <a href="Categorical_quotient" title="Categorical quotient">quotient</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X/G}">

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<annotation encoding="application/x-tex">{\displaystyle X/G}</annotation>

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</math><img src="./7cb103556c45e5f866285543c6b149b0f42f9121.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.969ex; height:2.843ex;" alt="{\displaystyle X/G}" loading="lazy"> exists as an <a href="Algebraic_space" title="Algebraic space">algebraic space</a> (for example, by the <a href="Keel%E2%80%93Mori_theorem" title="Keel–Mori theorem">Keel–Mori theorem</a>). The canonical map

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<annotation encoding="application/x-tex">{\displaystyle [X/G]\to X/G}</annotation>

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</math><img src="./361d409e5d9f33aac9b9163f2cb4bbd87c68fc03.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.846ex; height:2.843ex;" alt="{\displaystyle [X/G]\to X/G}" loading="lazy">,</dd></dl>

that sends a bundle P over T to a corresponding T-point,<a href="#cite_note-1">[1]</a> 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 xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X/G}">

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<annotation encoding="application/x-tex">{\displaystyle X/G}</annotation>

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</math><img src="./7cb103556c45e5f866285543c6b149b0f42f9121.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.969ex; height:2.843ex;" alt="{\displaystyle X/G}" loading="lazy"> exists.)

In general, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [X/G]}">

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<annotation encoding="application/x-tex">{\displaystyle [X/G]}</annotation>

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</math><img src="./6c10708eac95ed8fcce0dd132f18c6d844536ebf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.263ex; height:2.843ex;" alt="{\displaystyle [X/G]}" loading="lazy"> is an <a href="Artin_stack" class="mw-redirect" title="Artin stack">Artin stack</a> (also called algebraic stack). If the stabilizers of the <a href="Geometric_point" class="mw-redirect" title="Geometric point">geometric points</a> are finite and reduced, then it is a <a href="Deligne%E2%80%93Mumford_stack" title="Deligne–Mumford stack">Deligne–Mumford stack</a>.

<a href="Burt_Totaro" title="Burt Totaro">Burt Totaro</a> (<a href="#CITEREFTotaro2004">2004</a>) 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 <a href="Resolution_property" class="mw-redirect" title="Resolution property">resolution property</a>; i.e., every coherent sheaf is a quotient of a vector bundle. Earlier, <a href="Robert_Wayne_Thomason" title="Robert Wayne Thomason">Robert Wayne Thomason</a> proved that a quotient stack has the resolution property.

Remark: It is possible to approach the construction from the point of view of <a href="Simplicial_sheaf" class="mw-redirect" title="Simplicial sheaf">simplicial sheaves</a>.<a href="#cite_note-2">[2]</a> See also: <a href="Simplicial_diagram" title="Simplicial diagram">simplicial diagram</a>.

<div class="mw-heading mw-heading2"><h2 id="Examples">Examples</h2></div>

An effective quotient <a href="Orbifold" title="Orbifold">orbifold</a>, e.g., <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [M/G]}">

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<annotation encoding="application/x-tex">{\displaystyle [M/G]}</annotation>

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</math><img src="./375110835b60a898de6e63062ea75344f6de53ef.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.725ex; height:2.843ex;" alt="{\displaystyle [M/G]}" loading="lazy"> where the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}">

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<annotation encoding="application/x-tex">{\displaystyle G}</annotation>

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</math><img src="./f5f3c8921a3b352de45446a6789b104458c9f90b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}" loading="lazy"> action has only finite stabilizers on the smooth space <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}">

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<annotation encoding="application/x-tex">{\displaystyle M}</annotation>

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</math><img src="./f82cade9898ced02fdd08712e5f0c0151758a0dd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}" loading="lazy">, is an example of a quotient stack.<a href="#cite_note-3">[3]</a>

If <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X=S}">

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<annotation encoding="application/x-tex">{\displaystyle X=S}</annotation>

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</math><img src="./172d09e1e067fe81720250a49f71e75a878cfee4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.578ex; height:2.176ex;" alt="{\displaystyle X=S}" loading="lazy"> with trivial action of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}">

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<annotation encoding="application/x-tex">{\displaystyle G}</annotation>

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</math><img src="./f5f3c8921a3b352de45446a6789b104458c9f90b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}" loading="lazy"> (often <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S}">

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<annotation encoding="application/x-tex">{\displaystyle S}</annotation>

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</math><img src="./4611d85173cd3b508e67077d4a1252c9c05abca2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.499ex; height:2.176ex;" alt="{\displaystyle S}" loading="lazy"> is a point), then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [S/G]}">

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<annotation encoding="application/x-tex">{\displaystyle [S/G]}</annotation>

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</math><img src="./58495295cc8b13a63005a9e07ebd15ab143fb107.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.782ex; height:2.843ex;" alt="{\displaystyle [S/G]}" loading="lazy"> is called the classifying stack of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}">

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<annotation encoding="application/x-tex">{\displaystyle G}</annotation>

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</math><img src="./f5f3c8921a3b352de45446a6789b104458c9f90b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}" loading="lazy"> (in analogy with the <a href="Classifying_space" title="Classifying space">classifying space</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}">

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<annotation encoding="application/x-tex">{\displaystyle G}</annotation>

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</math><img src="./f5f3c8921a3b352de45446a6789b104458c9f90b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}" loading="lazy">) and is usually denoted by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle BG}">

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<annotation encoding="application/x-tex">{\displaystyle BG}</annotation>

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</math><img src="./773ca20b2080cb3766062a5451a01d2220e9b067.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.591ex; height:2.176ex;" alt="{\displaystyle BG}" loading="lazy">. <a href="Borel's_theorem" title="Borel's theorem">Borel's theorem</a> describes the <a href="Cohomology_ring" title="Cohomology ring">cohomology ring</a> of the classifying stack.

<div class="mw-heading mw-heading3"><h3 id="Moduli_of_line_bundles">Moduli of line bundles</h3></div>

One of the basic examples of quotient stacks comes from the moduli stack <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B\mathbb {G} _{m}}">

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<annotation encoding="application/x-tex">{\displaystyle B\mathbb {G} _{m}}</annotation>

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</math><img src="./85ebc4811253bfca5349df9577c2d3b0b23d9b58.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.247ex; height:2.509ex;" alt="{\displaystyle B\mathbb {G} _{m}}" loading="lazy"> of line bundles <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [*/\mathbb {G} _{m}]}">

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<annotation encoding="application/x-tex">{\displaystyle [*/\mathbb {G} _{m}]}</annotation>

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</math><img src="./d3424e9b5b264bd4bb5ff8db067a682e06323baa.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.102ex; height:2.843ex;" alt="{\displaystyle [*/\mathbb {G} _{m}]}" loading="lazy"> over <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\text{Sch}}}">

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<annotation encoding="application/x-tex">{\displaystyle {\text{Sch}}}</annotation>

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</math><img src="./361ab28a0bd8cfe0331fe93a226feb21d4fd7236.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.617ex; height:2.176ex;" alt="{\displaystyle {\text{Sch}}}" loading="lazy">, or <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [S/\mathbb {G} _{m}]}">

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<annotation encoding="application/x-tex">{\displaystyle [S/\mathbb {G} _{m}]}</annotation>

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</math><img src="./dbf618746df81249cee01a144c41a4aaab165e0b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.439ex; height:2.843ex;" alt="{\displaystyle [S/\mathbb {G} _{m}]}" loading="lazy"> over <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\text{Sch}}/S}">

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<annotation encoding="application/x-tex">{\displaystyle {\text{Sch}}/S}</annotation>

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</math><img src="./6f8f7f8883cd14e4dd3081f72183668412627c6b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.279ex; height:2.843ex;" alt="{\displaystyle {\text{Sch}}/S}" loading="lazy"> for the trivial <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {G} _{m}}">

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<annotation encoding="application/x-tex">{\displaystyle \mathbb {G} _{m}}</annotation>

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</math><img src="./338f0e58c3bbaf1b4f67cd73a3fac1f29c40e39c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.483ex; height:2.509ex;" alt="{\displaystyle \mathbb {G} _{m}}" loading="lazy">-action on <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S}">

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<annotation encoding="application/x-tex">{\displaystyle S}</annotation>

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</math><img src="./4611d85173cd3b508e67077d4a1252c9c05abca2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.499ex; height:2.176ex;" alt="{\displaystyle S}" loading="lazy">. For any scheme (or <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S}">

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<annotation encoding="application/x-tex">{\displaystyle S}</annotation>

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</math><img src="./4611d85173cd3b508e67077d4a1252c9c05abca2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.499ex; height:2.176ex;" alt="{\displaystyle S}" loading="lazy">-scheme) <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

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<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

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</math><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy">, the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

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<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

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</math><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy">-points of the moduli stack are the groupoid of principal <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {G} _{m}}">

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<annotation encoding="application/x-tex">{\displaystyle \mathbb {G} _{m}}</annotation>

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</math><img src="./338f0e58c3bbaf1b4f67cd73a3fac1f29c40e39c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.483ex; height:2.509ex;" alt="{\displaystyle \mathbb {G} _{m}}" loading="lazy">-bundles <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P\to X}">

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<annotation encoding="application/x-tex">{\displaystyle P\to X}</annotation>

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</math><img src="./706af59e3c62c6fa253d5f87c0810d24703b99ff.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.34ex; height:2.176ex;" alt="{\displaystyle P\to X}" loading="lazy">.

<div class="mw-heading mw-heading3"><h3 id="Moduli_of_line_bundles_with_n-sections">Moduli of line bundles with n-sections</h3></div>

There is another closely related moduli stack given by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [\mathbb {A} ^{n}/\mathbb {G} _{m}]}">

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<annotation encoding="application/x-tex">{\displaystyle [\mathbb {A} ^{n}/\mathbb {G} _{m}]}</annotation>

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</math><img src="./965da9456d85124750027716df5c549d6ca25715.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.836ex; height:2.843ex;" alt="{\displaystyle [\mathbb {A} ^{n}/\mathbb {G} _{m}]}" loading="lazy"> which is the moduli stack of line bundles with <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}">

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<annotation encoding="application/x-tex">{\displaystyle n}</annotation>

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</math><img src="./a601995d55609f2d9f5e233e36fbe9ea26011b3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" loading="lazy">-sections. This follows directly from the definition of quotient stacks evaluated on points. For a scheme <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

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<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

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</math><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy">, the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

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<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

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</math><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy">-points are the groupoid whose objects are given by the set<blockquote><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [\mathbb {A} ^{n}/\mathbb {G} _{m}](X)=\left\{{\begin{matrix}P&\to &\mathbb {A} ^{n}\\\downarrow &&\\X\end{matrix}}:{\begin{aligned}&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{aligned}}\right\}}">

<msup>

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<mrow>

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<mtext> equivariant and</mtext>

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</mtd>

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<mtd>

<mtext> is a principal </mtext>

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<mtext>-bundle</mtext>

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<annotation encoding="application/x-tex">{\displaystyle [\mathbb {A} ^{n}/\mathbb {G} _{m}](X)=\left\{{\begin{matrix}P&\to &\mathbb {A} ^{n}\\\downarrow &&\\X\end{matrix}}:{\begin{aligned}&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{aligned}}\right\}}</annotation>

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</math><img src="./e845f04daa50c893f43bf368a6eedada63e5c433.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.005ex; width:67.56ex; height:9.176ex;" alt="{\displaystyle [\mathbb {A} ^{n}/\mathbb {G} _{m}](X)=\left\{{\begin{matrix}P&\to &\mathbb {A} ^{n}\\\downarrow &&\\X\end{matrix}}:{\begin{aligned}&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{aligned}}\right\}}" loading="lazy"></blockquote>The morphism in the top row corresponds to the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle n}</annotation>

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</math><img src="./a601995d55609f2d9f5e233e36fbe9ea26011b3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" loading="lazy">-sections of the associated line bundle over <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

</semantics>

</math><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy">. This can be found by noting giving a <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {G} _{m}}">

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle \mathbb {G} _{m}}</annotation>

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</math><img src="./338f0e58c3bbaf1b4f67cd73a3fac1f29c40e39c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.483ex; height:2.509ex;" alt="{\displaystyle \mathbb {G} _{m}}" loading="lazy">-equivariant map <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \phi :P\to \mathbb {A} ^{1}}">

<msup>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \phi :P\to \mathbb {A} ^{1}}</annotation>

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</math><img src="./ae44433ff7dd7179990f998a83904e41f3afe702.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.414ex; height:3.009ex;" alt="{\displaystyle \phi :P\to \mathbb {A} ^{1}}" loading="lazy"> and restricting it to the fiber <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P|_{x}}">

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle P|_{x}}</annotation>

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</math><img src="./7f7f4442a95c0ff3461c2c4b015691b4ca2c761b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.565ex; height:3.009ex;" alt="{\displaystyle P|_{x}}" loading="lazy"> gives the same data as a section <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \sigma }</annotation>

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</math><img src="./59f59b7c3e6fdb1d0365a494b81fb9a696138c36.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle \sigma }" loading="lazy"> of the bundle. This can be checked by looking at a chart and sending a point <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\in X}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x\in X}</annotation>

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</math><img src="./3e580967f68f36743e894aa7944f032dda6ea01d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.15ex; height:2.176ex;" alt="{\displaystyle x\in X}" loading="lazy"> to the map <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \phi _{x}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \phi _{x}}</annotation>

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</math><img src="./c6c310bb476ad5fc326d5e7153c205c0f4e9d94f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.558ex; height:2.509ex;" alt="{\displaystyle \phi _{x}}" loading="lazy">, noting the set of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {G} _{m}}">

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {G} _{m}}</annotation>

</semantics>

</math><img src="./338f0e58c3bbaf1b4f67cd73a3fac1f29c40e39c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.483ex; height:2.509ex;" alt="{\displaystyle \mathbb {G} _{m}}" loading="lazy">-equivariant maps <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P|_{x}\to \mathbb {A} ^{1}}">

<msub>

</mrow>

</mrow>

</msub>

<msup>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle P|_{x}\to \mathbb {A} ^{1}}</annotation>

</semantics>

</math><img src="./8a4f8ed5dd8a9ce10971abf683981d4476d626bc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:9.911ex; height:3.343ex;" alt="{\displaystyle P|_{x}\to \mathbb {A} ^{1}}" loading="lazy"> is isomorphic to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {G} _{m}}">

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {G} _{m}}</annotation>

</semantics>

</math><img src="./338f0e58c3bbaf1b4f67cd73a3fac1f29c40e39c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.483ex; height:2.509ex;" alt="{\displaystyle \mathbb {G} _{m}}" loading="lazy">. This construction then globalizes by gluing affine charts together, giving a global section of the bundle. Since <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {G} _{m}}">

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {G} _{m}}</annotation>

</semantics>

</math><img src="./338f0e58c3bbaf1b4f67cd73a3fac1f29c40e39c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.483ex; height:2.509ex;" alt="{\displaystyle \mathbb {G} _{m}}" loading="lazy">-equivariant maps to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {A} ^{n}}">

<msup>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {A} ^{n}}</annotation>

</semantics>

</math><img src="./d64ccca0c6fbd18be5d4ec26e26efd89e9c0d76e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.897ex; height:2.343ex;" alt="{\displaystyle \mathbb {A} ^{n}}" loading="lazy"> is equivalently an <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle n}</annotation>

</semantics>

</math><img src="./a601995d55609f2d9f5e233e36fbe9ea26011b3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" loading="lazy">-tuple of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {G} _{m}}">

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {G} _{m}}</annotation>

</semantics>

</math><img src="./338f0e58c3bbaf1b4f67cd73a3fac1f29c40e39c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.483ex; height:2.509ex;" alt="{\displaystyle \mathbb {G} _{m}}" loading="lazy">-equivariant maps to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {A} ^{1}}">

<msup>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {A} ^{1}}</annotation>

</semantics>

</math><img src="./67530a5bd8c23c0be226ac63ddf5f6b2619e682d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.732ex; height:2.676ex;" alt="{\displaystyle \mathbb {A} ^{1}}" loading="lazy">, the result holds.

<div class="mw-heading mw-heading3"><h3 id="Moduli_of_formal_group_laws">Moduli of formal group laws</h3></div>

Example:<a href="#cite_note-4">[4]</a> Let L be the <a href="Lazard_ring" class="mw-redirect" title="Lazard ring">Lazard ring</a>; i.e., <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L=\pi _{*}\operatorname {MU} }">

<msub>

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<annotation encoding="application/x-tex">{\displaystyle L=\pi _{*}\operatorname {MU} }</annotation>

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</math><img src="./8c5120245eea89ae9c7b1530a83e727db356b4c3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.293ex; margin-bottom: -0.379ex; width:11.322ex; height:2.509ex;" alt="{\displaystyle L=\pi _{*}\operatorname {MU} }" loading="lazy">. Then the quotient stack <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [\operatorname {Spec} L/G]}">

</mrow>

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<annotation encoding="application/x-tex">{\displaystyle [\operatorname {Spec} L/G]}</annotation>

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</math><img src="./84edc305e4a9fb0c7a81330c4b19dd68ac6ce1d6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.903ex; height:2.843ex;" alt="{\displaystyle [\operatorname {Spec} L/G]}" loading="lazy"> by

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle G}</annotation>

</semantics>

</math><img src="./f5f3c8921a3b352de45446a6789b104458c9f90b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}" loading="lazy">,

</mrow>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msup>

</mrow>

</msup>

<msub>

</mrow>

</msub>

<msup>

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<annotation encoding="application/x-tex">{\displaystyle G(R)=\{g\in R[\![t]\!]|g(t)=b_{0}t+b_{1}t^{2}+\cdots ,b_{0}\in R^{\times }\}}</annotation>

</semantics>

</math><img src="./45cbd3f0df279acba0a61104b7a2cf47d2776cb0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:51.536ex; height:3.176ex;" alt="{\displaystyle G(R)=\{g\in R[\![t]\!]|g(t)=b_{0}t+b_{1}t^{2}+\cdots ,b_{0}\in R^{\times }\}}" loading="lazy">,</dd></dl>

is called the <a href="Moduli_stack_of_formal_group_laws" title="Moduli stack of formal group laws">moduli stack of formal group laws</a>, denoted by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {M}}_{\text{FG}}}">

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathcal {M}}_{\text{FG}}}</annotation>

</semantics>

</math><img src="./2740fce135af7d32ec0fcd00242d14b0066f02f9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.386ex; height:2.509ex;" alt="{\displaystyle {\mathcal {M}}_{\text{FG}}}" loading="lazy">.

<ul><li><a href="Homotopy_quotient" class="mw-redirect" title="Homotopy quotient">Homotopy quotient</a></li>

<li><a href="Moduli_stack_of_principal_bundles" title="Moduli stack of principal bundles">Moduli stack of principal bundles</a> (which, roughly, is an infinite product of classifying stacks.)</li>

<li><a href="Group-scheme_action" title="Group-scheme action">Group-scheme action</a></li>

<li><a href="Moduli_of_algebraic_curves" title="Moduli of algebraic curves">Moduli of algebraic curves</a></li></ul>

<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>

/* start https://en.wikipedia.org/ */

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<li id="cite_note-1"><a href="#cite_ref-1">^</a> The T-point is obtained by completing the diagram <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T\leftarrow P\to X\to X/G}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle T\leftarrow P\to X\to X/G}</annotation>

</semantics>

</math><img src="./f198aa3612e96bfaf6a18d3a1659d52ac9bd49d5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.173ex; height:2.843ex;" alt="{\displaystyle T\leftarrow P\to X\to X/G}" loading="lazy">.

</li>

/* start https://en.wikipedia.org/ */

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</style><cite id="CITEREFJardine2015" class="citation book cs1"><a href="Rick_Jardine" title="Rick Jardine">Jardine, John F.</a> (2015). Local homotopy theory. Springer Monographs in Mathematics. New York: Springer-Verlag. section 9.2. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2F978-1-4939-2300-7">10.1007/978-1-4939-2300-7</a>. <a href="MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=3309296">3309296</a>.</cite>

</li>

<li id="cite_note-3"><a href="#cite_ref-3">^</a> <cite class="citation book cs1">"Definition 1.7". Orbifolds and Stringy Topology. Cambridge Tracts in Mathematics. p. 4.</cite>

</li>

<li id="cite_note-4"><a href="#cite_ref-4">^</a> Taken from <a rel="nofollow" class="external free" href="http://www.math.harvard.edu/~lurie/252xnotes/Lecture11.pdf">http://www.math.harvard.edu/~lurie/252xnotes/Lecture11.pdf</a>

</li>

</ol></div></div>

<ul><li><cite id="CITEREFDeligneMumford1969" class="citation cs2"><a href="Pierre_Deligne" title="Pierre Deligne">Deligne, Pierre</a>; <a href="David_Mumford" title="David Mumford">Mumford, David</a> (1969), <a rel="nofollow" class="external text" href="http://www.numdam.org/item?id=PMIHES_1969__36__75_0">"The irreducibility of the space of curves of given genus"</a>, <a href="Publications_Math%C3%A9matiques_de_l'IH%C3%89S" title="Publications Mathématiques de l'IHÉS">Publications Mathématiques de l'IHÉS</a>, 36 (36): 75–109, <a href="CiteSeerX_(identifier)" class="mw-redirect" title="CiteSeerX (identifier)">CiteSeerX</a> <a rel="nofollow" class="external text" href="https://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.589.288">10.1.1.589.288</a>, <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2FBF02684599">10.1007/BF02684599</a>, <a href="MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=0262240">0262240</a></cite></li>

<li><cite id="CITEREFTotaro2004" class="citation journal cs1"><a href="Burt_Totaro" title="Burt Totaro">Totaro, Burt</a> (2004). "The resolution property for schemes and stacks". <a href="Crelle's_Journal" title="Crelle's Journal">Journal für die reine und angewandte Mathematik</a>. 577: 1–22. <a href="ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<a rel="nofollow" class="external text" href="https://arxiv.org/abs/math/0207210">math/0207210</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1515%2Fcrll.2004.2004.577.1">10.1515/crll.2004.2004.577.1</a>. <a href="MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=2108211">2108211</a>.</cite></li></ul>

Some other references are

<ul><li><cite id="CITEREFBehrend1991" class="citation thesis cs1"><a href="Kai_Behrend" title="Kai Behrend">Behrend, Kai</a> (1991). <a rel="nofollow" class="external text" href="http://www.math.ubc.ca/~behrend/thesis.pdf">The Lefschetz trace formula for the moduli stack of principal bundles</a> (PDF) (Thesis). University of California, Berkeley.</cite></li>

<li><cite id="CITEREFEdidin" class="citation web cs1">Edidin, Dan. <a rel="nofollow" class="external text" href="http://www.math.missouri.edu/~edidin/Papers/mfile.pdf">"Notes on the construction of the moduli space of curves"</a> (PDF).</cite></li></ul></div><div><div class="zim-footer">

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