ebab7d0da53c51438720d60c6e2e467c - /page/wikibook/entry.mu

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Stack (mathematics)

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In <a href="Mathematics" title="Mathematics">mathematics</a> a stack or 2-sheaf is, roughly speaking, a <a href="Sheaf_(mathematics)" title="Sheaf (mathematics)">sheaf</a> that takes values in <a href="Category_(mathematics)" title="Category (mathematics)">categories</a> rather than sets. Stacks are used to formalise some of the main constructions of <a href="Descent_theory" class="mw-redirect" title="Descent theory">descent theory</a>, and to construct fine moduli stacks when <a href="Fine_moduli_space" class="mw-redirect" title="Fine moduli space">fine moduli spaces</a> do not exist.

Descent theory is concerned with generalisations of situations where <a href="Isomorphism" title="Isomorphism">isomorphic</a>, compatible geometrical objects (such as <a href="Vector_bundle" title="Vector bundle">vector bundles</a> on <a href="Topological_space" title="Topological space">topological spaces</a>) can be "glued together" within a restriction of the topological basis. In a more general set-up the restrictions are replaced with <a href="Pullback_(category_theory)" title="Pullback (category theory)">pullbacks</a>; <a href="Fibred_category" title="Fibred category">fibred categories</a> then make a good framework to discuss the possibility of such gluing. The intuitive meaning of a stack is that it is a fibred category such that "all possible gluings work". The specification of gluings requires a definition of coverings with regard to which the gluings can be considered. It turns out that the general language for describing these coverings is that of a <a href="Grothendieck_topology" title="Grothendieck topology">Grothendieck topology</a>. Thus a stack is formally given as a fibred category over another base category, where the base has a Grothendieck topology and where the fibred category satisfies a few axioms that ensure existence and uniqueness of certain gluings with respect to the Grothendieck topology.

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

Stacks are the underlying structure of algebraic stacks (also called Artin stacks) and Deligne–Mumford stacks, which generalize <a href="Scheme_(mathematics)" title="Scheme (mathematics)">schemes</a> and <a href="Algebraic_space" title="Algebraic space">algebraic spaces</a> and which are particularly useful in studying <a href="Moduli_space" title="Moduli space">moduli spaces</a>. There are inclusions: <blockquote>schemes ⊆ algebraic spaces ⊆ Deligne–Mumford stacks ⊆ algebraic stacks (Artin stacks) ⊆ stacks.</blockquote><a href="#CITEREFEdidin2003">Edidin (2003)</a> and <a href="#CITEREFFantechi2001">Fantechi (2001)</a> give a brief introductory accounts of stacks, <a href="#CITEREFGómez2001">Gómez (2001)</a>, <a href="#CITEREFOlsson2007">Olsson (2007)</a> and <a href="#CITEREFVistoli2005">Vistoli (2005)</a> give more detailed introductions, and <a href="#CITEREFLaumonMoret-Bailly2000">Laumon & Moret-Bailly (2000)</a> describes the more advanced theory.

<div class="mw-heading mw-heading2"><h2 id="Motivation_and_history">Motivation and history</h2></div>

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La conclusion pratique à laquelle je suis arrivé dès maintenant, c'est que chaque fois que en vertu de mes critères, une variété de modules (ou plutôt, un schéma de modules) pour la classification des variations (globales, ou infinitésimales) de certaines structures (variétés complètes non singulières, fibrés vectoriels, etc.) ne peut exister, malgré de bonnes hypothèses de platitude, propreté, et non singularité éventuellement, la raison en est seulement l'existence d'automorphismes de la structure qui empêche la technique de descente de marcher.

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<div style="padding-bottom: 0; padding-top: 0.5em"><cite class="left-aligned" style="">Grothendieck's letter to Serre, 1959 Nov 5.</cite></div>

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The concept of stacks has its origin in the definition of effective descent data in <a href="#CITEREFGrothendieck1959">Grothendieck (1959)</a>.

In a 1959 letter to Serre, Grothendieck observed that a fundamental obstruction to constructing good moduli spaces is the existence of <a href="Automorphism" title="Automorphism">automorphisms</a>. A major motivation for stacks is that if a moduli space for some problem does not exist because of the existence of automorphisms, it may still be possible to construct a moduli stack.

<a href="#CITEREFMumford1965">Mumford (1965)</a> studied the <a href="Picard_group" title="Picard group">Picard group</a> of the <a href="Moduli_stack_of_elliptic_curves" title="Moduli stack of elliptic curves">moduli stack of elliptic curves</a>, before stacks had been defined. Stacks were first defined by Giraud (<a href="#CITEREFGiraud1966">1966</a>, <a href="#CITEREFGiraud1971">1971</a>), and the term "stack" was introduced by <a href="#CITEREFDeligneMumford1969">Deligne & Mumford (1969)</a> for the original French term "champ" meaning "field". In this paper they also introduced Deligne–Mumford stacks, which they called algebraic stacks, though the term "algebraic stack" now usually refers to the more general Artin stacks introduced by <a href="Michael_Artin" title="Michael Artin">Artin</a> (<a href="#CITEREFArtin1974">1974</a>).

When defining quotients of schemes by group actions, it is often impossible for the quotient to be a scheme and still satisfy desirable properties for a quotient. For example, if a few points have non-trivial stabilisers, then the <a href="Categorical_quotient" title="Categorical quotient">categorical quotient</a> will not exist among schemes, but it will exist as a stack.

In the same way, <a href="Moduli_space" title="Moduli space">moduli spaces</a> of curves, vector bundles, or other geometric objects are often best defined as stacks instead of schemes. Constructions of moduli spaces often proceed by first constructing a larger space parametrizing the objects in question, and then <a href="Quotient_stack" title="Quotient stack">quotienting by group action</a> to account for objects with automorphisms which have been overcounted.

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

<div class="mw-heading mw-heading3"><h3 id="Abstract_stacks">Abstract stacks</h3></div>

A category <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c}">

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

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</math><img src="./86a67b81c2de995bd608d5b2df50cd8cd7d92455.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.007ex; height:1.676ex;" alt="{\displaystyle c}" loading="lazy"> with a <a href="Functor" title="Functor">functor</a> to a category <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C}">

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

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</math><img src="./4fc55753007cd3c18576f7933f6f089196732029.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.766ex; height:2.176ex;" alt="{\displaystyle C}" loading="lazy"> is called a <a href="Fibered_category" class="mw-redirect" title="Fibered category">fibered category</a> over <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C}">

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

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</math><img src="./4fc55753007cd3c18576f7933f6f089196732029.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.766ex; height:2.176ex;" alt="{\displaystyle C}" loading="lazy"> if for any morphism <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F:X\to Y}">

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

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</math><img src="./b81de3fd0af85cab5904d269f0e3f02088cea565.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:11.045ex; height:2.176ex;" alt="{\displaystyle F:X\to Y}" loading="lazy"> in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C}">

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

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</math><img src="./4fc55753007cd3c18576f7933f6f089196732029.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.766ex; height:2.176ex;" alt="{\displaystyle C}" loading="lazy"> and any object <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y}">

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

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</math><img src="./b8a6208ec717213d4317e666f1ae872e00620a0d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.155ex; height:2.009ex;" alt="{\displaystyle y}" loading="lazy"> of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c}">

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

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</math><img src="./86a67b81c2de995bd608d5b2df50cd8cd7d92455.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.007ex; height:1.676ex;" alt="{\displaystyle c}" loading="lazy"> with image <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y}">

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

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</math><img src="./961d67d6b454b4df2301ac571808a3538b3a6d3f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:1.773ex; height:2.009ex;" alt="{\displaystyle Y}" loading="lazy"> (under the functor), there is a pullback <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f:x\to y}">

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<annotation encoding="application/x-tex">{\displaystyle f:x\to y}</annotation>

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</math><img src="./f1e5e985afba6236c1695a35213292b54583fac9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.315ex; height:2.509ex;" alt="{\displaystyle f:x\to y}" loading="lazy"> of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y}">

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

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</math><img src="./b8a6208ec717213d4317e666f1ae872e00620a0d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.155ex; height:2.009ex;" alt="{\displaystyle y}" loading="lazy"> by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F}">

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

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</math><img src="./545fd099af8541605f7ee55f08225526be88ce57.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.741ex; height:2.176ex;" alt="{\displaystyle F}" loading="lazy">. This means a morphism with image <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F}">

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

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</math><img src="./545fd099af8541605f7ee55f08225526be88ce57.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.741ex; height:2.176ex;" alt="{\displaystyle F}" loading="lazy"> such that any morphism <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g:z\to y}">

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<annotation encoding="application/x-tex">{\displaystyle g:z\to y}</annotation>

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</math><img src="./6f78f199237ade24bb5ad9c3a09b205e22b34825.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.911ex; height:2.176ex;" alt="{\displaystyle g:z\to y}" loading="lazy"> with image <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G=F\circ H}">

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

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</math><img src="./04644b0adb05a94719a3b2bb40206089f2bee46c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:10.924ex; height:2.176ex;" alt="{\displaystyle G=F\circ H}" loading="lazy"> can be factored as <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g=f\circ h}">

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<annotation encoding="application/x-tex">{\displaystyle g=f\circ h}</annotation>

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</math><img src="./4dcf9aea24f238cfc5c201a3bca568c75a223a61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.027ex; height:2.509ex;" alt="{\displaystyle g=f\circ h}" loading="lazy"> by a unique morphism <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h:z\to x}">

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<annotation encoding="application/x-tex">{\displaystyle h:z\to x}</annotation>

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</math><img src="./7f302c533b4a9122fd1772f54f158e4b1a0d71d6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:9.308ex; height:2.176ex;" alt="{\displaystyle h:z\to x}" loading="lazy"> in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c}">

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

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</math><img src="./86a67b81c2de995bd608d5b2df50cd8cd7d92455.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.007ex; height:1.676ex;" alt="{\displaystyle c}" loading="lazy"> such that the functor maps <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h}">

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

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</math><img src="./b26be3e694314bc90c3215047e4a2010c6ee184a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.339ex; height:2.176ex;" alt="{\displaystyle h}" loading="lazy"> to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H}">

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

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</math><img src="./75a9edddcca2f782014371f75dca39d7e13a9c1b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle H}" loading="lazy">. The element <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x=F^{*}y}">

<msup>

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<annotation encoding="application/x-tex">{\displaystyle x=F^{*}y}</annotation>

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</math><img src="./e0700ebc70b684e1c060bc77ec2de3d0abdd1e6c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.453ex; height:2.676ex;" alt="{\displaystyle x=F^{*}y}" loading="lazy"> is called the pullback of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y}">

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

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</math><img src="./b8a6208ec717213d4317e666f1ae872e00620a0d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.155ex; height:2.009ex;" alt="{\displaystyle y}" loading="lazy"> along <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F}">

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

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</math><img src="./545fd099af8541605f7ee55f08225526be88ce57.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.741ex; height:2.176ex;" alt="{\displaystyle F}" loading="lazy"> and is unique up to canonical isomorphism.

The category c is called a <a href="Prestack" title="Prestack">prestack</a> over a category C with a <a href="Grothendieck_topology" title="Grothendieck topology">Grothendieck topology</a> if it is fibered over C and for any object U of C and objects x, y of c with image U, the functor from the over category C/U to sets taking F:V→U to Hom(F*x,F*y) is a sheaf. This terminology is not consistent with the terminology for sheaves: prestacks are the analogues of separated presheaves rather than presheaves. Some authors require this as a property of stacks, rather than of prestacks.

The category c is called a stack over the category C with a Grothendieck topology if it is a prestack over C and every descent datum is effective. A descent datum consists roughly of a covering of an object V of C by a family Vi, elements xi in the fiber over Vi, and morphisms fji between the restrictions of xi and xj to Vij=Vi×VVj satisfying the compatibility condition fki = fkjfji. The descent datum is called effective if the elements xi are essentially the pullbacks of an element x with image V.

A stack is called a stack in groupoids or a (2,1)-sheaf if it is also fibered in groupoids, meaning that its fibers (the inverse images of objects of C) are groupoids. Some authors use the word "stack" to refer to the more restrictive notion of a stack in groupoids.

<div class="mw-heading mw-heading3"><h3 id="Algebraic_stacks">Algebraic stacks</h3></div>

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</style><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="Algebraic_stack" title="Algebraic stack">Algebraic stack</a></div>

An algebraic stack or Artin stack is a stack in groupoids X over the fppf site such that the diagonal map of X is representable and there exists a smooth surjection from (the stack associated to) a scheme to X.

A morphism Y<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \rightarrow }">

</mstyle>

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

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</math><img src="./53e574cc3aa5b4bf5f3f5906caf121a378eef08b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.324ex; height:1.843ex;" alt="{\displaystyle \rightarrow }" loading="lazy"> X of stacks is representable if, for every morphism S <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \rightarrow }">

</mstyle>

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

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</math><img src="./53e574cc3aa5b4bf5f3f5906caf121a378eef08b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.324ex; height:1.843ex;" alt="{\displaystyle \rightarrow }" loading="lazy"> X from (the stack associated to) a scheme to X, the <a href="Fiber_product" class="mw-redirect" title="Fiber product">fiber product</a> Y ×X S is isomorphic to (the stack associated to) an <a href="Algebraic_space" title="Algebraic space">algebraic space</a>. The fiber product of stacks is defined using the usual <a href="Universal_property" title="Universal property">universal property</a>, and changing the requirement that diagrams commute to the requirement that they 2-commute. See also <a href="Morphism_of_algebraic_stacks" title="Morphism of algebraic stacks">morphism of algebraic stacks</a> for further information.

The motivation behind the representability of the diagonal is the following: the diagonal morphism <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta :{\mathfrak {X}}\to {\mathfrak {X}}\times {\mathfrak {X}}}">

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<annotation encoding="application/x-tex">{\displaystyle \Delta :{\mathfrak {X}}\to {\mathfrak {X}}\times {\mathfrak {X}}}</annotation>

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</math><img src="./eee6ac1e1d348d5f83c9d540e16712c3f8d89274.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:15.341ex; height:2.176ex;" alt="{\displaystyle \Delta :{\mathfrak {X}}\to {\mathfrak {X}}\times {\mathfrak {X}}}" loading="lazy"> is representable if and only if for any pair of morphisms of algebraic spaces <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X,Y\to {\mathfrak {X}}}">

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

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</math><img src="./4354c41cfa7c3d1f6b980ef9e4b4cb03d50ad1f5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.072ex; height:2.509ex;" alt="{\displaystyle X,Y\to {\mathfrak {X}}}" loading="lazy">, their fiber product <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X\times _{\mathfrak {X}}Y}">

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

</semantics>

</math><img src="./8cb620c5b9c892a48acce4144fceb43e44688b74.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.008ex; height:2.509ex;" alt="{\displaystyle X\times _{\mathfrak {X}}Y}" loading="lazy"> is representable.

A <a href="Deligne%E2%80%93Mumford_stack" title="Deligne–Mumford stack">Deligne–Mumford stack</a> is an algebraic stack X such that there is an étale surjection from a scheme to X. Roughly speaking, Deligne–Mumford stacks can be thought of as algebraic stacks whose objects have no infinitesimal automorphisms.

<div class="mw-heading mw-heading4"><h4 id="Local_structure_of_algebraic_stacks">Local structure of algebraic stacks</h4></div>

Since the inception of algebraic stacks it was expected that they are locally quotient stacks of the form <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [{\text{Spec}}(A)/G]}">

</mrow>

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<annotation encoding="application/x-tex">{\displaystyle [{\text{Spec}}(A)/G]}</annotation>

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</math><img src="./cd13239580e1219dfbcaaf6b34b55690e02e3a12.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.485ex; height:2.843ex;" alt="{\displaystyle [{\text{Spec}}(A)/G]}" loading="lazy"> where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}">

</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"> is a <a href="Reductive_group" title="Reductive group">linearly reductive algebraic group</a>. This was recently proved to be the case:<a href="#cite_note-1">[1]</a> given a quasi-separated algebraic stack <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {X}}}">

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./073868db564a5e806f61f8b2d526ba975cb5c077.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.671ex; height:2.176ex;" alt="{\displaystyle {\mathfrak {X}}}" loading="lazy"> locally of finite type over an algebraically closed field <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}">

</mstyle>

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

</semantics>

</math><img src="./c3c9a2c7b599b37105512c5d570edc034056dd40.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}" loading="lazy"> whose stabilizers are affine, and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\in {\mathfrak {X}}(k)}">

</mrow>

</mstyle>

</mrow>

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

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</math><img src="./e677faf62c952d15d69e9912eeb5279ce5bf7f15.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.862ex; height:2.843ex;" alt="{\displaystyle x\in {\mathfrak {X}}(k)}" loading="lazy"> a smooth and closed point with linearly reductive stabilizer group <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G_{x}}">

<msub>

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

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</math><img src="./c2f78101676239fc50bea86722dedf1512180e2a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.999ex; height:2.509ex;" alt="{\displaystyle G_{x}}" loading="lazy">, there exists an <a href="%C3%89tale_morphism" title="Étale morphism">etale cover</a> of the <a href="GIT_quotient" title="GIT quotient">GIT quotient</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (U,u)\to (N_{x}//G_{x},0)}">

<msub>

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</msub>

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<msub>

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<annotation encoding="application/x-tex">{\displaystyle (U,u)\to (N_{x}//G_{x},0)}</annotation>

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</math><img src="./190bfd765a3d6f5da8e799dff182a210020ca2cd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.938ex; height:2.843ex;" alt="{\displaystyle (U,u)\to (N_{x}//G_{x},0)}" loading="lazy">, where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N_{x}=(J_{x}/J_{x}^{2})^{\vee }}">

<msub>

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</msub>

<msub>

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</msub>

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</msubsup>

<msup>

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<annotation encoding="application/x-tex">{\displaystyle N_{x}=(J_{x}/J_{x}^{2})^{\vee }}</annotation>

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</math><img src="./3b7ba9b9b9039faa687188041c9e003f3ea3dc5a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.48ex; height:3.009ex;" alt="{\displaystyle N_{x}=(J_{x}/J_{x}^{2})^{\vee }}" loading="lazy">, such that the diagram<blockquote><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{matrix}([W/G_{x}],w)&\to &([N_{x}/G_{x}],0)\\\downarrow &&\downarrow \\(U,u)&\to &(N_{x}//G_{x},0)\end{matrix}}}">

<mtr>

<mtd>

</mrow>

<msub>

</mrow>

</msub>

</mtd>

<mtd>

</mtd>

<mtd>

<msub>

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<msub>

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<mtr>

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<mtd>

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<mtr>

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<mtd>

</mtd>

<mtd>

<msub>

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</msub>

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</mrow>

<msub>

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</msub>

</mtd>

</mtr>

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</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\begin{matrix}([W/G_{x}],w)&\to &([N_{x}/G_{x}],0)\\\downarrow &&\downarrow \\(U,u)&\to &(N_{x}//G_{x},0)\end{matrix}}}</annotation>

</semantics>

</math><img src="./e0bb8f45224933caaddef21330dfcf18e768ff2b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.171ex; width:32.618ex; height:9.509ex;" alt="{\displaystyle {\begin{matrix}([W/G_{x}],w)&\to &([N_{x}/G_{x}],0)\\\downarrow &&\downarrow \\(U,u)&\to &(N_{x}//G_{x},0)\end{matrix}}}" loading="lazy"></blockquote>is cartesian, and there exists an etale morphism<blockquote><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f:([W/G_{x}],w)\to ({\mathfrak {X}},x)}">

</mrow>

<msub>

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</msub>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f:([W/G_{x}],w)\to ({\mathfrak {X}},x)}</annotation>

</semantics>

</math><img src="./0681ecbf15bd6fb34ff0ebb3afc64186df2e9dc5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:25.072ex; height:2.843ex;" alt="{\displaystyle f:([W/G_{x}],w)\to ({\mathfrak {X}},x)}" loading="lazy"></blockquote>inducing an isomorphism of the stabilizer groups at <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle w}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./88b1e0c8e1be5ebe69d18a8010676fa42d7961e6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.664ex; height:1.676ex;" alt="{\displaystyle w}" loading="lazy"> and <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="./87f9e315fd7e2ba406057a97300593c4802b53e4.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 x}" loading="lazy">.

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

<div class="mw-heading mw-heading3"><h3 id="Elementary_examples">Elementary examples</h3></div>

<ul><li>Every sheaf <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {F}}:C^{op}\to Sets}">

</mrow>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathcal {F}}:C^{op}\to Sets}</annotation>

</semantics>

</math><img src="./15670fab157ff329fa2a72f0b1b373ace921b780.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:15.645ex; height:2.343ex;" alt="{\displaystyle {\mathcal {F}}:C^{op}\to Sets}" loading="lazy"> from a category <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C}">

</mstyle>

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

</semantics>

</math><img src="./4fc55753007cd3c18576f7933f6f089196732029.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.766ex; height:2.176ex;" alt="{\displaystyle C}" loading="lazy"> with a Grothendieck topology can canonically be turned into a stack. For an object <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X\in {\text{Ob}}(C)}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X\in {\text{Ob}}(C)}</annotation>

</semantics>

</math><img src="./1940d050a44ce70846cbc8ce6185e3462f951524.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.497ex; height:2.843ex;" alt="{\displaystyle X\in {\text{Ob}}(C)}" loading="lazy">, instead of a set <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {F}}(X)}">

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./0863c8da8bca35eee88c4898e0439e26d1c3d62f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.716ex; height:2.843ex;" alt="{\displaystyle {\mathcal {F}}(X)}" loading="lazy"> there is a groupoid whose objects are the elements of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {F}}(X)}">

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./0863c8da8bca35eee88c4898e0439e26d1c3d62f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.716ex; height:2.843ex;" alt="{\displaystyle {\mathcal {F}}(X)}" loading="lazy"> and the arrows are the identity morphism.</li>

<li>More concretely, let <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./b26be3e694314bc90c3215047e4a2010c6ee184a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.339ex; height:2.176ex;" alt="{\displaystyle h}" loading="lazy"> be a contravariant functor</li></ul>

</mrow>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle h:(Sch/S)^{op}\to Sets}</annotation>

</semantics>

</math><img src="./6322c9b92cbd46e02ebb2dfa3605a23fa046dc23.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.576ex; height:2.843ex;" alt="{\displaystyle h:(Sch/S)^{op}\to Sets}" loading="lazy"></blockquote></dd>

<dd>Then, this functor <a href="Grothendieck_construction" class="mw-redirect" title="Grothendieck construction">determines</a> the following category <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./75a9edddcca2f782014371f75dca39d7e13a9c1b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle H}" loading="lazy">

<ol><li>an object is a pair <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (X\to S,x)}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./0fb491b9fa1321a201354ef881b9af623d9d0a36.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.266ex; height:2.843ex;" alt="{\displaystyle (X\to S,x)}" loading="lazy"> consisting of a scheme <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"> in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (Sch/S)^{op}}">

</mrow>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (Sch/S)^{op}}</annotation>

</semantics>

</math><img src="./277212580e421536b061024babb7e9d2977f942f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.173ex; height:2.843ex;" alt="{\displaystyle (Sch/S)^{op}}" loading="lazy"> and an element <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\in h(X)}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./c2d39d254310d9b8f7d0b3368bf81f41b3bbc5ac.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.299ex; height:2.843ex;" alt="{\displaystyle x\in h(X)}" loading="lazy"></li>

<li>a morphism <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (X\to S,x)\to (Y\to S,y)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (X\to S,x)\to (Y\to S,y)}</annotation>

</semantics>

</math><img src="./24659fcfd653d1571a0da0d6ef0f74768b76fe51.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:25.766ex; height:2.843ex;" alt="{\displaystyle (X\to S,x)\to (Y\to S,y)}" loading="lazy"> consists of a morphism <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \phi :X\to Y}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./b98600ef34bb9c70ab402e2f46de411a750ca3ab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.69ex; height:2.509ex;" alt="{\displaystyle \phi :X\to Y}" loading="lazy"> in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (Sch/S)}">

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./95e76452fed2d0886d17aca9a495af615154e85e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.316ex; height:2.843ex;" alt="{\displaystyle (Sch/S)}" loading="lazy"> such that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h(\phi )(y)=x}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle h(\phi )(y)=x}</annotation>

</semantics>

</math><img src="./6f85c7007cee1c9ede2667ed1683a6d1532c15ac.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.927ex; height:2.843ex;" alt="{\displaystyle h(\phi )(y)=x}" loading="lazy">.</li></ol></dd>

<dd>Via the forgetful functor <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p:H\to (Sch/S)}">

</mrow>

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle p:H\to (Sch/S)}</annotation>

</semantics>

</math><img src="./1c7f639b01a4c05cf3261bb95c14c3038b5bdf39.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.089ex; width:17.19ex; height:2.843ex;" alt="{\displaystyle p:H\to (Sch/S)}" loading="lazy">, the category <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./75a9edddcca2f782014371f75dca39d7e13a9c1b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle H}" loading="lazy"> is a <a href="Fibered_category" class="mw-redirect" title="Fibered category">category fibered</a> over <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (Sch/S)}">

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./95e76452fed2d0886d17aca9a495af615154e85e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.316ex; height:2.843ex;" alt="{\displaystyle (Sch/S)}" loading="lazy">. For example, if <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"> is a scheme in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (Sch/S)}">

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./95e76452fed2d0886d17aca9a495af615154e85e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.316ex; height:2.843ex;" alt="{\displaystyle (Sch/S)}" loading="lazy">, then it determines the contravariant functor <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h=\operatorname {Hom} (-,X)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle h=\operatorname {Hom} (-,X)}</annotation>

</semantics>

</math><img src="./0157f49a8879077f15617b9e21fc62ad8b141137.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.91ex; height:2.843ex;" alt="{\displaystyle h=\operatorname {Hom} (-,X)}" loading="lazy"> and the corresponding fibered category is the <style data-mw-deduplicate="TemplateStyles:r1238216509">

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</style>stack associated to X. Stacks (or prestacks) can be constructed as a variant of this construction. In fact, any scheme <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"> with a <a href="Quasi-compact_morphism" title="Quasi-compact morphism">quasi-compact diagonal</a> is an algebraic stack associated to the scheme <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">.</dd></dl>

<div class="mw-heading mw-heading3"><h3 id="Stacks_of_objects">Stacks of objects</h3></div>

<ul><li>A <a href="Group-stack" title="Group-stack">group-stack</a>.</li>

<li>The <a href="Moduli_stack_of_vector_bundles" title="Moduli stack of vector bundles">moduli stack of vector bundles</a>: the category of vector bundles V→S is a stack over the category of topological spaces S. A morphism from V→S to W→T consists of continuous maps from S to T and from V to W (linear on fibers) such that the obvious square commutes. The condition that this is a fibered category follows because one can take pullbacks of vector bundles over continuous maps of topological spaces, and the condition that a descent datum is effective follows because one can construct a vector bundle over a space by gluing together vector bundles on elements of an open cover.</li>

<li>The stack of quasi-coherent sheaves on schemes (with respect to the <a href="Fpqc-topology" class="mw-redirect" title="Fpqc-topology">fpqc-topology</a> and weaker topologies)</li>

<li>The stack of affine schemes on a base scheme (again with respect to the fpqc topology or a weaker one)</li></ul>

<div class="mw-heading mw-heading3"><h3 id="Constructions_with_stacks">Constructions with stacks</h3></div>

<div class="mw-heading mw-heading4"><h4 id="Stack_quotients">Stack quotients</h4></div>

If <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"> is a scheme <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (Sch/S)}">

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./95e76452fed2d0886d17aca9a495af615154e85e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.316ex; height:2.843ex;" alt="{\displaystyle (Sch/S)}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}">

</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"> is a smooth <a href="Affine_group" title="Affine group">affine group</a> scheme acting on <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">, then there is a <a href="Quotient_algebraic_stack" class="mw-redirect" title="Quotient algebraic stack">quotient algebraic stack</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [X/G]}">

</mrow>

</mstyle>

</mrow>

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

</semantics>

</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">,<a href="#cite_note-2">[2]</a> taking a scheme <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y\to S}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./f981ce77ed205b077417caa14476a742c538c8b7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.887ex; height:2.176ex;" alt="{\displaystyle Y\to S}" loading="lazy"> to the groupoid of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}">

</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">-torsors over the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S}">

</mstyle>

</mrow>

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

</semantics>

</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 Y}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./961d67d6b454b4df2301ac571808a3538b3a6d3f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:1.773ex; height:2.009ex;" alt="{\displaystyle Y}" loading="lazy"> with <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}">

</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">-equivariant maps to <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">. Explicitly, given a space <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"> with a <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}">

</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">-action, form the stack <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [X/G]}">

</mrow>

</mstyle>

</mrow>

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

</semantics>

</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">, which (intuitively speaking) <a href="Pseudo-functor" title="Pseudo-functor">sends</a> a space <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./961d67d6b454b4df2301ac571808a3538b3a6d3f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:1.773ex; height:2.009ex;" alt="{\displaystyle Y}" loading="lazy"> to the groupoid of pullback diagrams<blockquote><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [X/G](Y)={\begin{Bmatrix}Z&{\xrightarrow {\Phi }}&X\\\downarrow &&\downarrow \\Y&{\xrightarrow {\phi }}&[X/G]\end{Bmatrix}}}">

</mrow>

<mrow>

<mtr>

<mtd>

</mtd>

<mtd>

<mover>

</mpadded>

</mover>

</mrow>

</mtd>

<mtd>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

<mtd>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

<mtd>

<mover>

</mpadded>

</mover>

</mrow>

</mtd>

<mtd>

</mrow>

</mtd>

</mtr>

</mtable>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle [X/G](Y)={\begin{Bmatrix}Z&{\xrightarrow {\Phi }}&X\\\downarrow &&\downarrow \\Y&{\xrightarrow {\phi }}&[X/G]\end{Bmatrix}}}</annotation>

</semantics>

</math><img src="./30133b958e911b0db07a2c5dc115442ffdd25011.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.118ex; margin-top: -0.392ex; margin-bottom: -0.22ex; width:33.019ex; height:12.009ex;" alt="{\displaystyle [X/G](Y)={\begin{Bmatrix}Z&{\xrightarrow {\Phi }}&X\\\downarrow &&\downarrow \\Y&{\xrightarrow {\phi }}&[X/G]\end{Bmatrix}}}" loading="lazy"></blockquote>where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Phi }">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./aed80a2011a3912b028ba32a52dfa57165455f24.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \Phi }" loading="lazy"> is a <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}">

</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">-equivariant morphism of spaces and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Z\to Y}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./a2f836de5648fddbae09be6abce0cc80eb9ba3f0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.068ex; height:2.176ex;" alt="{\displaystyle Z\to Y}" loading="lazy"> is a principal <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}">

</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">-bundle. The morphisms in this category are just morphisms of diagrams where the arrows on the right-hand side are equal and the arrows on the left-hand side are morphisms of principal <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}">

</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">-bundles.

<div class="mw-heading mw-heading4"><h4 id="Classifying_stacks">Classifying stacks</h4></div>

A special case of this when X is a point gives the <a href="Classifying_stack" class="mw-redirect" title="Classifying stack">classifying stack</a> BG of a smooth affine group scheme G: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\textbf {B}}G:=[pt/G].}">

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\textbf {B}}G:=[pt/G].}</annotation>

</semantics>

</math><img src="./15410eabfc27396ed1c30ce67dd9c15414ab2ff7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.412ex; height:2.843ex;" alt="{\displaystyle {\textbf {B}}G:=[pt/G].}" loading="lazy"> It is named so since the category <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {B} G(Y)}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {B} G(Y)}</annotation>

</semantics>

</math><img src="./bc28d8e9743f060246282ecad1e5eba257803332.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.31ex; height:2.843ex;" alt="{\displaystyle \mathbf {B} G(Y)}" loading="lazy">, the fiber over Y, is precisely the category <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {Bun} _{G}(Y)}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \operatorname {Bun} _{G}(Y)}</annotation>

</semantics>

</math><img src="./d89aa4de057e8b3bd702c8293e560998c5df03db.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.337ex; height:2.843ex;" alt="{\displaystyle \operatorname {Bun} _{G}(Y)}" loading="lazy"> of principal <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}">

</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">-bundles over <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./961d67d6b454b4df2301ac571808a3538b3a6d3f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:1.773ex; height:2.009ex;" alt="{\displaystyle Y}" loading="lazy">. Note that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {Bun} _{G}(Y)}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \operatorname {Bun} _{G}(Y)}</annotation>

</semantics>

</math><img src="./d89aa4de057e8b3bd702c8293e560998c5df03db.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.337ex; height:2.843ex;" alt="{\displaystyle \operatorname {Bun} _{G}(Y)}" loading="lazy"> itself can be considered as a stack, the <a href="Moduli_stack_of_principal_bundles" title="Moduli stack of principal bundles">moduli stack of principal G-bundles on Y</a>.

An important subexample from this construction is <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {B} GL_{n}}">

</mrow>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./58363ac18080a6c4fc2973a705249d71e810ac6e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.529ex; height:2.509ex;" alt="{\displaystyle \mathbf {B} GL_{n}}" loading="lazy">, which is the moduli stack of principal <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle GL_{n}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./a41d6b4f9b104c2e8f659319ddbd61792518ace2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.628ex; height:2.509ex;" alt="{\displaystyle GL_{n}}" loading="lazy">-bundles. Since the data of a principal <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle GL_{n}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./a41d6b4f9b104c2e8f659319ddbd61792518ace2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.628ex; height:2.509ex;" alt="{\displaystyle GL_{n}}" loading="lazy">-bundle is equivalent to the data of a rank <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"> vector bundle, this is isomorphic to the <a href="Moduli_stack_of_vector_bundles" title="Moduli stack of vector bundles">moduli stack of rank <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"> vector bundles <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Vect_{n}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./3f10ef0b02be00f4a16557c39f4897580bd7820f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.936ex; height:2.509ex;" alt="{\displaystyle Vect_{n}}" loading="lazy"></a>.

<div class="mw-heading mw-heading5"><h5 id="Moduli_stack_of_line_bundles">Moduli stack of line bundles</h5></div>

The moduli stack of line bundles is <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B\mathbb {G} _{m}}">

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</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"> since every line bundle is canonically isomorphic to a principal <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">-bundle. Indeed, given a line bundle <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./103168b86f781fe6e9a4a87b8ea1cebe0ad4ede8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.583ex; height:2.176ex;" alt="{\displaystyle L}" loading="lazy"> over a scheme <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S}">

</mstyle>

</mrow>

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

</semantics>

</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">, the relative spec<blockquote><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\underline {\text{Spec}}}_{S}({\text{Sym}}_{S}(L^{\vee }))\to S}">

<msub>

</munder>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\underline {\text{Spec}}}_{S}({\text{Sym}}_{S}(L^{\vee }))\to S}</annotation>

</semantics>

</math><img src="./7af1c07e63df3d2b0e928b751e0bb65f60f056f1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.081ex; margin-bottom: -0.757ex; width:23.336ex; height:4.009ex;" alt="{\displaystyle {\underline {\text{Spec}}}_{S}({\text{Sym}}_{S}(L^{\vee }))\to S}" loading="lazy"></blockquote>gives a geometric line bundle. By removing the image of the zero section, one obtains a principal <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">-bundle. Conversely, from the representation <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle id:\mathbb {G} _{m}\to {\text{Aut}}(\mathbb {A} ^{1})}">

<msub>

</mrow>

</mrow>

</msub>

</mrow>

<msup>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle id:\mathbb {G} _{m}\to {\text{Aut}}(\mathbb {A} ^{1})}</annotation>

</semantics>

</math><img src="./042019f37c9f76557486d26bd1bf9fbd2575c77b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.535ex; height:3.176ex;" alt="{\displaystyle id:\mathbb {G} _{m}\to {\text{Aut}}(\mathbb {A} ^{1})}" loading="lazy">, the associated line bundle can be reconstructed.

<div class="mw-heading mw-heading4"><h4 id="Gerbes">Gerbes</h4></div>

A <a href="Gerbe" title="Gerbe">gerbe</a> is a stack in groupoids that is locally nonempty, for example the trivial gerbe <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle BG}">

</mstyle>

</mrow>

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

</semantics>

</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"> that assigns to each scheme the groupoid of principal <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}">

</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">-bundles over the scheme, for some group <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}">

</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">.

<div class="mw-heading mw-heading4"><h4 id="Relative_spec_and_proj">Relative spec and proj</h4></div>

If A is a quasi-coherent <a href="Sheaf_of_algebras" title="Sheaf of algebras">sheaf of algebras</a> in an algebraic stack X over a scheme S, then there is a stack Spec(A) generalizing the construction of the spectrum Spec(A) of a commutative ring A. An object of Spec(A) is given by an S-scheme T, an object x of X(T), and a morphism of sheaves of algebras from x*(A) to the coordinate ring O(T) of T.

If A is a quasi-coherent sheaf of graded algebras in an algebraic stack X over a scheme S, then there is a stack Proj(A) generalizing the construction of the projective scheme Proj(A) of a graded ring A.

<div class="mw-heading mw-heading3"><h3 id="Moduli_stacks">Moduli stacks</h3></div>

<div class="mw-heading mw-heading4"><h4 id="Moduli_of_curves">Moduli of curves</h4></div>

<ul><li><a href="#CITEREFMumford1965">Mumford (1965)</a> studied the <a href="Moduli_stack_of_elliptic_curves" title="Moduli stack of elliptic curves">moduli stack M1,1 of elliptic curves</a>, and showed that its Picard group is cyclic of order 12. For elliptic curves over the <a href="Complex_number" title="Complex number">complex numbers</a> the corresponding stack is similar to a quotient of the <a href="Upper_half-plane" title="Upper half-plane">upper half-plane</a> by the action of the <a href="Modular_group" title="Modular group">modular group</a>.</li>

<li>The <a href="Moduli_space_of_algebraic_curves" class="mw-redirect" title="Moduli space of algebraic curves">moduli space of algebraic curves</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {M}}_{g}}">

<msub>

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</msub>

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

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</math><img src="./43d9ec21af6dbbe97151cd1f34d8d153e57c38d5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.812ex; height:2.843ex;" alt="{\displaystyle {\mathcal {M}}_{g}}" loading="lazy"> defined as a universal family of smooth curves of given <a href="Genus_(mathematics)" title="Genus (mathematics)">genus</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./d3556280e66fe2c0d0140df20935a6f057381d77.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.116ex; height:2.009ex;" alt="{\displaystyle g}" loading="lazy"> does not exist as an algebraic variety because in particular there are curves admitting nontrivial automorphisms. However there is a moduli stack <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {M}}_{g}}">

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

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

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</math><img src="./43d9ec21af6dbbe97151cd1f34d8d153e57c38d5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.812ex; height:2.843ex;" alt="{\displaystyle {\mathcal {M}}_{g}}" loading="lazy">, which is a good substitute for the non-existent fine moduli space of smooth genus <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./d3556280e66fe2c0d0140df20935a6f057381d77.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.116ex; height:2.009ex;" alt="{\displaystyle g}" loading="lazy"> curves. More generally there is a moduli stack <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {M}}_{g,n}}">

<msub>

</mrow>

</mrow>

</msub>

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</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathcal {M}}_{g,n}}</annotation>

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</math><img src="./6daf7fff3319e7dbfb35a6084c2f8ed9f4c9f7e4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:5.256ex; height:2.843ex;" alt="{\displaystyle {\mathcal {M}}_{g,n}}" loading="lazy"> of genus <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./d3556280e66fe2c0d0140df20935a6f057381d77.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.116ex; height:2.009ex;" alt="{\displaystyle g}" loading="lazy"> curves with <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}">

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</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"> marked points. In general this is an algebraic stack, and is a Deligne–Mumford stack for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g\geq 2}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle g\geq 2}</annotation>

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</math><img src="./b84b2394f64ae9927e09ee51b848be213edecafe.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.377ex; height:2.509ex;" alt="{\displaystyle g\geq 2}" loading="lazy"> or <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g=1,n\geq 1}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle g=1,n\geq 1}</annotation>

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</math><img src="./90e09543e119fa9f02ab479c693cffd805a6afc1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.066ex; height:2.509ex;" alt="{\displaystyle g=1,n\geq 1}" loading="lazy"> or <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g=0,n\geq 3}">

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<annotation encoding="application/x-tex">{\displaystyle g=0,n\geq 3}</annotation>

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</math><img src="./55d47fc2105798ef0ea13f88da01a6cefef474fd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.066ex; height:2.509ex;" alt="{\displaystyle g=0,n\geq 3}" loading="lazy"> (in other words when the automorphism groups of the curves are finite). This moduli stack has a completion consisting of the moduli stack of stable curves (for given <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g}">

</mstyle>

</mrow>

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

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</math><img src="./d3556280e66fe2c0d0140df20935a6f057381d77.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.116ex; height:2.009ex;" alt="{\displaystyle g}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}">

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</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">), which is proper over Spec Z. For example, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {M}}_{0}}">

<msub>

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</mrow>

</msub>

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<annotation encoding="application/x-tex">{\displaystyle {\mathcal {M}}_{0}}</annotation>

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</math><img src="./03f0f07b65e6c14f037c9bfd1aa5a47024661c40.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.845ex; height:2.509ex;" alt="{\displaystyle {\mathcal {M}}_{0}}" loading="lazy"> is the classifying stack <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B{\text{PGL}}(2)}">

</mrow>

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<annotation encoding="application/x-tex">{\displaystyle B{\text{PGL}}(2)}</annotation>

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</math><img src="./aa15e8e580d56416b106a2bc79fb98fac2fd184e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.596ex; height:2.843ex;" alt="{\displaystyle B{\text{PGL}}(2)}" loading="lazy"> of the projective general linear group. (There is a subtlety in defining <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {M}}_{1}}">

<msub>

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</mrow>

</msub>

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<annotation encoding="application/x-tex">{\displaystyle {\mathcal {M}}_{1}}</annotation>

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</math><img src="./ed3e437e9a86b2372c117efc506f112d55466c4e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.845ex; height:2.509ex;" alt="{\displaystyle {\mathcal {M}}_{1}}" loading="lazy">, as one has to use algebraic spaces rather than schemes to construct it.)</li></ul>

<div class="mw-heading mw-heading4"><h4 id="Kontsevich_moduli_spaces">Kontsevich moduli spaces</h4></div>

Another widely studied class of moduli spaces are the Kontsevich moduli spaces parameterizing the space of stable maps between curves of a fixed genus to a fixed space <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"> whose image represents a fixed cohomology class. These moduli spaces are denoted<a href="#cite_note-:0-3">[3]</a><blockquote><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\overline {\mathcal {M}}}_{g,n}(X,\beta )}">

<msub>

<mover>

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<annotation encoding="application/x-tex">{\displaystyle {\overline {\mathcal {M}}}_{g,n}(X,\beta )}</annotation>

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</math><img src="./310886dbf457225357bacd64770da6a1c77b724c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:11.526ex; height:3.676ex;" alt="{\displaystyle {\overline {\mathcal {M}}}_{g,n}(X,\beta )}" loading="lazy"></blockquote>and can have wild behavior, such as being reducible stacks whose components are non-equal dimension. For example,<a href="#cite_note-:0-3">[3]</a> the moduli stack <blockquote><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\overline {\mathcal {M}}}_{1,0}(\mathbb {P} ^{2},3[H])}">

<msub>

<mover>

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<msup>

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<annotation encoding="application/x-tex">{\displaystyle {\overline {\mathcal {M}}}_{1,0}(\mathbb {P} ^{2},3[H])}</annotation>

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</math><img src="./533c98cdc3a131b01460e70ea86da25b64a7dd29.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:15.077ex; height:3.676ex;" alt="{\displaystyle {\overline {\mathcal {M}}}_{1,0}(\mathbb {P} ^{2},3[H])}" loading="lazy"></blockquote>has smooth curves parametrized by an open subset <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U\subset \mathbb {P} ^{9}=\mathbb {P} (\Gamma (\mathbb {P} ^{2},{\mathcal {O}}(3)))}">

<msup>

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</msup>

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<msup>

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<annotation encoding="application/x-tex">{\displaystyle U\subset \mathbb {P} ^{9}=\mathbb {P} (\Gamma (\mathbb {P} ^{2},{\mathcal {O}}(3)))}</annotation>

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</math><img src="./f9346978fbc8bd7d860e8bd23592d0144a0b6e4e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:25.276ex; height:3.176ex;" alt="{\displaystyle U\subset \mathbb {P} ^{9}=\mathbb {P} (\Gamma (\mathbb {P} ^{2},{\mathcal {O}}(3)))}" loading="lazy">. On the boundary of the moduli space, where curves may degenerate to reducible curves, there is a substack parametrizing reducible curves with a genus <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 0}</annotation>

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</math><img src="./2aae8864a3c1fec9585261791a809ddec1489950.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 0}" loading="lazy"> component and a genus <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1}">

</mstyle>

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

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</math><img src="./92d98b82a3778f043108d4e20960a9193df57cbf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 1}" loading="lazy"> component intersecting at one point, and the map sends the genus <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1}">

</mstyle>

</mrow>

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

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</math><img src="./92d98b82a3778f043108d4e20960a9193df57cbf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 1}" loading="lazy"> curve to a point. Since all such genus <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1}">

</mstyle>

</mrow>

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

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</math><img src="./92d98b82a3778f043108d4e20960a9193df57cbf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 1}" loading="lazy"> curves are parametrized by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./458a728f53b9a0274f059cd695e067c430956025.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.783ex; height:2.176ex;" alt="{\displaystyle U}" loading="lazy">, and there is an additional <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./92d98b82a3778f043108d4e20960a9193df57cbf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 1}" loading="lazy"> dimensional choice of where these curves intersect on the genus <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1}">

</mstyle>

</mrow>

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

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</math><img src="./92d98b82a3778f043108d4e20960a9193df57cbf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 1}" loading="lazy"> curve, the boundary component has dimension <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 10}">

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle 10}</annotation>

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</math><img src="./4ec811eb07dcac7ea67b413c5665390a1671ecb0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.325ex; height:2.176ex;" alt="{\displaystyle 10}" loading="lazy">.

<div class="mw-heading mw-heading4"><h4 id="Other_moduli_stacks">Other moduli stacks</h4></div>

<ul><li>A <a href="Picard_stack" class="mw-redirect" title="Picard stack">Picard stack</a> generalizes a <a href="Picard_variety" class="mw-redirect" title="Picard variety">Picard variety</a>.</li>

<li>The <a href="Moduli_stack_of_formal_group_laws" title="Moduli stack of formal group laws">moduli stack of formal group laws</a> classifies <a href="Formal_group_law" title="Formal group law">formal group laws</a>.</li>

<li>An <a href="Ind-scheme" title="Ind-scheme">ind-scheme</a> such as an infinite projective space and a <a href="Formal_scheme" title="Formal scheme">formal scheme</a> is a stack.</li>

<li>A moduli stack of shtukas is used in <a href="Geometric_Langlands_program" class="mw-redirect" title="Geometric Langlands program">geometric Langlands program</a>. (See also <a href="Shtuka" class="mw-redirect" title="Shtuka">shtukas</a>.)</li></ul>

<div class="mw-heading mw-heading3"><h3 id="Geometric_stacks">Geometric stacks</h3></div>

<div class="mw-heading mw-heading4"><h4 id="Weighted_projective_stacks">Weighted projective stacks</h4></div>

Constructing <a href="Weighted_projective_space" title="Weighted projective space">weighted projective spaces</a> involves taking the <a href="GIT_quotient" title="GIT quotient">quotient variety</a> of some <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {A} ^{n+1}-\{0\}}">

<msup>

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<annotation encoding="application/x-tex">{\displaystyle \mathbb {A} ^{n+1}-\{0\}}</annotation>

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</math><img src="./a080f3ac3bcb8244a5423a72c8c434cc904aa622.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.325ex; height:3.176ex;" alt="{\displaystyle \mathbb {A} ^{n+1}-\{0\}}" loading="lazy"> by a <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {G} _{m}}">

<msub>

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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. In particular, the action sends a tuple<blockquote><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g\cdot (x_{0},\ldots ,x_{n})\mapsto (g^{a_{0}}x_{0},\ldots ,g^{a_{n}}x_{n})}">

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<annotation encoding="application/x-tex">{\displaystyle g\cdot (x_{0},\ldots ,x_{n})\mapsto (g^{a_{0}}x_{0},\ldots ,g^{a_{n}}x_{n})}</annotation>

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</math><img src="./0e3b0e0b0859064329b5698af14f51a34c2256e9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:36.485ex; height:2.843ex;" alt="{\displaystyle g\cdot (x_{0},\ldots ,x_{n})\mapsto (g^{a_{0}}x_{0},\ldots ,g^{a_{n}}x_{n})}" loading="lazy"></blockquote>and the quotient of this action gives the weighted projective space <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {WP} (a_{0},\ldots ,a_{n})}">

</mrow>

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<annotation encoding="application/x-tex">{\displaystyle \mathbb {WP} (a_{0},\ldots ,a_{n})}</annotation>

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</math><img src="./3f48691c5e7e7edb5328348d5ae4c4ddf1456489.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.464ex; height:2.843ex;" alt="{\displaystyle \mathbb {WP} (a_{0},\ldots ,a_{n})}" loading="lazy">. Since this can instead be taken as a stack quotient, the weighted projective stack<a href="#cite_note-4">[4]</a> pg 30 is<blockquote><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\textbf {WP}}(a_{0},\ldots ,a_{n}):=[\mathbb {A} ^{n}-\{0\}/\mathbb {G} _{m}]}">

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<annotation encoding="application/x-tex">{\displaystyle {\textbf {WP}}(a_{0},\ldots ,a_{n}):=[\mathbb {A} ^{n}-\{0\}/\mathbb {G} _{m}]}</annotation>

</semantics>

</math><img src="./9491319ed6388ab9a2e13ae6a343d2d26ce8c5f2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:35.218ex; height:2.843ex;" alt="{\displaystyle {\textbf {WP}}(a_{0},\ldots ,a_{n}):=[\mathbb {A} ^{n}-\{0\}/\mathbb {G} _{m}]}" loading="lazy"></blockquote>Taking the vanishing locus of a weighted polynomial in a line bundle <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f\in \Gamma ({\textbf {WP}}(a_{0},\ldots ,a_{n}),{\mathcal {O}}(a))}">

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<annotation encoding="application/x-tex">{\displaystyle f\in \Gamma ({\textbf {WP}}(a_{0},\ldots ,a_{n}),{\mathcal {O}}(a))}</annotation>

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</math><img src="./a7bfb4e9b02abb2c26b3ea31aae3cd4799fcc8cc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:29.614ex; height:2.843ex;" alt="{\displaystyle f\in \Gamma ({\textbf {WP}}(a_{0},\ldots ,a_{n}),{\mathcal {O}}(a))}" loading="lazy"> gives a stacky weighted <a href="Projective_variety" title="Projective variety">projective variety</a>.

<div class="mw-heading mw-heading4"><h4 id="Stacky_curves">Stacky curves</h4></div>

<a href="Stacky_curve" title="Stacky curve">Stacky curves</a>, or orbicurves, can be constructed by taking the stack quotient of a morphism of curves by the monodromy group of the cover over the generic points. For example, take a projective morphism<blockquote><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\text{Proj}}(\mathbb {C} [x,y,z]/(x^{5}+y^{5}+z^{5}))\to {\text{Proj}}(\mathbb {C} [x,y])}">

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<annotation encoding="application/x-tex">{\displaystyle {\text{Proj}}(\mathbb {C} [x,y,z]/(x^{5}+y^{5}+z^{5}))\to {\text{Proj}}(\mathbb {C} [x,y])}</annotation>

</semantics>

</math><img src="./66984dbed9f3884447dd8bf48ca55a36dce48f70.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:46.47ex; height:3.176ex;" alt="{\displaystyle {\text{Proj}}(\mathbb {C} [x,y,z]/(x^{5}+y^{5}+z^{5}))\to {\text{Proj}}(\mathbb {C} [x,y])}" loading="lazy"></blockquote>which is generically <a href="%C3%89tale_morphism" title="Étale morphism">etale</a>. The stack quotient of the domain by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu _{5}}">

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<annotation encoding="application/x-tex">{\displaystyle \mu _{5}}</annotation>

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</math><img src="./deb3071ec6231e0eb038d5886e6cd7a8ad96ba4c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.456ex; height:2.176ex;" alt="{\displaystyle \mu _{5}}" loading="lazy"> gives a stacky <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {P} ^{1}}">

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

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</math><img src="./5e38808c13ecc929e128079b70c1642554e2a4b3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.475ex; height:2.676ex;" alt="{\displaystyle \mathbb {P} ^{1}}" loading="lazy"> with stacky points that have stabilizer group <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Z} /5}">

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<annotation encoding="application/x-tex">{\displaystyle \mathbb {Z} /5}</annotation>

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</math><img src="./c2ce0b4c522174f9be7c2ca76b286e0bc04d985a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.875ex; height:2.843ex;" alt="{\displaystyle \mathbb {Z} /5}" loading="lazy"> at the fifth roots of unity in the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x/y}">

</mrow>

</mstyle>

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

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</math><img src="./c9f79732fa39115855a4cf9b64d2a63971a41046.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.648ex; height:2.843ex;" alt="{\displaystyle x/y}" loading="lazy">-chart. This is because these are the points where the cover ramifies.

<div class="mw-heading mw-heading4"><h4 id="Non-affine_stack">Non-affine stack</h4></div>

An example of a non-affine stack is given by the half-line with two stacky origins. This can be constructed as the colimit of two inclusion of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [\mathbb {G} _{m}/(\mathbb {Z} /2)]\to [\mathbb {A} ^{1}/(\mathbb {Z} /2)]}">

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<annotation encoding="application/x-tex">{\displaystyle [\mathbb {G} _{m}/(\mathbb {Z} /2)]\to [\mathbb {A} ^{1}/(\mathbb {Z} /2)]}</annotation>

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</math><img src="./9256d8847ab16529807936cc3bbf79b7c94f32fe.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:26.111ex; height:3.176ex;" alt="{\displaystyle [\mathbb {G} _{m}/(\mathbb {Z} /2)]\to [\mathbb {A} ^{1}/(\mathbb {Z} /2)]}" loading="lazy">.

<div class="mw-heading mw-heading2"><h2 id="Quasi-coherent_sheaves_on_algebraic_stacks">Quasi-coherent sheaves on algebraic stacks</h2></div>

<div role="note" class="hatnote navigation-not-searchable">Main article: <a href="Quasi-coherent_sheaf_on_an_algebraic_stack" class="mw-redirect" title="Quasi-coherent sheaf on an algebraic stack">Quasi-coherent sheaf on an algebraic stack</a></div>

On an algebraic stack one can construct a category of quasi-coherent sheaves similar to the category of quasi-coherent sheaves over a scheme.

A quasi-coherent sheaf is roughly one that looks locally like the <a href="Sheaf_of_a_module" class="mw-redirect" title="Sheaf of a module">sheaf of a module</a> over a ring. The first problem is to decide what one means by "locally": this involves the choice of a Grothendieck topology, and there are many possible choices for this, all of which have some problems and none of which seem completely satisfactory. The Grothendieck topology should be strong enough so that the stack is locally affine in this topology: schemes are locally affine in the Zariski topology so this is a good choice for schemes as Serre discovered, algebraic spaces and Deligne–Mumford stacks are locally affine in the etale topology so one usually uses the etale topology for these, while algebraic stacks are locally affine in the smooth topology so one can use the smooth topology in this case. For general algebraic stacks the etale topology does not have enough open sets: for example, if G is a smooth connected group then the only etale covers of the classifying stack BG are unions of copies of BG, which are not enough to give the right theory of quasicoherent sheaves.

Instead of using the smooth topology for algebraic stacks one often uses a modification of it called the <a href="Smooth_topology" title="Smooth topology">Lis-Et topology</a> (short for Lisse-Etale: lisse is the French term for smooth), which has the same open sets as the smooth topology but the open covers are given by etale rather than smooth maps. This usually seems to lead to an equivalent category of quasi-coherent sheaves, but is easier to use: for example it is easier to compare with the etale topology on algebraic spaces. The Lis-Et topology has a subtle technical problem: a morphism between stacks does not in general give a morphism between the corresponding topoi. (The problem is that while one can construct a pair of adjoint functors f*, f*, as needed for a geometric morphism of topoi, the functor f* is not left exact in general. This problem is notorious for having caused some errors in published papers and books.<a href="#cite_note-5">[5]</a>) This means that constructing the pullback of a quasicoherent sheaf under a morphism of stacks requires some extra effort.

It is also possible to use finer topologies. Most reasonable "sufficiently large" Grothendieck topologies seem to lead to equivalent categories of quasi-coherent sheaves, but the larger a topology is the harder it is to handle, so one generally prefers to use smaller topologies as long as they have enough open sets. For example, the big fppf topology leads to essentially the same category of quasi-coherent sheaves as the Lis-Et topology, but has a subtle problem: the natural embedding of quasi-coherent sheaves into OX modules in this topology is not exact (it does not preserve kernels in general).

<div class="mw-heading mw-heading2"><h2 id="Other_types_of_stack">Other types of stack</h2></div>

<a href="Differentiable_stack" title="Differentiable stack">Differentiable stacks</a> and topological stacks are defined in a way similar to algebraic stacks, except that the underlying category of affine schemes is replaced by the category of smooth manifolds or topological spaces.

More generally one can define the notion of an n-sheaf or n–1 stack, which is roughly a sort of sheaf taking values in n–1 categories. There are several inequivalent ways of doing this. 1-sheaves are the same as sheaves, and 2-sheaves are the same as stacks. They are called <a href="Higher_stack" title="Higher stack">higher stacks</a>.

A very similar and analogous extension is to develop the stack theory on non-discrete objects (i.e., a space is really a <a href="Spectrum_(topology)" title="Spectrum (topology)">spectrum</a> in algebraic topology). The resulting stacky objects are called <a href="Derived_stack" title="Derived stack">derived stacks</a> (or spectral stacks). <a href="Jacob_Lurie" title="Jacob Lurie">Jacob Lurie</a>'s under-construction book Spectral Algebraic Geometry studies a generalization that he calls a spectral Deligne–Mumford stack. By definition, it is a ringed <a href="%E2%88%9E-topos" title="∞-topos">∞-topos</a> that is étale-locally the <a href="%C3%89tale_spectrum" title="Étale spectrum">étale spectrum</a> of an <a href="E-infinity_ring" class="mw-redirect" title="E-infinity ring">E∞-ring</a> (this notion subsumes that of a <a href="Derived_scheme" title="Derived scheme">derived scheme</a>, at least in characteristic zero.)

<div class="mw-heading mw-heading2"><h2 id="Set-theoretical_problems">Set-theoretical problems</h2></div>

There are some minor set theoretical problems with the usual foundation of the theory of stacks, because stacks are often defined as certain functors to the category of sets and are therefore not sets. There are several ways to deal with this problem:

<ul><li>One can work with Grothendieck universes: a stack is then a functor between classes of some fixed Grothendieck universe, so these classes and the stacks are sets in a larger Grothendieck universe. The drawback of this approach is that one has to assume the existence of enough Grothendieck universes, which is essentially a <a href="Large_cardinal" title="Large cardinal">large cardinal</a> axiom.</li>

<li>One can define stacks as functors to the set of sets of sufficiently large rank, and keep careful track of the ranks of the various sets one uses. The problem with this is that it involves some additional rather tiresome bookkeeping.</li>

<li>One can use reflection principles from set theory stating that one can find set models of any finite fragment of the axioms of ZFC to show that one can automatically find sets that are sufficiently close approximations to the universe of all sets.</li>

<li>One can simply ignore the problem. This is the approach taken by many authors.</li></ul>

<ul><li><a href="Algebraic_stack" title="Algebraic stack">Algebraic stack</a></li>

<li><a href="Chow_group_of_a_stack" title="Chow group of a stack">Chow group of a stack</a></li>

<li><a href="Deligne%E2%80%93Mumford_stack" title="Deligne–Mumford stack">Deligne–Mumford stack</a></li>

<li><a href="Glossary_of_algebraic_geometry" title="Glossary of algebraic geometry">Glossary of algebraic geometry</a></li>

<li><a href="Pursuing_Stacks" title="Pursuing Stacks">Pursuing Stacks</a></li>

<li><a href="Quotient_space_of_an_algebraic_stack" title="Quotient space of an algebraic stack">Quotient space of an algebraic stack</a></li>

<li><a href="Ring_of_modular_forms" title="Ring of modular forms">Ring of modular forms</a></li>

<li><a href="Simplicial_presheaf" title="Simplicial presheaf">Simplicial presheaf</a></li>

<li><a href="Stacks_Project" title="Stacks Project">Stacks Project</a></li>

<li><a href="Toric_stack" title="Toric stack">Toric stack</a></li>

<li><a href="Generalized_space" title="Generalized space">Generalized space</a></li></ul>

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

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</style><cite id="CITEREFAlperHallRydh2020" class="citation journal cs1">Alper, Jarod; Hall, Jack; Rydh, David (2020). "A Luna étale slice theorem for algebraic stacks". Annals of Mathematics. 191 (3): 675–738. <a href="ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<a rel="nofollow" class="external text" href="https://arxiv.org/abs/1504.06467">1504.06467</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.4007%2Fannals.2020.191.3.1">10.4007/annals.2020.191.3.1</a>. <a href="Hdl_(identifier)" class="mw-redirect" title="Hdl (identifier)">hdl</a>:<a rel="nofollow" class="external text" href="https://hdl.handle.net/10150%2F641331">10150/641331</a>. <a href="ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0003-486X">0003-486X</a>. <a href="JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/10.4007/annals.2020.191.3.1">10.4007/annals.2020.191.3.1</a>. <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:3225788">3225788</a>.</cite>

</li>

<li id="cite_note-2"><a href="#cite_ref-2">^</a> <cite id="CITEREFHeinloth2009" class="citation cs2">Heinloth, Jochen (January 29, 2009), "Lectures on the Moduli Stack of Vector Bundles on a Curve", Affine Flag Manifolds and Principal Bundles, Basel: Springer Basel (published 2010), pp. 123–153, <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2F978-3-0346-0288-4_4">10.1007/978-3-0346-0288-4_4</a>, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-3-0346-0287-7</bdi></cite>

</li>

<li id="cite_note-:0-3">^ <a href="#cite_ref-:0_3-0">a</a> <a href="#cite_ref-:0_3-1">b</a> <cite id="CITEREFMassarenti" class="citation web cs1">Massarenti, Alez. <a rel="nofollow" class="external text" href="http://w3.impa.br/~massaren/files/smgwqc.pdf">"Moduli of Stable Maps, Gromov-Witten Invariants, and Quantum Cohomology"</a> (PDF). pp. 1–4. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20180123190525/http://w3.impa.br/~massaren/files/smgwqc.pdf">Archived</a> (PDF) from the original on 2018-01-23.</cite>

</li>

<li id="cite_note-4"><a href="#cite_ref-4">^</a> <cite id="CITEREFFantechiMannNironi2009" class="citation arxiv cs1">Fantechi, Barbara; Mann, Etienne; Nironi, Fabio (2009-09-22). "Smooth toric DM stacks". <a href="ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<a rel="nofollow" class="external text" href="https://arxiv.org/abs/0708.1254">0708.1254</a> [<a rel="nofollow" class="external text" href="https://arxiv.org/archive/math.AG">math.AG</a>].</cite>

</li>

<li id="cite_note-5"><a href="#cite_ref-5">^</a> See, for example, <cite class="citation journal cs1">Olsson, Martin (2007). "Sheaves on Artin stacks". <a href="Journal_f%C3%BCr_die_reine_und_angewandte_Mathematik" class="mw-redirect" title="Journal für die reine und angewandte Mathematik">Journal für die reine und angewandte Mathematik</a>. 2007 (603): 55–112. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1515%2FCRELLE.2007.012">10.1515/CRELLE.2007.012</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=2312554">2312554</a>. <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:15445962">15445962</a>.</cite>

</li>

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

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

<div class="mw-heading mw-heading3"><h3 id="Pedagogical">Pedagogical</h3></div>

<ul><li><cite id="CITEREFBehrendConradEdidinFulton2006" class="citation cs2">Behrend, Kai; Conrad, Brian; Edidin, Dan; Fulton, William; Fantechi, Barbara; Göttsche, Lothar; Kresch, Andrew (2006), <a rel="nofollow" class="external text" href="https://web.archive.org/web/20080505043444/http://www.math.unizh.ch/index.php?pr_vo_det&key1=1287&key2=580&no_cache=1">Algebraic stacks</a>, archived from <a rel="nofollow" class="external text" href="http://www.math.unizh.ch/index.php?pr_vo_det&key1=1287&key2=580&no_cache=1">the original</a> on 2008-05-05</cite></li>

<li><cite id="CITEREFGoméz1999" class="citation cs2">Goméz, Tomás (1999), Algebraic stacks, <a href="ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<a rel="nofollow" class="external text" href="https://arxiv.org/abs/math/9911199">math/9911199</a>, <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1999math.....11199G">1999math.....11199G</a></cite> is an expository article describing the basics of stacks with examples.</li>

<li><cite id="CITEREFEdidin2003" class="citation cs2">Edidin, Dan (2003), <a rel="nofollow" class="external text" href="http://www.ams.org/notices/200304/what-is.pdf">"What is... a Stack?"</a> (PDF), Notices of the AMS, 50 (4): 458–459</cite></li></ul>

<div class="mw-heading mw-heading3"><h3 id="Guides_to_the_literature">Guides to the literature</h3></div>

<ul><li><a rel="nofollow" class="external free" href="https://maths-people.anu.edu.au/~alperj/papers/stacks-guide.pdf">https://maths-people.anu.edu.au/~alperj/papers/stacks-guide.pdf</a></li>

<li><a rel="nofollow" class="external free" href="http://stacks.math.columbia.edu/tag/03B0">http://stacks.math.columbia.edu/tag/03B0</a></li></ul>

<div class="mw-heading mw-heading3"><h3 id="References_2">References</h3></div>

<ul><li><cite id="CITEREFArtin1974" class="citation cs2"><a href="Michael_Artin" title="Michael Artin">Artin, Michael</a> (1974), "Versal deformations and algebraic stacks", <a href="Inventiones_Mathematicae" title="Inventiones Mathematicae">Inventiones Mathematicae</a>, 27 (3): 165–189, <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1974InMat..27..165A">1974InMat..27..165A</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%2FBF01390174">10.1007/BF01390174</a>, <a href="ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0020-9910">0020-9910</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=0399094">0399094</a>, <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:122887093">122887093</a></cite></li>

<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="ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/1618-1913">1618-1913</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>, <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:16482150">16482150</a></cite></li>

<li><cite id="CITEREFFantechi2001" class="citation cs2">Fantechi, Barbara (2001), <a rel="nofollow" class="external text" href="http://www.mathematik.uni-bielefeld.de/~rehmann/ECM/cdrom/3ecm/pdfs/pant3/fantechi.pdf">"Stacks for everybody"</a> (PDF), <a href="European_Congress_of_Mathematics" title="European Congress of Mathematics">European Congress of Mathematics Volume I</a>, Progr. Math., vol. 201, Basel: Birkhäuser, pp. 349–359, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>3-7643-6417-3</bdi>, <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=1905329">1905329</a></cite></li>

<li><cite id="CITEREFGiraud1964" class="citation cs2"><a href="Jean_Giraud_(mathematician)" title="Jean Giraud (mathematician)">Giraud, Jean</a> (1964), <a rel="nofollow" class="external text" href="http://www.numdam.org/item?id=MSMF_1964__2__R3_0">"Méthode de la descente"</a>, Société Mathématique de France. Bulletin. Supplément. Mémoire, 2: viii+150, <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=0190142">0190142</a></cite></li>

<li><cite id="CITEREFGiraud1966" class="citation cs2">Giraud, Jean (1966), Cohomologie non abélienne de degré 2, thesis, Paris</cite><code class="cs1-code">{{citation}}</code>: CS1 maint: location missing publisher (link)</li>

<li><cite id="CITEREFGiraud1971" class="citation cs2">Giraud, Jean (1971), Cohomologie non abélienne, <a href="Springer_Science%2BBusiness_Media" title="Springer Science+Business Media">Springer</a>, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>3-540-05307-7</bdi></cite></li>

<li><cite id="CITEREFGómez2001" class="citation cs2">Gómez, Tomás L. (2001), "Algebraic stacks", Proceedings - Mathematical Sciences, 111 (1): 1–31, <a href="ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<a rel="nofollow" class="external text" href="https://arxiv.org/abs/math/9911199">math/9911199</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%2FBF02829538">10.1007/BF02829538</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=1818418">1818418</a>, <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:373638">373638</a></cite></li>

<li><cite id="CITEREFGrothendieck1959" class="citation journal cs1">Grothendieck, Alexander (1959). <a rel="nofollow" class="external text" href="http://www.numdam.org/item?id=SB_1958-1960__5__299_0">"Technique de descente et théorèmes d'existence en géométrie algébrique. I. Généralités. Descente par morphismes fidèlement plats"</a>. Séminaire Bourbaki. 5 (Exposé 190).</cite></li>

<li><cite id="CITEREFLaumonMoret-Bailly2000" class="citation cs2">Laumon, Gérard; Moret-Bailly, Laurent (2000), Champs algébriques, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics, vol. 39, Berlin, New York: <a href="Springer-Verlag" class="mw-redirect" title="Springer-Verlag">Springer-Verlag</a>, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-3-540-65761-3</bdi>, <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=1771927">1771927</a></cite> Unfortunately this book uses the incorrect assertion that morphisms of algebraic stacks induce morphisms of lisse-étale topoi. Some of these errors were fixed by <a href="#CITEREFOlsson2007">Olsson (2007)</a>.</li>

<li><cite id="CITEREFLaszloOlsson2008" class="citation cs2">Laszlo, Yves; Olsson, Martin (2008), "The six operations for sheaves on Artin stacks. I. Finite coefficients", Institut des Hautes Études Scientifiques. Publications Mathématiques, 107 (1): 109–168, <a href="ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<a rel="nofollow" class="external text" href="https://arxiv.org/abs/math/0512097">math/0512097</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%2Fs10240-008-0011-6">10.1007/s10240-008-0011-6</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=2434692">2434692</a>, <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:371801">371801</a></cite></li>

<li><cite id="CITEREFMumford1965" class="citation cs2">Mumford, David (1965), <a rel="nofollow" class="external text" href="http://www.mathcs.emory.edu/~brussel/mumford.html">"Picard groups of moduli problems"</a>, in Schilling, O. F. G. (ed.), Arithmetical Algebraic Geometry (Proc. Conf. Purdue Univ., 1963), New York: Harper & Row, pp. 33–81, <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=0201443">0201443</a></cite></li>

<li><cite id="CITEREFOlsson2007" class="citation cs2">Olsson, Martin Christian (2007), Geraschenko, Anton (ed.), <a rel="nofollow" class="external text" href="http://stacky.net/files/written/Stacks/Stacks.pdf">Course notes for Math 274: Stacks</a> (PDF)</cite></li>

<li><cite id="CITEREFOlsson2016" class="citation cs2">Olsson, Martin (2016), Algebraic spaces and stacks, Colloquium Publications, vol. 62, American Mathematical Society, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-1470427986</bdi></cite></li>

<li><cite id="CITEREFVistoli2005" class="citation cs2">Vistoli, Angelo (2005), "Grothendieck topologies, fibered categories and descent theory", Fundamental algebraic geometry, Math. Surveys Monogr., vol. 123, Providence, R.I.: Amer. Math. Soc., pp. 1–104, <a href="ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<a rel="nofollow" class="external text" href="https://arxiv.org/abs/math/0412512">math/0412512</a>, <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2004math.....12512V">2004math.....12512V</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=2223406">2223406</a></cite></li></ul>

<div class="mw-heading mw-heading2"><h2 id="Further_reading">Further reading</h2></div>

<ul><li><cite id="CITEREFMorava2012" class="citation arxiv cs1"><a href="Jack_Morava" title="Jack Morava">Morava, Jack</a> (2012). "Theories of anything". <a href="ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<a rel="nofollow" class="external text" href="https://arxiv.org/abs/1202.0684">1202.0684</a> [<a rel="nofollow" class="external text" href="https://arxiv.org/archive/math.CT">math.CT</a>].</cite></li></ul>

<div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2></div>

<ul><li><a rel="nofollow" class="external text" href="https://ncatlab.org/nlab/show/stack">stack</a> at the <a href="NLab" title="NLab">nLab</a></li>

<li><a rel="nofollow" class="external text" href="https://ncatlab.org/nlab/show/descent">descent</a> at the <a href="NLab" title="NLab">nLab</a></li>

<li><cite id="CITEREFde_Jong" class="citation cs2">de Jong, Aise Johan, <a rel="nofollow" class="external text" href="http://stacks.math.columbia.edu">Stacks Project</a></cite></li>

<li><cite id="CITEREFFulton" class="citation cs2">Fulton, William, <a rel="nofollow" class="external text" href="http://www.msri.org/publications/ln/msri/2002/introstacks/fulton/1/index.html">What is a stack?</a>, MSRI video lecture and notes</cite></li>

<li><cite id="CITEREFToën2007" class="citation cs2">Toën, Bertrand (2007), <a rel="nofollow" class="external text" href="http://ens.math.univ-montp2.fr/~toen/m2.html">Cours de Master 2 : Champs algébriques (2006-2007)</a></cite></li>

<li><a rel="nofollow" class="external text" href="https://mathoverflow.net/q/2124">"Good introductory references on algebraic stacks?"</a></li></ul></div><div><div class="zim-footer">

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