Editing Algebraic stack

{{short description|Generalization of algebraic spaces or schemes}}
In mathematics, an '''algebraic stack''' is a vast generalization of [[algebraic space]]s, or [[Scheme (mathematics)|schemes]], which are foundational for studying [[moduli theory]]. Many moduli spaces are constructed using techniques specific to algebraic stacks, such as [[Artin's criterion|Artin's representability theorem]], which is used to construct the [[Moduli space of algebraic curves|moduli space of pointed algebraic curves]] <math>\mathcal{M}_{g,n}</math> and the [[moduli stack of elliptic curves]]. Originally, they were introduced by [[Alexander Grothendieck]]<ref>{{cite arXiv|last1=A'Campo|first1=Norbert|last2=Ji|first2=Lizhen|last3=Papadopoulos|first3=Athanase|date=2016-03-07|title=On Grothendieck's construction of Teichmüller space|class=math.GT|eprint=1603.02229}}</ref> to keep track of automorphisms on moduli spaces, a technique which allows for treating these moduli spaces as if their underlying schemes or algebraic spaces are [[Smooth scheme|smooth]]. After Grothendieck developed the general theory of [[Descent (mathematics)|descent]],<ref>{{cite arXiv|last1=Grothendieck|first1=Alexander|last2=Raynaud|first2=Michele|date=2004-01-04|title=Revêtements étales et groupe fondamental (SGA 1). Expose VI: Catégories fibrées et descente|eprint=math.AG/0206203}}</ref> and [[Jean Giraud (mathematician)|Giraud]] the general theory of stacks,<ref>{{cite book |doi=10.1007/978-3-662-62103-5|chapter=II. Les champs|title=Cohomologie non abélienne|series=Grundlehren der mathematischen Wissenschaften|year=1971|last1=Giraud|first1=Jean|volume=179|pages=64–105|isbn=978-3-540-05307-1}}</ref> the notion of algebraic stacks was defined by [[Michael Artin]].<ref name=":0">{{Cite journal|last=Artin|first=M.|date=1974|title=Versal deformations and algebraic stacks|url=https://eudml.org/doc/142310|journal=Inventiones Mathematicae|volume=27|issue=3|pages=165–189|doi=10.1007/bf01390174|bibcode=1974InMat..27..165A|s2cid=122887093|issn=0020-9910}}</ref>

== Definition ==

=== Motivation ===
One of the motivating examples of an algebraic stack is to consider a [[groupoid scheme]] <math>(R,U,s,t,m)</math> over a fixed scheme <math>S</math>. For example, if <math>R = \mu_n\times_S\mathbb{A}^n_S</math> (where <math>\mu_n</math> is the [[group scheme]] of roots of unity), <math>U = \mathbb{A}^n_S</math>, <math>s = \text{pr}_U</math> is the projection map, <math>t</math> is the group action<blockquote><math>\zeta_n \cdot (x_1,\ldots, x_n)=(\zeta_n  x_1,\ldots,\zeta_n x_n)</math></blockquote>and <math>m</math> is the multiplication map<blockquote><math>m: (\mu_n\times_S \mathbb{A}^n_S)\times_{\mu_n\times_S \mathbb{A}^n_S} (\mu_n\times_S \mathbb{A}^n_S) \to \mu_n\times_S \mathbb{A}^n_S</math></blockquote>on <math>\mu_n</math>. Then, given an <math>S</math>-scheme <math>\pi:X\to S</math>, the groupoid scheme <math>(R(X),U(X),s,t,m)</math> forms a groupoid (where <math>R,U</math> are their associated functors). Moreover, this construction is functorial on <math>(\mathrm{Sch}/S)</math> forming a contravariant [[2-functor]]<blockquote><math>(R(-),U(-),s,t,m): (\mathrm{Sch}/S)^\mathrm{op} \to \text{Cat}</math></blockquote>where <math>\text{Cat}</math> is the [[2 category|2-category]] of [[Small category|small categories]]. Another way to view this is as a [[fibred category]] <math>[U/R] \to (\mathrm{Sch}/S)</math> through the [[Grothendieck construction]]. Getting the correct technical conditions, such as the [[Grothendieck topology]] on <math>(\mathrm{Sch}/S)</math>, gives the definition of an algebraic stack. For instance, in the associated groupoid of <math>k</math>-points for a field <math>k</math>, over the origin object <math>0 \in \mathbb{A}^n_S(k)</math> there is the groupoid of automorphisms <math>\mu_n(k)</math>. However, in order to get an algebraic stack from <math>[U/R]</math>, and not just a stack, there are additional technical hypotheses required for <math>[U/R]</math>.<ref>{{Cite web|title=Section 92.16 (04T3): From an algebraic stack to a presentation—The Stacks project|url=https://stacks.math.columbia.edu/tag/04T3|access-date=2020-08-29|website=stacks.math.columbia.edu}}</ref>

=== Algebraic stacks ===
It turns out using the [[Fppf topology|fppf-topology]]<ref>{{Cite web|title=Section 34.7 (021L): The fppf topology—The Stacks project|url=https://stacks.math.columbia.edu/tag/021L|access-date=2020-08-29|website=stacks.math.columbia.edu}}</ref> (faithfully flat and locally of finite presentation) on <math>(\mathrm{Sch}/S)</math>, denoted <math>(\mathrm{Sch}/S)_{fppf}</math>, forms the basis for defining algebraic stacks. Then, an '''algebraic stack'''<ref>{{Cite web|title=Section 92.12 (026N): Algebraic stacks—The Stacks project|url=https://stacks.math.columbia.edu/tag/026N|access-date=2020-08-29|website=stacks.math.columbia.edu}}</ref> is a fibered category<blockquote><math>p: \mathcal{X} \to (\mathrm{Sch}/S)_{fppf}</math></blockquote>such that

# <math>\mathcal{X}</math> is a [[category fibered in groupoids]], meaning the [[overcategory]] for some <math>\pi:X\to S</math> is a groupoid
# The diagonal map <math>\Delta:\mathcal{X} \to \mathcal{X}\times_S\mathcal{X}</math> of fibered categories is representable as algebraic spaces
# There exists an <math>fppf</math> scheme <math>U \to S</math> and an associated 1-morphism of fibered categories <math>\mathcal{U} \to \mathcal{X}</math> which is surjective and smooth called an '''atlas'''.

==== Explanation of technical conditions ====

===== Using the fppf topology =====
First of all, the fppf-topology is used because it behaves well with respect to [[Descent theory|descent]]. For example, if there are schemes <math>X,Y \in \operatorname{Ob}(\mathrm{Sch}/S)</math> and <math>X \to Y</math>can be refined to an fppf-cover of <math>Y</math>, if <math>X</math> is flat, locally finite type, or locally of finite presentation, then <math>Y</math> has this property.<ref>{{Cite web|title=Lemma 35.11.8 (06NB)—The Stacks project|url=https://stacks.math.columbia.edu/tag/06NB|access-date=2020-08-29|website=stacks.math.columbia.edu}}</ref> this kind of idea can be extended further by considering properties local either on the target or the source of a morphism <math>f:X\to Y</math>. For a cover <math>\{X_i \to X\}_{i \in I}</math> we say a property <math>\mathcal{P}</math> is '''local on the source''' if<blockquote><math>f:X\to Y</math> has <math>\mathcal{P}</math> if and only if each <math>X_i \to Y</math> has <math>\mathcal{P}</math>.</blockquote>There is an analogous notion on the target called '''local on the target'''. This means given a cover <math>\{Y_i \to Y \}_{i \in I}</math><blockquote><math>f:X\to Y</math> has <math>\mathcal{P}</math> if and only if each <math>X\times_YY_i \to Y_i</math> has <math>\mathcal{P}</math>.</blockquote>For the fppf topology, having an immersion is local on the target.<ref>{{Cite web|title=Section 35.21 (02YL): Properties of morphisms local in the fppf topology on the target—The Stacks project|url=https://stacks.math.columbia.edu/tag/02YL|access-date=2020-08-29|website=stacks.math.columbia.edu}}</ref> In addition to the previous properties local on the source for the fppf topology, <math>f</math> being universally open is also local on the source.<ref>{{Cite web|title=Section 35.25 (036M): Properties of morphisms local in the fppf topology on the source—The Stacks project|url=https://stacks.math.columbia.edu/tag/036M|access-date=2020-08-29|website=stacks.math.columbia.edu}}</ref> Also, being locally Noetherian and Jacobson are local on the source and target for the fppf topology.<ref>{{Cite web|title=Section 35.13 (034B): Properties of schemes local in the fppf topology—The Stacks project|url=https://stacks.math.columbia.edu/tag/034B|access-date=2020-08-29|website=stacks.math.columbia.edu}}</ref> This does not hold in the fpqc topology, making it not as "nice" in terms of technical properties. Even though this is true, using algebraic stacks over the fpqc topology still has its use, such as in [[chromatic homotopy theory]]. This is because the [[Moduli stack of formal group laws]] <math>\mathcal{M}_{fg}</math> is an fpqc-algebraic stack<ref>{{Cite web|last=Goerss|first=Paul|title=Quasi-coherent sheaves on the Moduli Stack of Formal Groups|url=https://sites.math.northwestern.edu/~pgoerss/papers/modfg.pdf|url-status=live|archive-url=https://web.archive.org/web/20200829022756/https://sites.math.northwestern.edu/~pgoerss/papers/modfg.pdf|archive-date=29 August 2020}}</ref><sup>pg 40</sup>.

===== Representable diagonal =====
By definition, a 1-morphism <math>f:\mathcal{X} \to \mathcal{Y}</math> of categories fibered in groupoids is '''representable by algebraic spaces'''<ref>{{Cite web|title=Section 92.9 (04SX): Morphisms representable by algebraic spaces—The Stacks project|url=https://stacks.math.columbia.edu/tag/04SX|access-date=2020-08-29|website=stacks.math.columbia.edu}}</ref> if for any fppf morphism <math>U \to S</math> of schemes and any 1-morphism <math>y: (Sch/U)_{fppf} \to \mathcal{Y}</math>, the associated category fibered in groupoids<blockquote><math>(Sch/U)_{fppf}\times_{\mathcal{Y}} \mathcal{X}</math></blockquote>is '''representable as an algebraic space''',<ref>{{Cite web|title=Section 92.7 (04SU): Split categories fibred in groupoids—The Stacks project|url=https://stacks.math.columbia.edu/tag/04SU|access-date=2020-08-29|website=stacks.math.columbia.edu}}</ref><ref>{{Cite web|title=Section 92.8 (02ZV): Categories fibred in groupoids representable by algebraic spaces—The Stacks project|url=https://stacks.math.columbia.edu/tag/02ZV|access-date=2020-08-29|website=stacks.math.columbia.edu}}</ref> meaning there exists an algebraic space<blockquote><math>F:(Sch/S)^{op}_{fppf} \to Sets</math></blockquote>such that the associated fibered category <math>\mathcal{S}_F \to (Sch/S)_{fppf}</math><ref><math>Sets \to Cat</math> is the embedding sending a set <math>S</math> to the category of objects <math>S</math> and only identity morphisms. Then, the Grothendieck construction can be applied to give a category fibered in groupoids</ref> is equivalent to <math>(Sch/U)_{fppf}\times_{\mathcal{Y}} \mathcal{X}</math>. There are a number of equivalent conditions for representability of the diagonal<ref>{{Cite web|title=Lemma 92.10.11 (045G)—The Stacks project|url=https://stacks.math.columbia.edu/tag/045G|access-date=2020-08-29|website=stacks.math.columbia.edu}}</ref> which help give intuition for this technical condition, but one of main motivations is the following: for a scheme <math>U</math> and objects <math>x, y \in \operatorname{Ob}(\mathcal{X}_U)</math> the sheaf <math>\operatorname{Isom}(x,y)</math> is representable as an algebraic space. In particular, the stabilizer group for any point on the stack <math>x : \operatorname{Spec}(k) \to \mathcal{X}_{\operatorname{Spec}(k)}</math> is representable as an algebraic space.

Another important equivalence of having a representable diagonal is the technical condition that the intersection of any two algebraic spaces in an algebraic stack is an algebraic space. Reformulated using fiber products<blockquote><math>\begin{matrix}
Y \times_{\mathcal{X}}Z & \to & Y \\
\downarrow & & \downarrow \\
Z & \to & \mathcal{X}
\end{matrix}</math></blockquote>the representability of the diagonal is equivalent to <math>Y \to \mathcal{X}</math> being representable for an algebraic space <math>Y</math>. This is because given morphisms <math>Y \to \mathcal{X}, Z \to \mathcal{X}</math> from algebraic spaces, they extend to maps <math>\mathcal{X}\times\mathcal{X}</math> from the diagonal map. There is an analogous statement for algebraic spaces which gives representability of a sheaf on <math>(F/S)_{fppf}</math> as an algebraic space.<ref>{{Cite web|title=Section 78.5 (046I): Bootstrapping the diagonal—The Stacks project|url=https://stacks.math.columbia.edu/tag/046I|access-date=2020-08-29|website=stacks.math.columbia.edu}}</ref>

Note that an analogous condition of representability of the diagonal holds for some formulations of [[higher stacks]]<ref>{{cite arXiv|last=Simpson|first=Carlos|date=1996-09-17|title=Algebraic (geometric) ''n''-stacks|eprint=alg-geom/9609014}}</ref> where the fiber product is an <math>(n-1)</math>-stack for an <math>n</math>-stack <math>\mathcal{X}</math>.

==== Surjective and smooth atlas ====

===== 2-Yoneda lemma =====
The existence of an <math>fppf</math> scheme <math>U \to S</math> and a 1-morphism of fibered categories <math>\mathcal{U} \to \mathcal{X}</math> which is surjective and smooth depends on defining a smooth and surjective morphisms of fibered categories. Here <math>\mathcal{U}</math> is the algebraic stack from the representable functor <math>h_U</math> on <math>h_U: (Sch/S)_{fppf}^{op} \to Sets</math> upgraded to a category fibered in groupoids where the categories only have trivial morphisms. This means the set<blockquote><math>h_U(T) = \text{Hom}_{(Sch/S)_{fppf}}(T,U)</math></blockquote>is considered as a category, denoted <math>h_\mathcal{U}(T)</math>, with objects in <math>h_U(T)</math> as <math>fppf</math> morphisms<blockquote><math>f:T \to U</math></blockquote>and morphisms are the identity morphism. Hence<blockquote><math>h_{\mathcal{U}}:(Sch/S)_{fppf}^{op} \to Groupoids</math></blockquote>is a 2-functor of groupoids. Showing this 2-functor is a sheaf is the content of the [[2-Yoneda lemma]]. Using the Grothendieck construction, there is an associated category fibered in groupoids denoted <math>\mathcal{U} \to \mathcal{X}</math>.

===== Representable morphisms of categories fibered in groupoids =====
To say this morphism <math>\mathcal{U} \to \mathcal{X}</math> is smooth or surjective, we have to introduce representable morphisms.<ref>{{Cite web|title=Section 92.6 (04ST): Representable morphisms of categories fibred in groupoids—The Stacks project|url=https://stacks.math.columbia.edu/tag/04ST|access-date=2020-10-03|website=stacks.math.columbia.edu}}</ref> A morphism <math>p:\mathcal{X} \to \mathcal{Y}</math> of categories fibered in groupoids over <math>(Sch/S)_{fppf}</math> is said to be representable if given an object <math>T \to S</math> in <math>(Sch/S)_{fppf}</math> and an object <math>t \in \text{Ob}(\mathcal{Y}_T)</math> the [[2-fibered product]] <blockquote>'''<math>(Sch/T)_{fppf}\times_{t,\mathcal{Y}} \mathcal{X}_T</math>'''</blockquote>is representable by a scheme. Then, we can say the morphism of categories fibered in groupoids <math>p</math> is '''smooth and surjective''' if the associated morphism<blockquote>'''<math>(Sch/T)_{fppf}\times_{t,\mathcal{Y}} \mathcal{X}_T \to (Sch/T)_{fppf}</math>'''</blockquote>of schemes is smooth and surjective.

=== Deligne–Mumford stacks ===
Algebraic stacks, also known as '''Artin stacks''', are by definition equipped with a smooth surjective atlas <math>\mathcal{U} \to \mathcal{X}</math>, where <math>\mathcal{U}</math> is the stack associated to some scheme <math>U \to S</math>. If the atlas <math>\mathcal{U}\to \mathcal{X}</math> is moreover étale, then <math>\mathcal{X}</math> is said to be a '''[[Deligne–Mumford stack]]'''. The subclass of Deligne-Mumford stacks is useful because it provides the correct setting for many natural stacks considered, such as the [[moduli stack of algebraic curves]]. In addition, they are strict enough that object represented by <u>points in Deligne-Mumford stacks do not have [[infinitesimal]] automorphisms</u>. This is very important because infinitesimal automorphisms make studying the deformation theory of Artin stacks very difficult. For example, the deformation theory of the Artin stack <math>BGL_n = [*/GL_n]</math>, the moduli stack of rank <math>n</math> vector bundles, has infinitesimal automorphisms controlled partially by the [[Lie algebra]] <math>\mathfrak{gl}_n</math>. This leads to an infinite sequence of deformations and obstructions in general, which is one of the motivations for studying [[Moduli space of stable bundles|moduli of stable bundles]]. Only in the special case of the [[deformation theory of line bundles]] <math>[*/GL_1] = [*/\mathbb{G}_m]</math> is the deformation theory tractable, since the associated Lie algebra is [[Abelian Lie algebra|abelian]].

Note that many stacks cannot be naturally represented as Deligne-Mumford stacks because it only allows for finite covers, or, algebraic stacks with finite covers. Note that because every Etale cover is flat and locally of finite presentation, algebraic stacks defined with the fppf-topology subsume this theory; but, it is still useful since many stacks found in nature are of this form, such as the [[Moduli of algebraic curves|moduli of curves]] <math>\mathcal{M}_g</math>. Also, the differential-geometric analogue of such stacks are called [[orbifold]]s. The Etale condition implies the 2-functor<blockquote><math>B\mu_n:(\mathrm{Sch}/S)^\text{op} \to \text{Cat}</math></blockquote>sending a scheme to its groupoid of <math>\mu_n</math>-[[Torsor (algebraic geometry)|torsors]] is representable as a stack over the Etale topology, but the Picard-stack <math>B\mathbb{G}_m</math> of <math>\mathbb{G}_m</math>-torsors (equivalently the category of line bundles) is not representable. Stacks of this form are representable as stacks over the fppf-topology.

Another reason for considering the fppf-topology versus the etale topology is over characteristic <math>p</math> the [[Kummer sequence]]<blockquote><math>0 \to \mu_p \to \mathbb{G}_m \to \mathbb{G}_m \to 0</math></blockquote>is exact only as a sequence of fppf sheaves, but not as a sequence of etale sheaves.

=== Defining algebraic stacks over other topologies ===
Using other Grothendieck topologies on <math>(F/S)</math> gives alternative theories of algebraic stacks which are either not general enough, or don't behave well with respect to exchanging properties from the base of a cover to the total space of a cover. It is useful to recall there is the following hierarchy of generalization<blockquote><math>\text{fpqc} \supset \text{fppf} \supset \text{smooth} \supset \text{etale} \supset \text{Zariski}</math></blockquote>of big topologies on <math>(F/S)</math>.

== Structure sheaf ==
The structure sheaf of an algebraic stack is an object pulled back from a universal structure sheaf <math>\mathcal{O}</math> on the site <math>(Sch/S)_{fppf}</math>.<ref>{{Cite web|title=Section 94.3 (06TI): Presheaves—The Stacks project|url=https://stacks.math.columbia.edu/tag/06TI|access-date=2020-10-01|website=stacks.math.columbia.edu}}</ref> This '''universal structure sheaf'''<ref>{{Cite web|title=Section 94.6 (06TU): The structure sheaf—The Stacks project|url=https://stacks.math.columbia.edu/tag/06TU|access-date=2020-10-01|website=stacks.math.columbia.edu}}</ref> is defined as<blockquote><math>\mathcal{O}:(Sch/S)_{fppf}^{op} \to Rings, \text{ where } U/X \mapsto \Gamma(U,\mathcal{O}_U)</math></blockquote>and the associated structure sheaf on a category fibered in groupoids<blockquote><math>p:\mathcal{X} \to (Sch/S)_{fppf}</math></blockquote>is defined as<blockquote><math>\mathcal{O}_\mathcal{X} := p^{-1}\mathcal{O}</math></blockquote>where <math>p^{-1}</math> comes from the map of Grothendieck topologies. In particular, this means is <math>x \in \text{Ob}(\mathcal{X})</math> lies over <math>U</math>, so <math>p(x) = U</math>, then <math>\mathcal{O}_\mathcal{X}(x)=\Gamma(U,\mathcal{O}_U)</math>. As a sanity check, it's worth comparing this to a category fibered in groupoids coming from an <math>S</math>-scheme <math>X</math> for various topologies.<ref>{{Cite web|title=Section 94.8 (076N): Representable categories—The Stacks project|url=https://stacks.math.columbia.edu/tag/076N|access-date=2020-10-01|website=stacks.math.columbia.edu}}</ref> For example, if <blockquote><math>(\mathcal{X}_{Zar},\mathcal{O}_\mathcal{X}) = ((Sch/X)_{Zar}, \mathcal{O}_X)</math></blockquote>is a category fibered in groupoids over <math>(Sch/S)_{fppf}</math>, the structure sheaf for an open subscheme <math>U \to X</math> gives<blockquote><math>\mathcal{O}_\mathcal{X}(U) = \mathcal{O}_X(U) = \Gamma(U,\mathcal{O}_X)</math></blockquote>so this definition recovers the classic structure sheaf on a scheme. Moreover, for a [[quotient stack]] <math>\mathcal{X} = [X/G]</math>, the structure sheaf this just gives the <math>G</math>-invariant sections<blockquote><math>\mathcal{O}_{\mathcal{X}}(U) = \Gamma(U,u^*\mathcal{O}_X)^{G}</math></blockquote>for <math>u:U\to X</math> in <math>(Sch/S)_{fppf}</math>.<ref>{{Cite web|title=Lemma 94.13.2 (076S)—The Stacks project|url=https://stacks.math.columbia.edu/tag/076S|access-date=2020-10-01|website=stacks.math.columbia.edu}}</ref><ref>{{Cite web|title=Section 76.12 (0440): Quasi-coherent sheaves on groupoids—The Stacks project|url=https://stacks.math.columbia.edu/tag/0440|access-date=2020-10-01|website=stacks.math.columbia.edu}}</ref>

== Examples ==

=== Classifying stacks ===
{{See also|Quotient stack}}
Many classifying stacks for [[Algebraic group|algebraic groups]] are algebraic stacks. In fact, for an algebraic group space <math>G</math> over a scheme <math>S</math> which is flat of finite presentation, the stack <math>BG</math> is algebraic<ref name=":0" /><sup>theorem 6.1</sup>.

== See also ==

* [[Gerbe]]
* [[Chow group of a stack]]
* [[Cohomology of a stack]]
* [[Quotient stack]]
* [[Sheaf on an algebraic stack]]
* [[Toric stack]]
* [[Artin's criterion]]
* [[Pursuing Stacks]]
* [[Derived algebraic geometry]]

== References ==
<references />

== External links ==
=== Artin's Axioms ===
* https://stacks.math.columbia.edu/tag/07SZ - Look at "Axioms" and "Algebraic stacks"
* [https://web.archive.org/web/20201001012450/https://sites.math.washington.edu/~jarod/papers/mainz.pdf Artin Algebraization and Quotient Stacks - Jarod Alper]

=== Papers ===

* {{cite web |last=Alper |first=Jarod |year=2009 |title=A Guide to the Literature on Algebraic Stacks |url=https://pdfs.semanticscholar.org/816b/67f2a8163533db5720b9fa608f0bb9d8d3b0.pdf |archive-url=https://web.archive.org/web/20200213125356/https://pdfs.semanticscholar.org/816b/67f2a8163533db5720b9fa608f0bb9d8d3b0.pdf |url-status=dead |archive-date=2020-02-13 |s2cid=51803452}}
* {{cite journal |arxiv=1011.5484|doi=10.1016/j.aim.2013.12.002|doi-access=free|title=The Hilbert stack|year=2014|last1=Hall|first1=Jack|last2=Rydh|first2=David|journal=[[Advances in Mathematics]]|volume=253|pages=194–233|s2cid=55936583}}
* {{cite journal |last= Behrend |first=Kai A.|year=2003 |title=Derived ℓ-Adic Categories for Algebraic Stacks |url=http://www.math.ubc.ca/~behrend/ladic.pdf |journal=Memoirs of the American Mathematical Society |volume=163 |issue=774 |pages=1–93 |doi=10.1090/memo/0774|isbn= 978-1-4704-0372-0}}

=== Applications ===

* {{cite arXiv |eprint=1404.6416|last1=Lafforgue|first1=Vincent|title=Introduction to chtoucas for reductive groups and to the global Langlands parameterization|year=2014|class=math.AG}}
* {{cite book |doi=10.1007/978-3-540-37855-6_4|chapter=Les Schémas de Modules de Courbes Elliptiques|title=Modular Functions of One Variable II|series=Lecture Notes in Mathematics|year=1973|last1=Deligne|first1=P.|last2=Rapoport|first2=M.|volume=349|pages=143–316|isbn=978-3-540-06558-6}}
* {{cite journal |url=https://www.mscand.dk/article/view/12001 |title=The projectivity of the moduli space of stable curves, II: The stacks <math>\mathcal{M}_{g,n}</math> |doi=10.7146/math.scand.a-12001 |year=1983|last1=Knudsen|first1=Finn F.|journal=Mathematica Scandinavica|volume=52|page=161|doi-access=free}}
* {{cite arXiv |eprint=1911.00250|last1=Jiang|first1=Yunfeng|title=On the construction of moduli stack of projective Higgs bundles over surfaces|year=2019|class=math.AG}}

=== Other ===

* [https://stacks.math.columbia.edu/tag/04SL Examples of Stacks]
* [[arxiv:math/0412512|Notes on Grothendieck topologies, fibered categories and descent theory]]
* [https://web.archive.org/web/20200801212411/https://folk.uio.no/fredrme/algstacks.pdf Notes on algebraic stacks]

[[Category:Algebraic curves]]
[[Category:Moduli theory]]
[[Category:Algebraic geometry]]