Editing Algebraic stack (section)

{{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>