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In mathematics, an algebraic stack is a vast generalization of algebraic spaces, or schemes, which are foundational for studying moduli theory. Many moduli spaces are constructed using techniques specific to algebraic stacks, such as Artin's representability theorem, which is used to construct the moduli space of pointed algebraic curves and the moduli stack of elliptic curves. Originally, they were introduced by Grothendieck 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. But, through many generalizations the notion of algebraic stacks was finally discovered by Michael Artin.

Attributes	Values
rdf:type	Thing
rdfs:label	Algebraic stack (en) Champ algébrique (fr) 代数的スタック (ja)
rdfs:comment	In mathematics, an algebraic stack is a vast generalization of algebraic spaces, or schemes, which are foundational for studying moduli theory. Many moduli spaces are constructed using techniques specific to algebraic stacks, such as Artin's representability theorem, which is used to construct the moduli space of pointed algebraic curves and the moduli stack of elliptic curves. Originally, they were introduced by Grothendieck 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. But, through many generalizations the notion of algebraic stacks was finally discovered by Michael Artin. (en) En mathématiques, un champ algébrique est une catégorie généralisant la notion de schéma ; elle permet notamment l'étude des actions de groupes lorsqu'elles ne sont pas libres. (fr) 代数スタックとは、モジュライ理論の研究の基礎となる代数空間またはスキームの一般化である。多くのモジュライ空間は、アルティンの表現可能定理など、代数スタック固有の手法を駆使して構築される。これは、尖った代数曲線のモジュライ空間の構築に使用される。は楕円曲線のモジュラススタックで、それらはモジュライ空間の自己同型を追跡するためにはグロタンディークにより導入された。これは、モジュライ空間を基礎とするスキームや代数空間が滑らかであるかのように扱うことを可能とする。多くの一般化を通じ、代数スタックの概念がついにアルティンにより発見された。 (ja)
rdfs:seeAlso	Quotient stack
dcterms:subject	Moduli theory Algebraic curves Algebraic geometry
Wikipage page ID	717377 (xsd:integer)
Wikipage revision ID	1068676415 (xsd:integer)
Link from a Wikipage to another Wikipage	Algebraic space Moduli theory Derived algebraic geometry Quotient stack Pursuing Stacks Lie algebra Chow group of a stack Chromatic homotopy theory Gerbe 2-functor Torsor (algebraic geometry) Abelian Lie algebra Algebraic curves Algebraic geometry Advances in Mathematics Cohomology of a stack Moduli of algebraic curves Moduli stack of elliptic curves Moduli stack of formal group laws Small category Artin's criterion Fibred category Grothendieck construction Grothendieck topology Groupoid scheme Michael Artin Orbifold Scheme (mathematics) Toric stack Smooth scheme Sheaf on an algebraic stack Overcategory Fppf topology Category fibered in groupoids Descent theory 2 category Moduli space of algebraic curves Moduli stack of algebraic curves Moduli theory Deligne-Mumford stack dbr:2-Yoneda_lemma dbr:2-fibered_product dbr:Arxiv:math/0412512 dbr:Deformation_theory_of_line_bundles dbr:Higher_stacks dbr:Kummer_sequence dbr:Moduli_space_of_stable_bundles
Link from a Wikipage to an external page	https://stacks.math.columbia.edu/tag/04SL https://www.mscand.dk/article/view/12001 http://www.math.ubc.ca/~behrend/ladic.pdf https://stacks.math.columbia.edu/tag/07SZ https://web.archive.org/web/20200213125356/https:/pdfs.semanticscholar.org/816b/67f2a8163533db5720b9fa608f0bb9d8d3b0.pdf https://web.archive.org/web/20200801212411/https:/folk.uio.no/fredrme/algstacks.pdf https://web.archive.org/web/20201001012450/https:/sites.math.washington.edu/~jarod/papers/mainz.pdf https://pdfs.semanticscholar.org/816b/67f2a8163533db5720b9fa608f0bb9d8d3b0.pdf https://mathoverflow.net/questions/15082/fpqc-covers-of-stacks%3Frq=1 https://mathoverflow.net/questions/200990/do-algebraic-stacks-satisfy-fpqc-descent https://mathoverflow.net/questions/29268/stacks-in-the-fpqc-topology
sameAs	Algebraic stack Algebraic stack Algebraic stack Algebraic stack Algebraic stack
dbp:wikiPageUsesTemplate	dbt:Cite_arXiv dbt:Cite_book dbt:Cite_journal dbt:See_also dbt:Short_description
has abstract	In mathematics, an algebraic stack is a vast generalization of algebraic spaces, or schemes, which are foundational for studying moduli theory. Many moduli spaces are constructed using techniques specific to algebraic stacks, such as Artin's representability theorem, which is used to construct the moduli space of pointed algebraic curves and the moduli stack of elliptic curves. Originally, they were introduced by Grothendieck 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. But, through many generalizations the notion of algebraic stacks was finally discovered by Michael Artin. (en) En mathématiques, un champ algébrique est une catégorie généralisant la notion de schéma ; elle permet notamment l'étude des actions de groupes lorsqu'elles ne sont pas libres. (fr) 代数スタックとは、モジュライ理論の研究の基礎となる代数空間またはスキームの一般化である。多くのモジュライ空間は、アルティンの表現可能定理など、代数スタック固有の手法を駆使して構築される。これは、尖った代数曲線のモジュライ空間の構築に使用される。は楕円曲線のモジュラススタックで、それらはモジュライ空間の自己同型を追跡するためにはグロタンディークにより導入された。これは、モジュライ空間を基礎とするスキームや代数空間が滑らかであるかのように扱うことを可能とする。多くの一般化を通じ、代数スタックの概念がついにアルティンにより発見された。 (ja)
prov:wasDerivedFrom	wikipedia-en:Algebraic_stack?oldid=1068676415&ns=0
page length (characters) of wiki page	23872 (xsd:nonNegativeInteger)
foaf:isPrimaryTopicOf	wikipedia-en:Algebraic_stack
is Link from a Wikipage to another Wikipage of	Behrend's trace formula List of algebraic geometry topics Morphism of algebraic stacks David Mumford ELSV formula Lie algebroid List of incomplete proofs List of online encyclopedias Smooth topology Quasi-separated morphism Glossary of algebraic geometry Compact object (mathematics) Deligne–Mumford stack

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