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Derived algebraic geometry is a branch of mathematics that generalizes algebraic geometry to a situation where commutative rings, which provide local charts, are replaced by either differential graded algebras (over ), simplicial commutative rings or -ring spectra from algebraic topology, whose higher homotopy groups account for the non-discreteness (e.g., Tor) of the structure sheaf. Grothendieck's scheme theory allows the structure sheaf to carry nilpotent elements. Derived algebraic geometry can be thought of as an extension of this idea, and provides natural settings for intersection theory (or motivic homotopy theory) of singular algebraic varieties and cotangent complexes in deformation theory (cf. J. Francis), among the other applications.

Attributes	Values
rdf:type	organisation
rdfs:label	Derived algebraic geometry (en) 導来代数幾何学 (ja)
rdfs:comment	Derived algebraic geometry is a branch of mathematics that generalizes algebraic geometry to a situation where commutative rings, which provide local charts, are replaced by either differential graded algebras (over ), simplicial commutative rings or -ring spectra from algebraic topology, whose higher homotopy groups account for the non-discreteness (e.g., Tor) of the structure sheaf. Grothendieck's scheme theory allows the structure sheaf to carry nilpotent elements. Derived algebraic geometry can be thought of as an extension of this idea, and provides natural settings for intersection theory (or motivic homotopy theory) of singular algebraic varieties and cotangent complexes in deformation theory (cf. J. Francis), among the other applications. (en) 導来代数幾何学は、代数幾何学を、局所座標を与える可換環を（上の）次数付き微分代数に置き換えることで一般化する数学の一分野である。単純可換環または --代数的トポロジーからの環スペクトルとなる。その高次ホモトピー群は、構造層の非離散性（Torなど）を説明する。グロタンディークのスキーム理論は、構造層を冪零元へ運ぶことを可能にする。導来代数幾何学はこの考えの拡張と考えることができ、他の応用として、変形理論における特異代数多様体と非特異代数多様体の交叉理論（またはモチヴィックホモトピー理論）の自然な導出を引き起こす。（cf. J. Francis） (ja)
dct:subject	Ring theory Algebraic topology Algebraic geometry Homotopical algebra Scheme theory
Wikipage page ID	40778281 (xsd:integer)
Wikipage revision ID	1100520897 (xsd:integer)
Link from a Wikipage to another Wikipage	Quasi-coherent sheaf En-ring Motivic homotopy theory Deformation theory Algebraic K-theory Algebraic topology David Ben-Zvi ∞-topos Derivator Derived category Derived functor Derived scheme Derived stack Derived tensor product Intersection number Weibel's conjecture Ring theory Geometric Langlands correspondence Pursuing Stacks Étale spectrum Commutative ring Dennis Gaitsgory Triangulated category Algebraic geometry Algebraic topology Nilpotent element Intersection theory Cotangent complex Simplicial commutative ring Abelian category Algebraic geometry Homotopical algebra Eilenberg–MacLane space Higher Topos Theory Highly structured ring spectrum Homotopy hypothesis Tor functor Differential graded algebra Scheme theory Scheme theory Algebra over an operad Scheme-theoretic intersection Noncommutative algebraic geometry Stable ∞-category Serre's intersection formula ∞-category Étale Spec dbr:Arxiv:1111.4234 dbr:Arxiv:1208.6325 dbr:Arxiv:1803.09476 dbr:Arxiv:math/0703214
Link from a Wikipage to an external page	https://indico.math.cnrs.fr/event/2518/attachments/754/844/09102013a.pdf https://www.ams.org/jams/2010-23-04/S0894-0347-10-00669-7/S0894-0347-10-00669-7.pdf https://www.ams.org/notices/201107/rtx110700955p.pdf https://mathoverflow.net/q/219361 https://mathoverflow.net/q/30396 https://mathoverflow.net/questions/229022/why-do-people-say-dg-algebras-behave-badly-in-positive-characteristic http://www.math.uchicago.edu/~may/PAPERS/Esheaf.pdf http://math.uchicago.edu/~amathew/dag.html http://www.math.northwestern.edu/~jnkf/writ/thezrev.pdf https://arxiv.org/abs/1104.0181 https://web.archive.org/web/20140413142221/http:/www-personal.umich.edu/~erman/DAG.html https://web.archive.org/web/20200425203513/https:/faculty.math.illinois.edu/~rezk/sag-chapter.pdf http://ncatlab.org/nlab/show/derived+algebraic+geometry http://www.math.harvard.edu/~lurie/
sameAs	Derived algebraic geometry Derived algebraic geometry Derived algebraic geometry Derived algebraic geometry
dbp:wikiPageUsesTemplate	dbt:Citation dbt:Cite_arXiv dbt:Cite_book dbt:Cite_journal dbt:Harvtxt dbt:Reflist
has abstract	Derived algebraic geometry is a branch of mathematics that generalizes algebraic geometry to a situation where commutative rings, which provide local charts, are replaced by either differential graded algebras (over ), simplicial commutative rings or -ring spectra from algebraic topology, whose higher homotopy groups account for the non-discreteness (e.g., Tor) of the structure sheaf. Grothendieck's scheme theory allows the structure sheaf to carry nilpotent elements. Derived algebraic geometry can be thought of as an extension of this idea, and provides natural settings for intersection theory (or motivic homotopy theory) of singular algebraic varieties and cotangent complexes in deformation theory (cf. J. Francis), among the other applications. (en) 導来代数幾何学は、代数幾何学を、局所座標を与える可換環を（上の）次数付き微分代数に置き換えることで一般化する数学の一分野である。単純可換環または --代数的トポロジーからの環スペクトルとなる。その高次ホモトピー群は、構造層の非離散性（Torなど）を説明する。グロタンディークのスキーム理論は、構造層を冪零元へ運ぶことを可能にする。導来代数幾何学はこの考えの拡張と考えることができ、他の応用として、変形理論における特異代数多様体と非特異代数多様体の交叉理論（またはモチヴィックホモトピー理論）の自然な導出を引き起こす。（cf. J. Francis） (ja)
gold:hypernym	Branch
prov:wasDerivedFrom	wikipedia-en:Derived_algebraic_geometry?oldid=1100520897&ns=0
page length (characters) of wiki page	14310 (xsd:nonNegativeInteger)
foaf:isPrimaryTopicOf	wikipedia-en:Derived_algebraic_geometry
is Link from a Wikipage to another Wikipage of	Algebraic stack Bertrand Toën Derived category Derived scheme Derived stack Double complex Kähler differential

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