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In mathematics, the derived category D(A) of an abelian category A is a construction of homological algebra introduced to refine and in a certain sense to simplify the theory of derived functors defined on A. The construction proceeds on the basis that the objects of D(A) should be chain complexes in A, with two such chain complexes considered isomorphic when there is a chain map that induces an isomorphism on the level of homology of the chain complexes. Derived functors can then be defined for chain complexes, refining the concept of hypercohomology. The definitions lead to a significant simplification of formulas otherwise described (not completely faithfully) by complicated spectral sequences.

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rdf:type	company yago:WikicatCategory-theoreticCategories yago:Abstraction100002137 yago:Class107997703 yago:Collection107951464 yago:Group100031264
rdfs:label	Derived category (en) Derivierte Kategorie (de) Catégorie dérivée (fr) 유도 범주 (ko) 導来圏 (ja) 导出范畴 (zh)
rdfs:comment	Die derivierte Kategorie einer abelschen Kategorie ist ein wichtiges Objekt in der modernen homologischen Algebra. Sie wurde durch Grothendiecks Student Verdier eingeführt. (de) 호몰로지 대수학에서 유도 범주(誘導範疇, 영어: derived category)는 사슬 복합체의 범주에서, 호몰로지들이 같은 사슬 복합체들을 서로 동형으로 간주하도록 변형한 범주이다. (ko) 导出范畴是同调代数中的一种构造。导出范畴的概念推广并深化了传统同调代数中导出函子的理论。这一构造是格罗滕迪克在20世纪60年代初提出的，他的学生让-路易·韦迪耶在其指导下发展了相关理论。今天，导出范畴被广泛应用于代数几何和D-模理论。 (zh) In mathematics, the derived category D(A) of an abelian category A is a construction of homological algebra introduced to refine and in a certain sense to simplify the theory of derived functors defined on A. The construction proceeds on the basis that the objects of D(A) should be chain complexes in A, with two such chain complexes considered isomorphic when there is a chain map that induces an isomorphism on the level of homology of the chain complexes. Derived functors can then be defined for chain complexes, refining the concept of hypercohomology. The definitions lead to a significant simplification of formulas otherwise described (not completely faithfully) by complicated spectral sequences. (en) La catégorie dérivée d'une catégorie est une construction, originellement introduite par Jean-Louis Verdier dans sa thèse et reprise dans SGA 4½, qui permet notamment de raffiner et simplifier la théorie des foncteurs dérivés. Elle a amené à plusieurs développements importants, ainsi que des reformulations élégantes par exemple de la théorie des D-modules et des preuves de la (en) qui généralise le vingt-et-unième problème de Hilbert. En particulier, le langage des catégories dérivées permet de simplifier des problèmes exprimés en termes de suites spectrales. (fr) 数学においてアーベル圏の導来圏(どうらいけん、英: Derived category、仏: Catégorie dérivée) はホモロジー代数から構成されるもので、上に定義された導来函手の理論を精密化するとともに、ある意味で単純化するべく導入された。その構成は基本的には次の様に進む：まず圏の対象はの双対鎖複体であり、次に2つのその様な双対鎖複体の間にチェイン写像が存在してコホモロジーを取った段階で同型を誘導する場合に同型であると考えるのである。このとき、導来函手は双対鎖複体に対して定義され、の考えを精密化したものとなる。これらの定義により、煩雑なスペクトル系列を用いて（完全に忠実ではなく）記述されるよりほか無かった式は劇的に簡素化される。 (ja)
dcterms:subject	Homological algebra Categories in category theory
Wikipage page ID	985897 (xsd:integer)
Wikipage revision ID	1111688495 (xsd:integer)
Link from a Wikipage to another Wikipage	Epimorphism Module (mathematics) Monomorphism Subcategory Non-singular D-module Derived algebraic geometry Derived functor Derived noncommutative algebraic geometry Quasi-isomorphism Mathematics Class (set theory) Mirror symmetry (string theory) Equivalence of categories Étale cohomology Functor Microlocal analysis Triangulated category Dual abelian variety Localization of a category Riemann–Hilbert correspondence Alexander Grothendieck Algebraic geometry D-brane Field (mathematics) Direct image functor Fourier–Mukai transform Isomorphism Kan extension Mapping cone (homological algebra) Projective line Projective object Quiver (mathematics) Ring (mathematics) Group cohomology Jean-Louis Verdier Tensor product Hyperhomology Homological algebra Abelian category Abelian variety Cohen–Macaulay ring Coherent duality Coherent sheaf Coherent sheaf cohomology Cohomology Cokernel Homological algebra Homology (mathematics) Homotopy Homotopy category of chain complexes Homotopy theory Model category Categories in category theory Spectral sequence Grothendieck category Grothendieck spectral sequence Injective object Injective resolution Mikio Sato Chain complex Set (mathematics) Société Mathématique de France Scheme (mathematics) Sheaf (mathematics) Ext functor Field of fractions Exact category Exact sequence T-structure Sheaf cohomology Glossary of algebraic geometry Topological space Kronecker quiver Springer-Verlag Grothendieck's Séminaire de géométrie algébrique Higher direct image functor Global sections functor

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