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Derived category

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…

Standards, Projective and injective resolutions & Motivations

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Overview

Motivations

Definition

Remarks

Projective and injective resolutions

The relation to derived functors

Derived equivalence

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Derived category

Nodes91
Edges90
Triples72
Avg. degree1.98
Density0.021978
Components1

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Derived category

Top relations

related to References · 31
Derived category → Amsterdam, Annales Scientifiques, Astérisque, Bernhard, Catégories Abéliennes, Compositio Mathematica, Derived, Deriving DG, Des Catégories Dérivées, DG-algebras, Doorn, EMS PressKeller, Encyclopedia, Four, France, French, Handbook, Hazewinkel, ISBN, ISSN
related to Relation to the homotopy category · 9
Derived category → Consequently, From, If, It, Kom, Since, The, There, We
related to Definition · 7
Derived category → Examples, If, Kom, Let, Such, The, Xi
related to Constructing the derived category · 6
Derived category → However, If, The, There, This, When
related to Remarks · 6
Derived category → Also, Db, For, Grothendieck, If, The
related to Projective and injective resolutions · 5
Derived category → Classically, In, One, The, Therefore
related to Derived Hom-sets · 4
Derived category → As, In, To, We
related to The relation to derived functors · 3
Derived category → In, The, There
is a · 1
Derived category → natural framework to define and study derived functors

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Important terminology

derived category displaystyle categories morphisms homotopy mathcal abelian objects two complexes construction injective functors one complex theory bullet functor chain

Entity relationships Subject–Predicate–Object triples

SubjectPredicateObjectConfidenceSrc
Derived categoryis anatural framework to define and study derived functors0.90text
Derived categoryrelated to Constructing the derived categoryThere0.60section
Derived categoryrelated to Constructing the derived categoryWhen0.60section
Derived categoryrelated to Constructing the derived categoryThis0.60section
Derived categoryrelated to Constructing the derived categoryIf0.60section
Derived categoryrelated to Constructing the derived categoryThe0.60section
Derived categoryrelated to Constructing the derived categoryHowever0.60section
Derived categoryrelated to DefinitionLet0.60section
Derived categoryrelated to DefinitionExamples0.60section
Derived categoryrelated to DefinitionThe0.60section
Derived categoryrelated to DefinitionKom0.60section
Derived categoryrelated to DefinitionXi0.60section

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