Book contents
- Frontmatter
- Dedication
- Contents
- 0 Introduction
- 1 Basic Facts on Categories
- 2 Abelian Categories and Additive Functors
- 3 Differential Graded Algebra
- 4 Translations and Standard Triangles
- 5 Triangulated Categories and Functors
- 6 Localization of Categories
- 7 The Derived Category D(A,M)
- 8 Derived Functors
- 9 DG and Triangulated Bifunctors
- 10 Resolving Subcategories of K(A,M)
- 11 Existence of Resolutions
- 12 Adjunctions, Equivalences and Cohomological Dimension
- 13 Dualizing Complexes over Commutative Rings
- 14 Perfect and Tilting DG Modules over NC DG Rings
- 15 Algebraically Graded Noncommutative Rings
- 16 Derived Torsion over NC Graded Rings
- 17 Balanced Dualizing Complexes over NC Graded Rings
- 18 Rigid Noncommutative Dualizing Complexes
- References
- Index
8 - Derived Functors
Published online by Cambridge University Press: 15 November 2019
- Frontmatter
- Dedication
- Contents
- 0 Introduction
- 1 Basic Facts on Categories
- 2 Abelian Categories and Additive Functors
- 3 Differential Graded Algebra
- 4 Translations and Standard Triangles
- 5 Triangulated Categories and Functors
- 6 Localization of Categories
- 7 The Derived Category D(A,M)
- 8 Derived Functors
- 9 DG and Triangulated Bifunctors
- 10 Resolving Subcategories of K(A,M)
- 11 Existence of Resolutions
- 12 Adjunctions, Equivalences and Cohomological Dimension
- 13 Dualizing Complexes over Commutative Rings
- 14 Perfect and Tilting DG Modules over NC DG Rings
- 15 Algebraically Graded Noncommutative Rings
- 16 Derived Torsion over NC Graded Rings
- 17 Balanced Dualizing Complexes over NC Graded Rings
- 18 Rigid Noncommutative Dualizing Complexes
- References
- Index
Summary
Here we talk about derived functors. To make the definitions precise, we introduce 2-categorical notation. Suppose K and E are abstract categories, F : K → E is a functor and S ⊆ K is a multiplicatively closed set of morphisms. In this context we define the left and right derived functors of F w.r.t. S.These derived functors RF, LF : KS → E have universal properties, making each unique up to a unique isomorphism. Then we provide a general existence theorem for right and left abstract derived functors, in terms of the existence of suitable resolving subcategories J, P ⊆ K, respectively.In Section 8.4 we specialize to triangulated derived functors. Here K and E are triangulated categories, F : K → E is a triangulated functor and S ⊆ K is a multiplicatively closed set of cohomological origin. The right and left derived functors RF, LF : KS → E are defined like in the abstract setting, and their uniqueness is also proved the same way. Existence requires resolving subcategories P and J that are full triangulated subcategories of K. The chapter is concluded with a discussion of contravariant triangulated derived functors.
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- Information
- Derived Categories , pp. 186 - 215Publisher: Cambridge University PressPrint publication year: 2019