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
4 - Translations and Standard Triangles
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
We talk about the translation (or shift or suspension) functor and standard triangles in the DG category C(A,M). The translation T(M) of a DG module M is the usual one. A calculation shows that T is a DG functor from C(A,M) to itself. We introduce the degree -1 morphism tM : M → T(M), called the little t operator, which facilitates many calculations.
A morphism φ : M → N in Cstr(A,M) gives rise to the standard Cone(φ) = N ⊕ T(M) , whose differential is a matrix involving the degree 1 morphism φ ◦ (tM)-1. The standard cone sits inside the standard triangle associated to φ.A DG functor F : C(A,M) → C(B,N) gives rise to a T-additive functor F : Cstr(A,M) → Cstr(B,N), and it sends standard triangles in Cstr(A,M) to standard triangles in Cstr(B,N).
- Type
- Chapter
- Information
- Derived Categories , pp. 101 - 116Publisher: Cambridge University PressPrint publication year: 2019