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
3 - Differential Graded Algebra
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
A good understanding of DG (differential graded) algebra is essential in our approach to derived categories. By DG algebra, we mean DG rings, DG modules, DG categories and DG functors. The first section is on cohomologically graded rings and modules, with a discussion of the monoidal braiding (i.e. the Koszul sign rule). After that we study DG rings, DG modules and operations on them. We go on to discuss DG categories, DG functors between them and morphisms between DG functors. We recall the DG category C(M) of complexes in an abelian category M.A new feature we introduce is the DG category C(A,M) of DG A-modules in M, where A is a DG ring and M is an abelian category. This includes as special cases the category C(M) mentioned above, and the category C(A) of DG A-modules over a DG ring A. Another new feature is the distinction between the DG category C(A,M) and its strict subcategory Cstr(A,M), whose morphisms are the degree 0 cocycles, and it is an abelian category.
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- Derived Categories , pp. 62 - 100Publisher: Cambridge University PressPrint publication year: 2019