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
16 - Derived Torsion over NC Graded Rings
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
Let A be a connected graded ring over the base field K, with augmentation ideal m. In this chapter we study derived m-torsion, both for complexes of graded A-modules and for complexes of graded bimodules. The graded bimodule A* := HomK(A,K) is graded-injective and m-torsion on both sides. One of the main results is on the representability of the right derived m-torsion functor RΓm. Under quite general conditions the functor RΓm is isomorphic to the left derived tensor functor P⊗LA(-), where P := RΓm(A). We also prove the NC MGM Equivalence in the connected graded context and a theorem on symmetric derived m-torsion. The χ condition of Artin and Zhang is introduced in Section 16.5. We study how this condition interacts with symmetric derived m-torsion.
- Type
- Chapter
- Information
- Derived Categories , pp. 462 - 507Publisher: Cambridge University PressPrint publication year: 2019