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
18 - Rigid Noncommutative Dualizing Complexes
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 NC noetherian ring, with enveloping ring Aen.A NC DC over A is a complex R ∈ D(Aen) satisfying the conditions stated earlier. The NC square of R is a complex Sq(R) ∈ D(Aen). A NC rigidDC over A is a pair (R,ρ), where R is a NC DC and ρ : R → Sq(R) is an isomorphism in D(Aen). We prove that a rigid NC DC (R,ρ) is unique up to a unique rigid isomorphism. If the ring A admits a filtration such that the graded ring Gr(A) is noetherian connected and has a balanced DC, then A has a rigid DC. This material is due to Van den Bergh. If the graded ring Gr(A) is AS regular, then the rigid NC DC of A is R = A(μ)[n], where μ is a ring automorphism of A and n is an integer. The automorphism ν := μ−1 is called the Nakayama automorphism of A. Such a ring A is called an n-dimensional twisted Calabi--Yau ring. We state and prove the Van den Bergh Duality Theorem for Hochschild (co)homology and give an example of a Calabi--Yau category of fractional dimension.
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- Chapter
- Information
- Derived Categories , pp. 542 - 589Publisher: Cambridge University PressPrint publication year: 2019