Derived Categories

Abstract

This is the fourth (and last) prepublication version of a book on derived categories, that will be published by Cambridge University Press. The purpose of the book is to provide solid foundations for the theory of derived categories, and to present several applications of this theory in commutative and noncommutative algebra. The emphasis is on constructions and examples, rather than on axiomatics. Here are the topics covered in the book: - A review of standard facts on abelian categories. - Differential graded algebra (DG rings, DG modules, DG categories and DG functors). - Triangulated categories and triangulated functors between them. How they arise from the DG background. The homotopy category K(A,M) of DG A-modules in M. - Localization of categories. The derived category D(A,M), which is the localization of K(A,M) with respect to the quasi-isomorphisms. - Left and right derived functors of a triangulated functor. - K-injective, K-projective and K-flat DG modules. Their roles, and their existence in several important algebraic situations. - Dualizing and residue complexes over commutative noetherian rings, including Van den Bergh rigidity. - Perfect DG modules and tilting DG bimodules over NC (noncommutative) DG rings. - NC connected graded rings, including Artin-Schelter regular rings. Derived torsion for NC connected graded rings, its relation to the chi condition of Artin-Zhang, and the NC MGM Equivalence. Balanced dualizing complexes, their uniqueness, existence and trace functoriality. - NC rigid dualizing complexes, following Van den Bergh. The uniqueness and existence of these complexes, a few examples, and their relation to Calabi-Yau rings. Readers of this preview version are urged to write to the author with any comments regarding errors, suggestions or questions.

Fourth (and last) prepublication version. To be published by CUP. This version: 459 pages, many changes (large and small)

References in corpus (7)

• A tour about existence and uniqueness of dg enhancements and lifts
• Derived Categories
• Weak Proregularity, Weak Stability, and the Noncommutative MGM Equivalence
• Reduction of Hochschild cohomology over algebras finite over their center
• Dg algebras with enough idempotents, their dg modules and their derived categories
• Dedualizing complexes of bicomodules and MGM duality over coalgebras
• Rigid Dualizing Complexes via Differential Graded Algebras (Survey)

Cited by in corpus (8)

• Derived Categories
• Koszul complexes over Cohen-Macaulay rings
• Naive liftings of DG modules
• Open loci results for commutative DG-rings
• A proof of the Brown--Goodearl Conjecture for weak Hopf algebras
• Smooth flat maps over commutative DG-rings
• Duality for Witt-divisorial sheaves
• Maximal Cohen-Macaulay DG-complexes