Derived Categories

TL;DR

This book provides comprehensive foundations and applications of derived categories in algebra, emphasizing constructions and examples over axiomatic approaches, and covers both commutative and noncommutative algebraic contexts.

Contribution

It offers a detailed, example-driven exposition of derived categories, including new insights into dualizing complexes, tilting modules, and noncommutative algebraic structures.

Findings

Development of the theory of dualizing and residue complexes.

Introduction of derived torsion and MGM equivalence for NC rings.

Analysis of rigid dualizing complexes and their relation to Calabi-Yau rings.

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.