
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…
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