Title: Derived Categories
Author: Amnon Yekutieli
Publication status: Book, Cambridge University Press, 2019.

Dedicated to Alexander Grothendieck, in Memoriam

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 is a brief description of the book. After a review of standard facts on abelian categories, we study differential graded algebra in depth (including DG rings, DG modules, DG categories and DG functors).

We then move to triangulated categories and triangulated functors between them, and explain how they arise from the DG background. Specifically, we talk about the homotopy category K(A, M) of DG A-modules in M, where A is a DG ring and M is an abelian category. The derived category D(A, M) is the localization of K(A, M) with respect to the quasi-isomorphisms. We define the left and right derived functors of a triangulated functor, and prove their uniqueness. Existence of derived functors relies on having enough K-injective, K-projective or K-flat DG modules, as the case may be. We give constructions of resolutions by such DG modules in several important algebraic situations.

There is a detailed study of the derived Hom and tensor bifunctors, and the related adjunction formulas. We talk about cohomological dimensions of functors and how they are used.

The last sections of the book are more specialized. There is a section on dualizing and residue complexes over commutative noetherian rings, as defined by Grothendieck. We introduce Van den Bergh rigidity in this context. The algebro-geometric applications of commutative rigid dualizing complexes are outlined.

A section is devoted to perfect DG modules and tilting DG bimodules over NC (noncommutative) DG rings. This includes theorems on derived Morita theory and on the structure of the NC derived Picard group.

Three sections are on NC connected graded rings. In the first of them we give some basic definitions and results on algebraically graded rings, including Artin-Schelter regular rings. The second section is on derived torsion for NC connected graded rings, it relation to the χ condition of Artin-Zhang, and the NC MGM Equivalence. In the third section we introduce balanced dualizing complexes, and we prove their uniqueness, existence and trace functoriality. The final section of the book is on NC rigid dualizing complexes, following Van den Bergh. We prove the uniqueness and existence of these complexes, give a few examples, and discuss their relation to Calabi-Yau rings.

Feedback from readers (corrections and suggestions) is most welcome.

updated 22 Nov 2020