Title: Derived Categories
Author: Amnon Yekutieli
Publication status: to appear as book by Cambridge University Press; prepublcation version arXiv:1610.09640
Abstract: This is the second 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. This framework includes K(M), the homotopy category of unbounded complexes in M, and K(A), the homotopy category of DG A-modules.
The next topic is localization of categories, that we present in a very detailed way. We arrive at the derived category D(A, M), which 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 and existence. Here we introduce 2-categorical language, as an effective means to state the universal properties of the derived functors.
We introduce some resolving subcategories of K(A, M). These are the categories of K-injective, K-projective and K-flat DG modules. We study their roles, and prove their existence 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 book ends with two more specialized sections. There is a section on dualizing and residue complexes over commutative noetherian rings, as defined by Grothendieck. We recall the definitions of these complexes, and prove their existence (under suitable assumptions) and their classification. We introduce Van den Bergh rigidity for complexes, and prove uniqueness of rigid dualizing and residue complexes. Existence of rigid residue complexes, their functorial properties, and their application to Grothendieck Duality in algebraic geometry, are outlined.
The last section of the book [not yet written] is on derived categories in noncommutative algebra. We talk about perfect DG modules, dualizing DG modules and tilting DG modules, all over noncommutative DG rings. We present a few results on derived Morita theory, including Rickard’s Theorem.
Readers of this preview version are urged to write to the author with any comments regarding errors, suggestions or questions.
updated 13 Sep 2017