Derived Categories

The first systematic exposition of the theory of derived categories, with key applications in commutative and noncommutative algebra.

Amnon Yekutieli (Author)

9781108419338, Cambridge University Press

Hardback, published 19 December 2019

370 pages, 2 b/w illus. 155 exercises
23.4 x 15.8 x 3.8 cm, 0.99 kg

'Individuals hoping to learn about derived categories from the ground up (and willing to commit a significant amount of time to the process) will find that this book provides a solid foundation for the topic. Researchers already familiar with some of the theory may benefit from reading this linear development of derived categories, as it also offers a number of enlightening historical and contextual remarks along the way.' Peder Thompson, Mathematical Reviews

There have been remarkably few systematic expositions of the theory of derived categories since its inception in the work of Grothendieck and Verdier in the 1960s. This book is the first in-depth treatment of this important component of homological algebra. It carefully explains the foundations in detail before moving on to key applications in commutative and noncommutative algebra, many otherwise unavailable outside of research articles. These include commutative and noncommutative dualizing complexes, perfect DG modules, and tilting DG bimodules. Written with graduate students in mind, the emphasis here is on explicit constructions (with many examples and exercises) as opposed to axiomatics, with the goal of demystifying this difficult subject. Beyond serving as a thorough introduction for students, it will serve as an important reference for researchers in algebra, geometry and mathematical physics.

Introduction
1. Basic facts on categories
2. Abelian categories and additive functors
3. Differential graded algebra
4. Translations and standard triangles
5. Triangulated categories and functors
6. Localization of categories
7. The derived category D(A,M)
8. Derived functors
9. DG and triangulated bifunctors
10. Resolving subcategories of K(A,M)
11. Existence of resolutions
12. Adjunctions, equivalences and cohomological dimension
13. Dualizing complexes over commutative rings
14. Perfect and tilting DG modules over NC DG rings
15. Algebraically graded noncommutative rings
16. Derived torsion over NC graded rings
17. Balanced dualizing complexes over NC graded rings
18. Rigid noncommutative dualizing complexes
References
Index.

Subject Areas: Algebra [PBF]