Homological Theory of Representations

This book for advanced graduate students and researchers discusses representations of associative algebras and their homological theory.

Henning Krause (Author)

9781108838894, Cambridge University Press

Hardback, published 18 November 2021

375 pages
23.6 x 15.9 x 3.8 cm, 0.93 kg

'Krause's book is a tour de force, providing the current state-of-the-art methodology in homological algebra as it pertains to representation theory … this book is a gem.' James Turner, MAA Reviews

Modern developments in representation theory rely heavily on homological methods. This book for advanced graduate students and researchers introduces these methods from their foundations up and discusses several landmark results that illustrate their power and beauty. Categorical foundations include abelian and derived categories, with an emphasis on localisation, spectra, and purity. The representation theoretic focus is on module categories of Artin algebras, with discussions of the representation theory of finite groups and finite quivers. Also covered are Gorenstein and quasi-hereditary algebras, including Schur algebras, which model polynomial representations of general linear groups, and the Morita theory of derived categories via tilting objects. The final part is devoted to a systematic introduction to the theory of purity for locally finitely presented categories, covering pure-injectives, definable subcategories, and Ziegler spectra. With its clear, detailed exposition of important topics in modern representation theory, many of which were unavailable in one volume until now, it deserves a place in every representation theorist's library.

Introduction
Conventions and notations
Glossary
Standard functors and isomorphisms
Part I. Abelian and Derived Categories: 1. Localisation
2. Abelian categories
3.Triangulated categories
4. Derived categories
5. Derived categories of representations
Part II. Orthogonal Decompositions: 6. Gorenstein algebras, approximations and Serre duality
7. Tilting in exact categories
8. Polynomial representations
Part III. Derived Equivalences: 9. Derived equivalences
10. Examples of derived equivalences
Part IV. Purity: 11. Locally finitely presented categories
12. Purity
13. Endofiniteness
14. Krull–Gabriel dimension
References
Notation
Index.

Subject Areas: Algebra [PBF]