Homological Theory of Representations
£59.99
Part of Cambridge Studies in Advanced Mathematics
- Author: Henning Krause, Universität Bielefeld, Germany
- Date Published: November 2021
- availability: Available
- format: Hardback
- isbn: 9781108838894
£
59.99
Hardback
Other available formats:
eBook
Looking for an inspection copy?
This title is not currently available on inspection
-
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.
Read more- Explains the use of homological methods in representation theory of algebras
- Includes a self-contained account of landmark results in the area
- Collects much material previously unavailable in book form or at this level of detail
Reviews & endorsements
'This wide-ranging masterpiece offers the sophisticated abstraction of homological algebra and representation theory together with the meticulous analysis of many down-to-earth examples. Krause's clear style will delight specialists and beginners alike.' Paul Balmer, University of California, Los Angeles
See more reviews'This text makes an excellent addition to the literature on representation theory. The choice of topics includes most of what one would like to see in the homological end of the subject, especially triangulated categories, derived categories, and tilting. It's nice to see purity and Krull-Gabriel dimension treated well. The level is suitable for an advanced graduate student as well as researchers in related fields.' David Benson, University of Aberdeen
'Over the last fifty years, the representation theory of quivers and finite-dimensional algebras has seen an ever increasing use of tools from homological algebra and has, in turn, significantly contributed to this toolkit through developments like tilting theory, derived Morita theory, quasi-hereditary algebras ... This has led to increased interactions with module theory, non commutative (and commutative) algebraic geometry, Lie representation theory, K-theory, ... In the present volume, Henning Krause is the first to provide a comprehensive panorama of these developments. The presentation puts the emphasis on the theory without neglecting the fundamental examples. The overall organization is of great clarity. The essential notions are introduced with elegance and concision. Proofs taken from the literature are clarified and streamlined and in several instances, they are new. Obviously, this book is the fruit of a deep and prolonged reflection on the foundations of the subject. It will quickly become a standard text, indispensable to students and experienced researchers alike.' Bernhard Keller, Université de Paris
'This book is an excellent introduction to, though not an introductory textbook on, some of the major threads of research in the representation theory of algebra, broadly interpreted. The point of view is decidedly homological - derived categories and functor categories play a central role - and the writing is spare, demanding mathematically maturity and a good grasp of the basics in both representation theory and homological algebra. The rewards awaiting the reader are plentiful, including new insights on many classical results in the subject.' Srikanth B. Iyengar, University of Utah
'This book paints a compact and useful expository portrait of the landscape of representation theory and homological methods … The author has rendered a significant service to the mathematical community by providing this exposition. The book is clearly written and organised very well, and at the same time the book is kept as condensed as possible.' E. R. Alvares, MathSciNet
'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
Customer reviews
Not yet reviewed
Be the first to review
Review was not posted due to profanity
×Product details
- Date Published: November 2021
- format: Hardback
- isbn: 9781108838894
- length: 375 pages
- dimensions: 236 x 159 x 38 mm
- weight: 0.93kg
- availability: Available
Table of Contents
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.
Sorry, this resource is locked
Please register or sign in to request access. If you are having problems accessing these resources please email lecturers@cambridge.org
Register Sign in» Proceed
You are now leaving the Cambridge University Press website. Your eBook purchase and download will be completed by our partner www.ebooks.com. Please see the permission section of the www.ebooks.com catalogue page for details of the print & copy limits on our eBooks.
Continue ×Are you sure you want to delete your account?
This cannot be undone.
Thank you for your feedback which will help us improve our service.
If you requested a response, we will make sure to get back to you shortly.
×