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.

'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

'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 Source: 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 Source: MAA Reviews