Homological Theory of Representations

Description

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. Frontmatter Contents Introduction Conventions and Notations Glossary Standard Functors and Isomorphisms PART ONE ABELIAN AND DERIVED CATEGORIES 1 Localisation 2 Abelian Categories 3 Triangulated Categories 4 Derived Categories 5 Derived Categories of Representations PART TWO ORTHOGONAL DECOMPOSITIONS 6 Gorenstein Algebras, Approximations, Serre Duality 7 Tilting in Exact Categories 8 Polynomial Representations PART THREE DERIVED EQUIVALENCES 9 Derived Equivalences 10 Examples of Derived Equivalences PART FOUR PURITY 11 Locally Finitely Presented Categories 12 Purity 13 Endofiniteness 14 Krull–Gabriel Dimension References Notation Index