Homological Theory of Representations

The modern development of representation theory relies heavily on homological methods. The book provides for advanced students and researchers a self-contained introduction into these methods and discusses several landmark results which illustrate theirpower and beauty. The 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, which includes the representation theory of finite groups and finite quivers. Specific classes of algebras which are covered are Gorenstein and quasi-hereditary algebras. A prominent example are the Schur algebras which model polynomial representations of the general linear groups. The Morita theory of derived categories via tilting objects is another important subject of this book. 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--