Representation theory. A homological algebra point of view.
This book is intended as a text for first year master students who want to specialize on representation theory, more precisely: representations of finite-dimensional algebras and modular group representations, with special emphasis on homological methods. The book consists of six chapters, a first one on basic module theory and homological algebra, including self-injective and hereditary algebras. The second chapter deals with modular representations of finite groups. Apart from standard items like Green correspondence, Clifford theory, and blocks of finite type, the reader learns about Külshammer's generalized Reynolds ideals, which have been recognized as invariants under derived equivalence [\textit{A. Zimmermann}, Math. Proc. R. Ir. Acad. 107A, No. 1, 1-9 (2007; Zbl 1131.16003)]. The third chapter gives an introduction to the basics on derived categories and derived functors, including spectral sequences, while the short chapter 4 on Morita theory contains Külshammer's elegant proof of Puig's description of nilpotent blocks. The last two chapters deal with stable and derived equivalences. The description of blocks with cyclic defect by means of Brauer trees, the stable invariance of Külshammer ideals, and Reiten's stable invariance theorem for self-injective algebras are proved in chapter 5. More advanced topics are treated in chapter 6 on derived equivalences, which contains, e.g., a section on singularity categories, and a brief history of the Abelian defect conjecture. At the end of the book, the reader is invited to broaden and deepen his studies, filling the spots which had to be left blank on the 700 pages world map: He may be inspired to learn about Auslander-Reiten theory, tame blocks, splendid equivalences, geometric representation theory, to mention just a few topics which could be studied subsequent to this introductory text on representation theory.
modular representations of finite groups
representations of finite-dimensional algebras
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