Representation theory. A homological algebra point of view.
type: Item
instance of: scholarly article
title: Representation theory. A homological algebra point of view.
zbMATH Open document ID: 1306.20001
DOI: 10.1007/978-3-319-07968-4
author: Alexander Zimmermann
published in: Algebra and Applications
publication date: 2014-07-07T00:00:00Z
review text: 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.
reviewed by: Wolfgang Rump
Mathematics Subject Classification ID: 20-02
Mathematics Subject Classification ID: 20C20
Mathematics Subject Classification ID: 20J05
Mathematics Subject Classification ID: 16G10
Mathematics Subject Classification ID: 16D90
Mathematics Subject Classification ID: 16E35
zbMATH DE Number: 6314003
zbMATH Keywords: modular representations of finite groups
zbMATH Keywords: representations of finite-dimensional algebras
zbMATH Keywords: Abelian categories
zbMATH Keywords: triangulated categories
zbMATH Keywords: stable module categories
zbMATH Keywords: derived equivalences
zbMATH Keywords: Morita theory
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1007/978-3-319-07968-4
OpenAlex ID: W4246560108
description: scientific article