×

Modules and group algebras. Notes by Ruedi Suter. (English) Zbl 0883.20006

Lectures in Mathematics, ETH Zürich. Basel: Birkhäuser. xi, 91 p. (1996).
In the past modular representation theory of finite groups has been seen very much from the perspective of character theory (as R. Brauer did) or from module theory (in the sense of J. A. Green). But modular representation theory may also be seen – and this has been proved a powerful concept since the time complexity and varieties of finitely generated modules were discovered – in the light of homological algebra. This area is the theme of the booklet (91 pages) under review which arose from a Nachdiplom course at the ETH Zürich in 1995.
The first part of the book mostly deals with well-known material (the modules are assumed to be finitely generated) such as projective resolutions and cohomology, the stable category (which is triangulated in the sense of Verdier), various interpretations of the cup product, the computation of \(H^*(G,K)\) (\(K\) a field) for small groups \(G\).
The second part is devoted to more recent material, mainly due to the author, D. Benson, C. Peng, J. Rickard and W. Wheeler. It starts with relative projectivity “relative to modules” (a generalization of the well-known concept “relative to subgroups”) and related ideas in cohomology.
A short section on varieties follows. In the rest of the book the author puts much emphasis on the stable category of all \(KG\)-modules. Not to restrict to f.g. modules turned out to be very fruitful in the last years and the author gives some flavour of the new concepts. First he proves that the class of projective \(KG\)-modules coincides with that of the injectives, which as a consequence says that the stable category of all modules is triangulated too.
One of the most striking facts about this category is the existence of (nontrivial) idempotent modules, i.e. modules \(X\) – necessarily infinitely generated – with \(X\otimes X\cong X\) in the stable category, a fact first observed by J. Rickard [J. Lond. Math. Soc., II. Ser. 56, No. 1, 149-170 (1997)]. To each nonempty, closed, homogeneous subvariety \(V\) of the maximal ideal spectrum of \(H^*(G,K)\) idempotent modules \(E(V)\) and \(F(V)\) are constructed as a direct limit of modules associated to \(V\). Applications of the idempotent modules finally round off the subject.
The reading of the book requires some knowledge about categories and standard concepts in homological algebra. So the text does not serve as a selfcontained and comprehensive introduction to the field – here the books of D. Benson, L. Evans and C. A. Weibel are more helpful for the standard material – but as an illuminating and delightful guide from elementary backgrounds to recent advances involving infinitely generated modules. In the meantime many of the original articles on which the new material is based are in print.

MSC:

20C20 Modular representations and characters
20J05 Homological methods in group theory
20C05 Group rings of finite groups and their modules (group-theoretic aspects)
20-02 Research exposition (monographs, survey articles) pertaining to group theory
PDFBibTeX XMLCite