The notes in this volume were written as a part of a Nachdiplom course that I gave at the ETH in the summer semester of 1995. The aim of my lectures was the development of some of the basics of the interaction of homological algebra, or more specifically the cohomology of groups, and modular representation theory. Every time that I had given such a course in the past fifteen years, the choice of the material and the order of presentation of the results have followed more or less the same basic pattern. Such a course began with the fundamentals of group cohomology, and then investigated the structure of cohomology rings, and their maximal ideal spectra. Then the variety of a module was defined and related to actual module structure through the rank variety. Applications followed. The standard approach was used in my University of Essen Lecture Notes [e1] in 1984. Evens [E] and Benson [B2] have written it up in much clearer detail and included it as part of their books on the subject.

1 Augmentations, nilpotent ideals, and semisimplicity.- 2 Tensor products, Homs, and duality.- 3 Restriction and induction.- 4 Projective resolutions and cohomology.- 5 The stable category.- 6 Products in cohomology.- 7 Examples and diagrams.- 8 Relative projectivity.- 9 Relative projectivity and ideals in cohomology.- 10 Varieties and modules.- 11 Infinitely generated modules.- 12 Idempotent modules.- 13 Varieties and induced modules.- References.- List of symbols.