These notes grew out of a Nachdiplom course given at the ETH Zuerich in the summer semester of 1995. They aim at the development of some of the basics of the interaction of homological algebras, or more specifically the cohomology of groups and modular representation theory. The book presents an entirely new approach to the subject based upon the recent development in this field. Basically the shift has been towards a much more categorical view of representation theory, and an expansion of the viewpoint to include infinitely generated modules as well as the finitely generated ones. Some of the constructions in the category of all modules have had new and original applications for the category of finitely generated modules. The introduction to a fresh view of the module theory for finite groups is of interest to students and researchers in homotopy theory and group action as well as representation theory of finite groups.

Preface Notation 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 ideas in cohomology 10 Varieties and modules 11 Infinitely generated modules 12 Idempotent modules 13 Varieties and induced modules References List of symbols Index

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