## Modules and Group AlgebrasThe 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. |

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### Contents

Augmentations nilpotent ideals and semisimplicity | 2 |

Tensor products Homs and duality | 5 |

Restriction and induction | 9 |

Projective resolutions and cohomology | 12 |

The stable category | 17 |

Products in cohomology | 26 |

Examples and diagrams | 42 |

Relative projectivity | 53 |

Relative projectivity and ideals in cohomology | 58 |

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annihilates application assume Axiom chain map closed cohomology cohomology ring commutative diagram complex composition consider Corollary corresponding course cover defined Definition denote direct limit direct summand dual easy element equivalent exact sequence example EXERCISE exists fact factors field finitely follows functor given group algebras Hence Hom(M homogeneous homological homomorphism ideal indecomposable induced infinitely generated modules injective integer ISBN isomorphism kG-homomorphism kG-module kgmod komod kostmod Lemma look maximal ideal minimal projective resolution Moreover morphism natural nilpotent nonzero Note Notice object p-group positive proj projective module Proof Proposition prove relative representation represented restriction result rows splits stable category subgroup Suppose surjective tensor product Theorem theory triangle trivial V-projective V-split varieties VG(k VG(M