Derived Equivalences for Group Rings

Steffen König, Alexander Zimmermann (auth.)3540643117, 9783540643111

A self-contained introduction is given to J. Rickard’s Morita theory for derived module categories and its recent applications in representation theory of finite groups. In particular, Broué’s conjecture is discussed, giving a structural explanation for relations between the p-modular character table of a finite group and that of its “p-local structure”. The book is addressed to researchers or graduate students and can serve as material for a seminar. It surveys the current state of the field, and it also provides a “user’s guide” to derived equivalences and tilting complexes. Results and proofs are presented in the generality needed for group theoretic applications.

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Table of contents :
Introduction….Pages 1-4
Basic definitions and some examples….Pages 5-32
Rickard’s fundamental theorem….Pages 33-50
Some modular and local representation theory….Pages 51-80
Onesided tilting complexes for group rings….Pages 81-104
Tilting with additional structure: twosided tilting complexes….Pages 105-149
Historical remarks….Pages 151-154
On the construction of triangle equivalences….Pages 155-176
Triangulated categories in the modular representation theory of finite groups….Pages 177-198
The derived category of blocks with cyclic defect groups….Pages 199-220
On stable equivalences of Morita type….Pages 221-232