On the Local Structure of Morita and Rickard Equivalences between Brauer Blocks

Brauer had already introduced the defect of a block and opened the way towards a classification by solving all the problems in defects zero and one, and by providing some evidence for the finiteness of the set of blocks with a given defect. In 1959 he discovered the defect group, and in 1964 Dade determined the blocks with cyclic defect groups. In 1978 Alperin and Broué discovered the Brauer category, and Broué and the author determined the blocks having a nilpotent Brauer category. In 1979, the author discovered the source algebra which determines all the other current invariants, representing faithfully the block – and found its structure in the nilpotent blocks. Recently, the discovery by Rickard that all blocks with the same cyclic defect group and the same Brauer category have the same homotopic category focussed great interest on the new, loose relationship between blocks called Rickard equivalence. This book describes the source algebra of a block from the source algebra of a Rickard equivalent block and the source of the Rickard equivalence. Front Matter....Pages i-vii Introduction....Pages 1-7 General notation, terminology and quoted results....Pages 9-21 Noninjective induction of O G -interior algebras....Pages 23-35 Hecke O G -interior algebras and noninjective induction....Pages 37-45 On the local structure of Hecke O G -interior algebras....Pages 47-59 Morita stable equivalences between Brauer blocks....Pages 61-72 Basic Morita stable equivalences between Brauer blocks....Pages 73-88 The Morita stable equivalent class of a nilpotent block....Pages 89-91 The differential Z-grading O -algebra....Pages 93-101 D G -modules....Pages 103-111 D -algebras and D G -interior algebras....Pages 113-122 Induction of D G -interior algebras....Pages 123-131 Brauer sections in basic induced D G -interior algebras....Pages 133-149 Pointed groups on D G -interior algebras and Higman embeddings....Pages 151-174 Hecke D G -interior algebras and noninjective induction....Pages 175-180 On the local structure of Hecke D G -interior algebras....Pages 181-197 Brauer sections in basic Hecke D G -interior algebras....Pages 199-213 Rickard equivalences between Brauer blocks....Pages 215-228 Basic Rickard equivalences between Brauer blocks....Pages 229-242 Back Matter....Pages 243-264 Section 1. Introduction section 2. General notation, terminology and quoted results section 3. Noninjective induction of OG-interior algebras section 4. Hecke OG-interior algebras and noninjective induction section 5. On the local structure of Hecke OG-interior algebras section 6. Morita stable equivalences between Brauer blocks section 7. Basic Morita stable equivalences between Brauer blocks section 8. The Morita stable equivalent class of a nilpotent block section 9. The differential Z-grading O-algebra section 10. DG-modules section 11. D-algebras and DG-interior algebras section 12. Induction of DG-interior algebras section 13. Brauer sections in basic induced DG-interior algebras section 14. Pointed groups on DG-interior algebras and Higman embeddings section 15. Hecke DG-interior algebras and noninjective induction section 16. On the local structure of Hecke DG-interior algebras section 17. Brauer sections in basic Hecke DG-interior algebras section 18. Rickard equivalences between Brauer blocks section 19. Basic Rickard equivalences between Brauer blocks. "Although the book reaches the most advanced level, a special effort has been made to make it accessible to graduate students interested in finite groups or noncommutative algebras. A full chapter is devoted to reviewing the terminology, and the particular case of Morita equivalences is discussed separately, as an introduction. Two appendices, one on Weiss's criterion for permutation modules and the other on tensor induction of graded differential algebras, are of interest on their own."--Jacket "This book describes the source algebra of a block from the source algebra of a Rickard equivalent block and the source of the Rickard equivalence. This description requires a new induction procedure and the introduction of suitable graded differential algebras. It leads to strong consequences such as the facts that the nilpotent blocks form a union of classes and that the basic Rickard equivalences preserve defect groups and Brauer categories."