Ebook: 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.

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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 Brou? discovered the Brauer category, and Brou? 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.

---

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 Brou? discovered the Brauer category, and Brou? 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.

Content:
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

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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 Brou? discovered the Brauer category, and Brou? 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.

Content:
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
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