Finite Group Algebras and their Modules (London Mathematical Society Lecture Note Series, Series Number 84)

Originally published in 1983, the principal object of this book is to discuss in detail the structure of finite group rings over fields of characteristic, p, P-adic rings and, in some cases, just principal ideal domains, as well as modules of such group rings. The approach does not emphasize any particular point of view, but aims to present a smooth proof in each case to provide the reader with maximum insight. However, the trace map and all its properties have been used extensively. This generalizes a number of classical results at no extra cost and also has the advantage that no assumption on the field is required. Finally, it should be mentioned that much attention is paid to the methods of homological algebra and cohomology of groups as well as connections between characteristic 0 and characteristic p. LMS Lecture Note Series 84......Page 1 Finite Group Algebras and their Modules......Page 4 Contents......Page 6 Preface......Page 8 1. Idempotents in rings. Liftings.......Page 12 2. Projective and injective modules.......Page 16 3. The radical and artinian rings.......Page 18 4. Cartan invariants and blocks.......Page 22 5. Finite dimensional algebras.......Page 25 6. Duality.......Page 29 7. Symmetry.......Page 33 8. Loewy series and socle series.......Page 36 9. The p.i.m.'s.......Page 40 10. Ext.......Page 45 11. Orders.......Page 53 12. Modular systems and blocks.......Page 58 13. Centers.......Page 61 14. R-forms and liftable modules.......Page 66 15. Decomposition numbers and Brauer characters.......Page 70 16. Basic algebras and small blocks.......Page 77 17. Pure submodules.......Page 83 18. Examples.......Page 86 1. The trace maps and the Nakayama relations.......Page 93 2. Relative projectivity.......Page 104 3. Vertices and sources.......Page 114 4. Green Correspondence.......Page 123 5. Relative projective homomorphisms.......Page 128 6. Tensor products.......Page 133 7. The Green ring.......Page 149 8. Endomorphism rings.......Page 157 9. Almost split sequences.......Page 161 10. Inner products on the Green ring.......Page 166 11. Induction from normal subgroups.......Page 170 12. Permutation modules.......Page 183 13. Examples......Page 194 1. Blocks, defect groups and the Brauer map.......Page 200 2. Brauer's First Main Theorem.......Page 206 3. Blocks of groups with a normal subgroup.......Page 211 4. The Extended First Main Theorem.......Page 218 5. Defect groups and vertices.......Page 220 6. Generalized decomposition numbers.......Page 224 7. Subpairs.......Page 229 8. Characters in blocks.......Page 233 9. Vertices of simple modules.......Page 250 10. Defect groups.......Page 257 Appendix I: Extensions.......Page 268 Appendix II: Tor.......Page 271 Appendix III: Extensions of hte ring of coefficients.......Page 273 References......Page 276 Index......Page 284 This book is concerned with the structure of group algebras of finite groups over fields of characteristic [lowercase italic]p dividing the order of the group, or closely related rings such as rings of algebraic integers and in particular their [lowercase italic]p-adic completions, as well as modules and homomorphisms between them, or such group algebras. Our principal aim has been to present some of the more recent ideas which have enriched and improved this theory. This text is not restricted to particular methods, be they ring theoretic or character theoretic, while presenting approaches or proofs which are distinguished by being fast, elegant, illuminating, with potential for further advancement, or all of these at the same time. This text hopes to attract non-specialists, perhaps algebraic topologists and group theorists who might use the tools of modular representations more frequently CHAPTER II. INDECOMPOSABLE MODULES AND RELATIVE PROJECTIVITY1. The trace maps and the Nakayama relations; 2. Relative projectivity; 3. Vertices and sources; 4. Green Correspondence; 5. Relative projective homomorphisms; 6. Tensor products; 7. The Green ring; 8. Endomorphism rings; 9. Almost split sequences; 10. Inner products on the Green ring; 11. Induction from normal subgroups; 12. Permutation modules; 13. Examples; CHAPTER III. BLOCK THEORY; 1. Blocks, defect groups and the Brauer map; 2. Brauer's First Main Theorem; 3. Blocks of groups with a normal subgroup Cover; Series Page; Title; Copyright; CONTENTS; PREFACE; CHAPTER I. THE STRUCTURE OF GROUP ALGEBRAS; 1. Idempotents in rings. Liftings; 2. Projective and injective modules; 3. The radical and artinian rings; 4. Cartan invariants and blocks; 5. Finite dimensional algebras; 6. Duality; 7. Symmetry; 8 Loewy series and socle series; 9. The p.i.m.'s; 10. Ext; 11. Orders; 12. Modular systems and blocks; 13. Centers; 14. R-forms and liftable modules; 15. Decomposition numbers and Brauer characters; 16. Basic algebras and small blocks; 17. Pure submodules; 18. Examples 4. The Extended First Màin Theorem5. Defect groups and vertices; 6. Generalized decomposition numbers; 7. Subpairs; 8. Characters in blocks; 9. Vertices of simple modules; 10. Defect groups; Appendix I: Extensions; Appendix II: Tor; Appendix III: Extensions of the ring of coefficients; REFERENCES; INDEX