CONTENTS
Notation = xix
Introduction = 1
1. Background material on algebras and groups = 1
1A. Algebras over commutative rings. Integral closure = 1
1B. Background from group theory = 6
2. The functors Hom and Projective, injective, and flat modules = 13
2A. Homomorphisms = 14
2B. Tenser products = 23
2C. Categories and functors = 27
2D. Projective, infective, and flat modules = 29
3. Semisimple rings and modules. The Wedderburn and Morita Theorems = 40
3A. Finiteness conditions = 40
3B. Semisimple modules and rings = 42
3C. Semisimple algebras over fields. The theorems of Burnside and Frobenius-Schur = 50
3D. The Morita theorems = 55
3E. Tensor products of simple algebras and modules. The Skolem-Noether Theorem = 64
4. Dedekind domains = 73
4A. Localization = 73
4B. Ideal theory = 76
4C. Valuations, completions, localizations = 81
4D. Modules over Dedekind domains = 84
4E. Duals of lattices = 89
4F. Ideal class groups : global fields = 92
4G. Primary decompositions = 93
4H. Cyclotomic fields = 94
5. Radicals = 101
5A. Basic definitions = 101
5B. Radicals of artinian rings = 109
5C. Local rings = 111
6. Idempotents, indecomposable modules, and the KruIl-Schmidt-Azumaya Theorem. Projective covers and injective hulls = 119
6A. Idempotents = 119
6B. The Krull-Schmidt-Azumaya Theorem = 127
6C. Projective covers = 131
6D. Injective hulls = 134
7. Separable algebras and splitting fields = 142
7A. Separable algebras and modules = 142
7B. Splitting fields = 149
7C. Splitting fields for division algebras = 154
7D. Reduced norms = 158
8. Ext, Tor : cohomology of groups = 171
8A. Ext, Tor = 171
8B. Cohomology of groups = 179
8C. The Schur Criterion for split extensions = 184
8D. Tate cohomology groups = 187
Chapter 1. Group representations and character theory = 195
9. Orthogonality relations and central idempotents = 195
9A. Frobenius and symmetric algebras = 195
9B. Characters and central idempotents in split semisimple algebras : orthogonality relations = 202
9C. Orthogonality relations for characters of finite groups = 206
9D. The character table = 214
9E. Burnside's pa qb -Theorem = 221
10. Induced modules = 227
10A. Definition of induced modules. Frobenius Reciprocity = 227
10B. Mackey's Subgroup Theorem and Tensor Product Theorem = 235
10C. The Intertwining Number Theorem = 243
10D. Contragredient modules = 245
10E. Outer tensor products = 249
11. Decomposition of induced modules. Clifford theory and Hecke algebras = 259
11A. Clifford's Theorem = 259
11B. Applications of Clifford's Theorem to character theory = 262
11C. Decomposition of induced modules from normal subgroups = 267
11D. Hecke algebras and induced modules = 279
11E. Projective representations and central extensions = 291
12. Tensor algebras = 308
12A. Tensor algebras = 308
12B. Adams operators on the ring of virtual characters = 313
12C. Symmetric and skew-symmetric squares of induced modules = 318
13. Tensor induction and transfer = 331
13A. Tensor induction = 331
13B. Transfer and the determinant map = 337
13C. Normal p-complements and the transfer = 341
14. Special classes and exceptional characters = 343
14A. Frobenius groups = 344
14B. Special classes = 348
14C. Exceptional characters. Suzuki's CA-group Theorem = 353
14D. Characters of As . Examples of exceptional characters = 363
14E. The Brauer-Suzuki Theorem on generalized quaternion Sylow 2-groups = 366
14F. Centralizers of involutions and special classes = 370
15. The Artin and Brauer Induction Theorems = 377
15A. Artin's Induction Theorem revisited = 377
15B. Character rings and the Brauer Induction Theorem = 380
15C. Applications of the Brauer Induction Theorem 384
15D. Extensions of invariant characters = 388
15E. A criterion for existence of normal complements = 392
15F. A converse to Brauer's Theorem = 395
15G. The Aramata-Brauer Induction Theorem = 396
Chapter 2. Introduction to modular representations = 401
16. The decomposition map = 402
16A. Notation and terminology = 402
16B. Grothendieck groups = 403
16C. Reduction mod % and the decomposition map = 408
16D. Behavior of Grothendieck groups under extension of ground field = 414
17. Brauer characters = 417
17A. Splitting fields = 417
17B. Brauer characters = 419
18. The Cartan-Brauer triangle = 427
18A. The Cartan map and the Cartan-Brauer triangle = 428
18B. Properties of the Cartan-Brauer triangle (K sufficiently large) = 432
18C. Orthogonality relations for Brauer characters = 437
19. Vertices and sources = 448
19A. Relative projective and injective modules over group rings = 449
19B. Vertices and sources of indecomposable lattices = 453
19C. The Green Indecomposability Theorem = 459
20. The Green correspondence. Applications to character theory = 470
20A. The Green correspondence = 470
20B. Applications to character theory = 475
21. The induction theorem for arbitrary fields = 491
21A. The Witt-Berman Induction Theorem = 491
21B. The induction theorem over fields of characteristic p>0 = 500
21C. The Cartan-Brauer triangle (general case) = 503
22. Modular representations of p-solvable groups = 513
Chapter 3. Integral representations : Orders and lattices = 520
23. Lattices and orders = 522
24. Jordan-Zassenhaus Theorem = 534
25. Extensions of lattices = 538
26. Maximal and hereditary orders = 559
26A. Existence of maximal orders in separable algebras = 559
26B. Maximal orders are hereditary = 565
26C. Structure theorems for maximal and hereditary orders = 571
27. Group rings and maximal orders = 581
28. Twisted group rings and crossed product orders = 588
29. Annihilator of Ext = 603
29A. Annihilator of Ext, Higman ideal = 603
29B. Projectile endomorphisms = 609
Chapter 4. Local and global theory of integral representations = 617
30. Local theory = 618
30A. Reduction mod pk = 621
30B. Extension of the ground ring = 631
30C. Representations mod pk = 637
31. Genus = 642
31A. Basic properties = 643
31B. Id e ` les and class groups = 651
31C. Roiter's Theorem on genera = 659
32. Projective lattices over group rings : Swan's Theorem = 670
32A. Local case = 671
32B. Global case = 676
32C. Characters afforded by projectile lattices = 679
33. Finite representation type = 686
33A. Jones' Theorem. Jacobinski's criterion for group rings = 687
33B. Dade's Theorem = 691
33C. Commutative orders = 695
34. Examples of integral representations = 711
34A. Extensions of lattices = 712
34B. Cyclic p-groups = 719
34C. Cyclic groups of order p2 = 730
34D. An order in a matrix algebra = 742
34E. Dihedral and metacyclic groups = 747
35. Invertible ideals = 755
36. The Krull-Schmidt-Azumaya Theorem over discrete valuation rings = 767
37. Bass and Gorenstein orders = 776
Bibliography = 795
Index 813
2
CONTENTS
Chapter 5. Algebraic K-theory = 1
38. Grothendieck groups = 2
38A. Grothendieck groups. Frobenius functors = 2
38B. Grothendieck groups and projective class groups = 14
38C. Regular rings = 19
38D. Localization sequences = 31
39. Grothendieck groups of integral group rings = 44
39A. Localization sequences = 45
39B. Explicit calculations = 54
40. Whitehead groups = 61
40A. Introduction = 61
40B. Localization sequences = 65
40C. Elementary matrices = 73
40D. Unimodular rows and stably free modules = 77
41. Basic elements, stable range, and cancellation = 83
42. Mayer-Vietoris sequences = 101
43. K-theory of polynomial rings = 112
44. Relative K-theory = 120
45. S K1 of orders = 138
45A. Reduced norms = 138
45B. Maximal orders = 142
45C. Finiteness of S K1 = 151
45D. Profinite groups = 156
46. Whitehead groups of integral group rings = 163
47. Milnor's K2 -group = 184
47A. Steinberg groups and K2 = 184
47B. Relative K-theory = 190
47C. Symbols = 197
48. S K1 of integral group rings = 210
Chapter 6. Class groups of integral group rings and orders = 216
49. Locally free class groups = 217
49A. Basic-formulas = 217
49B. Functorial properties and the kernel group = 229
49C. Frobenius functor properties for class groups of group rings = 238
50. Class groups of integral group rings = 243
50A. Cyclic groups of squarefree order = 243
50B. The kernel group for p-groups = 254
50C. Metacyclic groups = 259
50D. Dihedral and quaternion 2-groups = 266
50E. An involution on class groups and kernel groups = 274
50F. Cyclic p-groups = 283
50G. Twisted group rings and crossed-product orders = 291
51. Jacobinski's Cancellation Theorem and the Eichler condition = 303
51A. The Eichler condition = 304
51B. The Eichler-Swan Theorem = 306
51C. Locally free cancellation = 322
52. The Horn description of the class group = 329
53. The Swan subgroup of the class group = 343
53A. The Swan subgroup = 343
53B. Rings of integers in tame extensions = 351
53C. Generalized Swan subgroups = 353
54. p-Adic logarithms and Taylor's Theorem = 356
55. Picard groups = 369
55A. Basic properties = 369
55B. Picard groups of orders = 376
55C. Locally free Picard groups = 382
55D. Radical reduction = 391
55E. Picard groups of group rings = 396
Chapter 7. The theory of blocks = 406
56. Introduction to block theory = 407
56A. Background and notation for block theory = 407
56B. Definition of p-blocks for a finite group G = 412
56C. A criterion for P.I.M.'s to belong to the same p-block = 414
56D. Central characters and blocks of KG-modules = 416
56E. The defect of a block = 422
57. The defect group of a p-block = 429
57A. G-algebras, the trace map. and defect groups = 429
57B. Defect groups as vertices = 437
57C. Defect groups as Sylow intersections = 440
58. The Brauer Correspondence = 445
58A. The Brauer map = 445
58B. Brauer's First Main Theorem = 448
58C. The Brauer Correspondence = 451
59. Applications of blocks to character theory = 462
59A. The Nagao Decomposition = 463
59B. Brauer's Second Main Theorem = 467
60. p-Sections and characters in blocks = 471
60A. Block orthogonality and p'sections = 471
60B. Determination of the principal block using block orthogonality = 473
60C. Applications to the classification of transitive permutation groups of degree p = 478
61. Refinements of the Brauer Correspondence = 484
61A. Blocks and normal subgroups = 484
61B. An extension of Brauer's First Main Theorem = 489
61C. Brauer's Third Main Theorem = 494
62. Blocks with cyclic defect groups = 495
62A. Preliminary results from homological algebra = 496
62B. Functorial properties of the Green Correspondence = 499
62C. Uniserial algebras and blocks of finite representation type = 504
62D. Modular representations in blocks wish cyclic defect groups = 512
62E. Periodic projective resolutions in blocks with cyclic defect groups = 522
63. Applications to group theory = 530
63A. The kernel of the principal block = 530
63B. The Brauer-Suzuki Theorem on quaternion Sylow 2-subgroups = 532
63C. Glauberman's Z* -Theorem = 545
Chapter 8. The representation theory of finite groups of Lie type = 549
64. Root systems and finite reflection groups = 550
64A. Finite groups generated by reflections. Root systems = 550
64B. Coxeter groups = 561
64C. Parabolic subgroups of finite Coxeter groups = 570
65. Finite groups with BN-pairs = 576
65A. The Bruhat decomposition = 576
65B. Examples of BN-pairs = 580
65C. Parabolic subgroups of finite groups with BN-pairs = 583
66. Homology representations of finite groups with BN-pairs = 586
66A. Homology representations of finite groups = 586
66B. The Coxeter poset of a finite g.g.r. = 600
66C. The combinatorial building and the Steinberg representation of a finite group with a BN-pair = 605
67. The Hecke algebra H(G. B) and the decomposition of ( IB \G = 609
67A. The structure of the Hecke algebra H(G. B) = 609
67B. The sign representation of H and the Steinberg representation of G = 614
67C. Representations of the Hecke algebra H for a BN-pair of rank 2 = 619
67D. The Feit-Higman Theorem on generalized polygons = 623
67E. The reflection representation of the Hecke algebra H = 630
68. Generic algebras and finite Coxeter groups = 635
68A. Generic algebras and the Deformation Theorem = 635
68B. Parametrization of characters in ( IB \G = 643
68C. Generic degrees = 648
69. Finite groups with split BN-pairs = 653
69A. The Levi Decomposition = 653
69B. Intersections of parabolic subgroups = 662
70. Cuspidal characters = 666
70A. Generalized restriction and induction = 666
70B. The philosophy of cusp forms = 676
70C. Formulas for character values = 681
71. A Duality Operation in eh CG. = 688
71A. Definition and basic properties of DG = 689
71B. The effects of DG on character degrees = 692
71C. The values of the Steinberg character = 697
72. Modular representations of finite groups of Lie type = 700
72A. The Ballard-Lusztig Theorem on characters of P.I.M.'s = 700
72B. The simple kG-modules = 706
Chapter 9. Rationality questions = 719
73. Unitary, orthogonal, and symplectic CG-modules = 720
73A. Rationality questions over the real field R = 720
73B. Induction theorems for real-rained characters = 727
74. The Schur Index = 732
74A. General theory = 732
74B. Schur indices for group algebras = 740
74C. The Benard-Schacher Theorem = 746
75. Representations and characters of the symmetric group = 762
75A. Specht modules and simple F Sn -modules = 762
75B. Solomon's Theorem and the irreducible characters of Sn = 774
76. The Artin exponent = 782
Chapter 10. Indecomposable modules = 790
77. Representations of graphs and Gabriel's Theorem = 790
77A. Representations of graphs and Coxeter functors = 790
77B. Representation categories of finite type (Gabriel's Theorem) = 799
78. Auslander-Reiten sequences = 806
78A. The Heller loop-space operator = 807
78B. Auslander-Reiten sequences for group algebras = 815
78C. Auslander-Reiten sequences for algebras = 822
79. Algebras of finite representation type = 830
Chapter 11. The Burnside ring and the representation ring of a finite group = 837
80. Permutation representations and Burnside rings = 838
80A. Burnside rings = 838
80B. G-sets and induction maps = 846
80C. Tenser induction and algebraic maps = 852
80D. Coition's Induction Theorem = 859
81. Representation rings = 868
81A. Preliminary results = 869
8lB. Conlon's Theorems = 878
81C. Species = 891
81D. Dual elements in the Green algebra = 898
81E. Semisimplicity of representation algebras = 906
81F. Nilpotent elements in representation algebras = 912
Bibliography = 921
Notation index = 943
Subject index = 947
