Separable Algebras

Cover......Page 1
Title page......Page 4
Contents......Page 8
Preface......Page 16
1. Rings and Modules......Page 24
1.1. Categories and Functors......Page 25
1.2. Progenerator Modules......Page 28
1.3. Exercises......Page 31
1.4. Nakayama’s Lemma......Page 32
1.6. Module Direct Summands of Rings......Page 34
1.7. Exercises......Page 36
2.1. The Ring of Polynomial Functions on a Module......Page 37
2.2. Resultant of Two Polynomials......Page 38
2.3. Polynomial Functions on an Algebraic Curve......Page 42
2.4. Exercises......Page 44
3.1. Tensor Product......Page 46
3.2. Exercises......Page 50
3.3. Hom Groups......Page 52
3.4. Hom Tensor Relations......Page 56
3.5. Exercises......Page 59
4. Direct Limit and Inverse Limit......Page 60
4.1. The Direct Limit......Page 61
4.2. The Inverse Limit......Page 63
4.3. Inverse Systems Indexed by Nonnegative Integers......Page 65
4.4. Exercises......Page 68
5. The Morita Theorems......Page 70
5.1. Exercises......Page 75
1. Localization of Modules and Rings......Page 76
1.1. Local to Global Lemmas......Page 77
1.2. Exercises......Page 80
2. The Prime Spectrum of a Commutative Ring......Page 81
2.1. Exercises......Page 86
3. Finitely Generated Projective Modules......Page 88
3.1. Exercises......Page 91
4. Faithfully Flat Modules and Algebras......Page 93
4.1. Exercises......Page 97
5. Chain Conditions......Page 98
5.1. Exercises......Page 101
6.1. Fundamental Theorem on Faithfully Flat Base Change......Page 103
6.2. Locally Free Finite Rank is Finitely Generated Projective......Page 106
6.3. Invertible Modules and the Picard Group......Page 108
6.4. Exercises......Page 111
1. The Jacobson Radical and Nakayama’s Lemma......Page 114
2. Semisimple Modules and Semisimple Rings......Page 117
2.1. Simple Rings and the Wedderburn-Artin Theorem......Page 120
2.2. Commutative Artinian Rings......Page 123
2.3. Exercises......Page 125
3. Integral Extensions......Page 126
4. Completion of a Linear Topological Module......Page 129
4.1. Graded Rings and Graded Modules......Page 133
4.2. Lifting of Idempotents......Page 135
1. Separable Algebra, the Definition......Page 138
1.1. Exercises......Page 142
2. Examples of Separable Algebras......Page 143
3. Separable Algebras Under a Change of Base Ring......Page 146
4. Homomorphisms of Separable Algebras......Page 150
4.1. Exercises......Page 156
5.1. Central Simple Equals Central Separable......Page 160
5.2. Unique Decomposition Theorems......Page 163
5.3. The Skolem-Noether Theorem......Page 166
5.4. Exercises......Page 167
6.1. Separable Extensions of Commutative Rings......Page 169
6.2. Separability and the Trace......Page 171
6.3. Twisted Form of the Trivial Extension......Page 175
6.4. Exercises......Page 176
7. Formally Unramified, Smooth and Étale Algebras......Page 178
1. Group Cohomology......Page 182
1.1. Cocycle and Coboundary Groups in Low Degree......Page 184
1.2. Applications and Computations......Page 186
1.3. Exercises......Page 193
2. The Tensor Algebra of a Module......Page 196
2.1. Exercises......Page 199
3.1. The Amitsur Complex......Page 200
3.2. The Descent of Elements......Page 201
3.3. Descent of Homomorphisms......Page 203
3.4. Descent of Modules......Page 204
3.5. Descent of Algebras......Page 209
4. Hochschild Cohomology......Page 211
5.1. The Definition and First Properties......Page 214
5.2. Twisted Forms......Page 218
Chapter 6. The Divisor Class Group......Page 222
1.1. Krull Dimension......Page 223
1.2. The Serre Criteria for Normality......Page 224
1.3. The Hilbert-Serre Criterion for Regularity......Page 225
1.4. Discrete Valuation Rings......Page 227
2. The Class Group of Weil Divisors......Page 229
2.1. Exercises......Page 233
3.1. Definition and First Properties......Page 236
3.2. Reflexive Lattices......Page 239
3.3. A Local to Global Theorem for Reflexive Lattices......Page 245
3.4. Exercises......Page 247
4. The Ideal Class Group......Page 249
4.1. Exercises......Page 255
5.1. Flat Extensions......Page 256
5.2. Finite Extensions......Page 258
5.3. Galois Descent of Divisor Classes......Page 259
5.4. The Class Group of a Regular Domain......Page 261
5.5. Exercises......Page 265
1. First Properties of Azumaya Algebras......Page 266
2. The Commutator Theorems......Page 272
3. The Brauer Group......Page 275
4. Splitting Rings......Page 277
5. Azumaya Algebras over a Field......Page 281
6. Azumaya Algebras up to Brauer Equivalence......Page 286
6.1. Exercises......Page 289
7. Noetherian Reduction of Azumaya Algebras......Page 290
7.1. Exercises......Page 296
8.1. Definition of the Picard Group......Page 297
8.2. The Skolem-Noether Theorem......Page 302
8.3. Exercise......Page 303
9. The Brauer Group Modulo an Ideal......Page 304
9.1. Lifting Azumaya Algebras......Page 307
9.2. The Brauer Group of a Commutative Artinian Ring......Page 309
1.1. The Definition and First Results......Page 310
1.2. A Noncommutative Binomial Theorem in Characteristic ......Page 314
1.3. Extensions of Derivations......Page 315
1.4. Exercises......Page 317
1.5. More Tests for Separability......Page 319
1.7. Exercises......Page 324
2. Differential Crossed Product Algebras......Page 326
2.1. Elementary -Algebras......Page 328
3.1. The Definition and Fundamental Exact Sequences......Page 331
3.2. More Tests for Separability......Page 335
3.3. Exercises......Page 339
4. Separably Generated Extension Fields......Page 340
4.1. Emmy Noether’s Normalization Lemma......Page 343
4.2. Algebraic Curves......Page 346
5.1. A Differential Criterion for Regularity......Page 348
5.2. A Jacobian Criterion for Regularity......Page 349
1. Complete Noetherian Rings......Page 352
2.1. Étale Algebras......Page 359
2.2. Formally Smooth Algebras......Page 362
2.4. An Example of Raynaud......Page 369
3.1. Quasi-finite Algebras......Page 371
3.3. Standard Étale Algebras......Page 373
3.4. Theorems of Permanence......Page 376
3.5. Étale Algebras over a Normal Ring......Page 378
3.6. Topological Invariance of Étale Coverings......Page 380
3.7. Étale Neighborhood of a Local Ring......Page 382
4. Ramified Radical Extensions......Page 384
4.1. Exercises......Page 387
Chapter 10. Henselization and Splitting Rings......Page 390
1.1. The Definition......Page 391
1.2. Henselian Noetherian Local Rings......Page 399
1.3. Exercises......Page 402
2. Henselization of a Local Ring......Page 403
2.1. Henselization of a Noetherian Local Ring......Page 404
2.2. Henselization of an Arbitrary Local Ring......Page 407
2.3. Strict Henselization of a Noetherian Local Ring......Page 408
3.1. Existence of Splitting Rings (Local Version)......Page 410
3.2. Local to Global Lemmas......Page 414
3.3. Splitting Rings for Azumaya Algebras......Page 417
4. Cech Cohomology......Page 418
4.1. The Definition......Page 419
4.2. The Brauer group and Amitsur Cohomology......Page 421
1. Invariants Attached to Elements in Azumaya Algebras......Page 430
1.1. The Characteristic Polynomial......Page 431
1.3. The Rank of an Element......Page 435
2. The Brauer Group is Torsion......Page 437
2.1. Applications to Division Algebras......Page 440
3.1. Definition, First Properties......Page 442
3.2. Localization and Completion of Maximal Orders......Page 445
3.3. When is a Maximal Order an Azumaya Algebra?......Page 447
3.4. Azumaya Algebras at the Generic Point......Page 449
3.5. Azumaya Algebras over a DVR......Page 451
3.6. Locally Trivial Azumaya Algebras......Page 453
3.7. An Example of Ojanguren......Page 454
3.8. Exercises......Page 457
4. Brauer Groups in Characteristic ......Page 459
4.1. The Brauer Group is -divisible......Page 460
4.2. Generators for the Subgroup Annihilated by ......Page 462
4.3. Exercises......Page 465
1. Crossed Product Algebras, the Definition......Page 468
2. Galois Extension, the Definition......Page 470
3. Induced Galois Extensions and Galois Extensions of Fields......Page 479
4. Galois Descent of Modules and Algebras......Page 482
5. The Fundamental Theorem of Galois Theory......Page 485
5.1. Fundamental Theorem for a Connected Galois Extension......Page 486
5.2. Exercises......Page 489
6.1. Embedding a Separable Algebra......Page 491
6.2. Embedding a Connected Separable Algebra......Page 493
7. Separable Polynomials......Page 496
8.1. The Separable Closure......Page 501
8.2. The Fundamental Theorem of Infinite Galois Theory......Page 506
8.3. Exercises......Page 507
9.1. Kummer Theory......Page 509
9.2. Artin-Schreier Extensions......Page 514
9.3. Exercises......Page 515
Chapter 13. Crossed Products and Galois Cohomology......Page 520
1. Crossed Product Algebras......Page 521
2. Generalized Crossed Product Algebras......Page 524
2.1. Exercises......Page 535
3.1. The Theorem and Its Corollaries......Page 536
3.2. Exercises......Page 543
3.3. Galois Cohomology Agrees with Amitsur Cohomology......Page 544
3.4. Galois Cohomology and the Brauer Group......Page 546
4. Cyclic Crossed Product Algebras......Page 548
4.2. Cyclic Algebras in Characteristic ......Page 551
4.3. The Brauer Group of a Henselian Local Ring......Page 553
4.4. Exercises......Page 554
5. Generalized Cyclic Crossed Product Algebras......Page 555
6. The Brauer Group of a Polynomial Ring......Page 564
6.1. The Brauer Group of a Graded Ring......Page 567
6.2. The Brauer Group of a Laurent Polynomial Ring......Page 568
6.3. Examples of Brauer Groups......Page 569
6.4. Exercises......Page 575
1. Corestriction......Page 580
1.1. Norms of Modules and Algebras......Page 584
1.2. Applications of Corestriction......Page 589
1.3. Corestriction and Galois Descent......Page 591
1.4. Corestriction and Amitsur Cohomology......Page 594
1.5. Corestriction and Galois Cohomology......Page 600
1.6. Corestriction and Generalized Crossed Products......Page 604
1.7. Exercises......Page 606
2. A Mayer-Vietoris Sequence for the Brauer Group......Page 607
2.1. Milnor’s Theorem......Page 608
2.2. Mayer-Vietoris Sequences......Page 614
2.3. Exercises......Page 621
3. Brauer Groups of Some Nonnormal Domains......Page 622
3.1. The Brauer Group of an Algebraic Curve......Page 623
3.2. Every Finite Abelian Group is a Brauer Group......Page 624
3.3. A Family of Nonnormal Subrings of [,]......Page 625
3.4. The Brauer Group of a Subring of a Global Field......Page 628
3.5. Exercises......Page 635
Acronyms......Page 638
Glossary of Notations......Page 640
Bibliography......Page 644
Index......Page 654
Back Cover......Page 663