Almost Free Modules: Set-theoretic Methods

Front Cover......Page 1
Almost Free Modules: Set-theoretic Methods......Page 4
Copyright Page......Page 5
TABLE OF CONTENTS......Page 18
PREFACE to the revised edition......Page 8
PREFACE to the first edition......Page 12
1. Homomorphisms and extensions......Page 24
2. Direct sums and products......Page 27
3. Linear topologies......Page 35
1. Ordinary set theory......Page 40
2. Filters and large cardinals......Page 46
3. Ultraproducts......Page 53
4. Clubs and stationary sets......Page 58
5. Games and trees......Page 65
6. Δ-systems and partitions......Page 68
Exercises......Page 72
Notes......Page 76
1. Introduction to slenderness......Page 78
2. Examples of slender modules and rings......Page 85
3. The Łoś-Eda theorem......Page 92
Exercises......Page 103
Notes......Page 106
0. Introduction to N1-free abelian groups......Page 108
1. κ-free modules......Page 111
2. N1-free abelian groups......Page 119
3. Compactness results......Page 130
Exercises......Page 139
Notes......Page 144
1. Structure theory......Page 146
2. Cotorsion groups......Page 158
Exercises......Page 163
Notes......Page 164
CHAPTER VI. MORE SET THEORY......Page 166
1. Prediction Principles......Page 167
2. Models of set theory......Page 175
3. L, the constructible universe......Page 182
4. MA and PFA......Page 192
5. PCF theory and I[λ]......Page 203
Exercises......Page 209
Notes......Page 212
CHAPTER VII. ALMOST FREE MODULES REVISISTED (IV, VI)......Page 214
0. N1-free abelian groups revisited......Page 215
1. κ- free modules revisited......Page 216
2. κ-free abelian groups......Page 222
3. Transversals, λ-systems and NPT......Page 232
3A. Reshuffling λ-systems......Page 244
4. Hereditarily separable groups......Page 260
5. NPT and the construction of almost free groups......Page 272
Exercises......Page 279
Notes......Page 285
CHAPTER VIII. N1-SEPARABLE GROUPS (VI, VII.0, 1)......Page 287
1. Constructions and definitions......Page 288
2. N1-separable groups under Martin's axiom......Page 299
3. N1-separable groups under PFA......Page 306
Exercises......Page 311
Notes......Page 314
1. Perps and products......Page 315
2. Countable products of the integers......Page 321
3. Uncountable products of the integers......Page 324
4. Radicals and large cardinals......Page 327
Exercises......Page 335
Notes......Page 337
1. The Reid class......Page 339
2. Types in the Reid class......Page 343
Notes......Page 350
1. Inverse and direct limits......Page 351
2. Completions......Page 359
3. Density and dual bases......Page 364
4. Groups of continuous functions......Page 369
5. Sheaves of abelian groups......Page 378
Exercises......Page 382
Notes......Page 384
1. Ext and Diamond......Page 386
2. Ext, MA and Proper forcing......Page 396
3. Baer modules......Page 402
4. The structure of Ext......Page 406
5. The structure of Ext when Hom = 0......Page 417
Exercises......Page 420
Notes......Page 422
0. Whitehead groups and uniformization......Page 424
1. The basic construction and its applications......Page 428
2. The necessity of uniformization......Page 435
3. The diversity of Whitehead groups......Page 452
4. Monochromatic uniformization and hereditarily separable groups......Page 457
Exercises......Page 460
Notes......Page 463
CHAPTER XIV. THE BLACK BOX AND ENDOMORPHISM RINGS (V, VI)......Page 464
1. Introducing the Black Box......Page 465
2. Proof of the Black Box......Page 473
3. Endomorphism rings of cotorsion-free groups......Page 477
4. Endomorphism rings of separable groups......Page 483
5. Weak realizability of endomorphism rings and the Kaplansky Test problems......Page 492
Exercises......Page 495
Notes......Page 497
CHAPTER XV. SOME CONSTRUCTIONS IN ZFC (VII, VIII, XIV)......Page 499
1. A rigid N1-free group of cardinality N1......Page 500
2. Nn-separable groups with the Corner pathology......Page 505
3. Absolutely indecomposable modules......Page 510
4. The existence of l-separable groups......Page 515
Notes......Page 520
1. Orthogonal classes and splitters......Page 521
2. Cotorsion theories......Page 529
3. Almost free splitters......Page 535
4. The Black Box and Ext......Page 539
Exercises......Page 546
Notes......Page 548
1. Invariants of dual groups......Page 550
2. Tree groups......Page 556
3. Criteria for being a dual group......Page 561
4. Some non-reflexive groups......Page 566
5. Dual groups in L......Page 573
Notes......Page 580
APPENDIX: OPEN AND SOLVED PROBLEMS......Page 582
Bibliography......Page 586
Index......Page 616