Almost Free Modules : Set-Theoretic Methods

This is an extended treatment of the set-theoretic techniques which have transformed the study of abelian group and module theory over the last 15 years. Part of the book is new work which does not appear elsewhere in any form. In addition, a large body of material which has appeared previously (in scattered and sometimes inaccessible journal articles) has been extensively reworked and in many cases given new and improved proofs. The set theory required is carefully developed with algebraists in mind, and the independence results are derived from explicitly stated axioms. The book contains exercises and a guide to the literature and is suitable for use in graduate courses or seminars, as well as being of interest to researchers in algebra and logic. Front Cover......Page 1 Almost Free Modules: Set-theoretic Methods......Page 5 Copyright Page......Page 6 TABLE OF CONTENTS......Page 14 1. Homomorphisms and extensions......Page 18 2. Direct sums and products......Page 21 3. Linear topologies......Page 28 1. Ordinary set theory......Page 33 2. Filters and large cardinals......Page 39 3. Ultraproducts......Page 46 4. Cubs and stationary sets......Page 51 5. Games and trees......Page 58 Exercises......Page 62 Notes......Page 67 1. Introduction to slenderness......Page 68 2. Examples of slender modules and rings......Page 75 3. The Loś-Eda theorem......Page 82 Exercises......Page 94 Notes......Page 97 1. k-free modules......Page 99 2. N1-free abelian groups......Page 108 3. Compactness results......Page 119 Exercises......Page 129 Notes......Page 133 1. Structure theory......Page 135 2. Cotorsion groups......Page 147 Exercises......Page 152 Notes......Page 154 1. Prediction principles......Page 155 2. Models of set theory......Page 163 3. L, the constructible universe......Page 170 4. MA and PFA......Page 180 Exercises......Page 192 Notes......Page 195 1. k-free modules revisited......Page 197 2. k-free abelian groups......Page 204 3. Transversals and λ-systems......Page 215 3A. Reshuffling λ-systems......Page 226 4. Hereditarily separable groups......Page 242 Exercises......Page 254 Notes......Page 261 1. Constructions and definitions......Page 263 2. N1-separable groups under Martin’s axiom......Page 275 3. N1-separable groups under PFA......Page 282 Exercises......Page 288 Notes......Page 290 1. Perps and products......Page 292 2. Countable products of the integers......Page 298 3. Uncountable products of the integers......Page 302 4. Radicals and large cardinals......Page 305 Exercises......Page 312 Notes......Page 315 1. The Reid class......Page 316 2. Types in the Reid class......Page 320 Notes......Page 327 1. Inverse and direct limits......Page 329 2. Completions......Page 337 3. Density and dual bases......Page 342 4. Groups of continuous functions......Page 346 Exercises......Page 356 Notes......Page 358 1. The vanishing of Ext......Page 360 2. The rank of Ext......Page 371 3. Uniformization and W-groups......Page 383 Exercises......Page 399 Notes......Page 403 1. Black Box......Page 405 2. Proof of the Black Box......Page 412 3. Endomorphism rings of cotorsion-free groups......Page 416 4. Endomorphism rings of separable groups......Page 423 Exercises......Page 431 Notes......Page 434 1. Invariants of dual groups......Page 436 2. Tree groups......Page 441 3. Criteria for being a dual group......Page 446 4. Some non-reflexive dual groups......Page 451 5. More dual groups......Page 459 Notes......Page 468 OPEN PROBLEMS......Page 469 BIBLIOGRAPHY......Page 472 INDEX......Page 493