This book provides a comprehensive exposition of the use of set-theoretic methods in abelian group theory, module theory, and homological algebra, including applications to Whitehead's Problem, the structure of Ext and the existence of almost-free modules over non-perfect rings. This second edition is completely revised and udated to include major developments in the decade since the first edition. Among these are applications to cotorsion theories and covers, including a proof of the Flat Cover Conjecture, as well as the use of Shelah's pcf theory to constuct almost free groups. As with the first edition, the book is largely self-contained, and designed to be accessible to both graduate students and researchers in both algebra and logic. They will find there an introduction to powerful techniques which they may find useful in their own work.
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1. Homomorphisms and extensions.
2. Direct sums and products.
3. Linear topologies.
II. SET THEORY
1. Ordinary set theory.
2. Filters and large cardinals.
4. Clubs and stationary sets.
5. Games and trees.
6. ▵-systems and partitions.
III. SLENDER MODULES
1.Introduction to slenderness.
2.Examples of slender modules and rings.
3.The Łoś-Eda theorem.
IV. ALMOST FREE MODULES
0. Introduction to ℵ1free abelian groups.
1. &kgr;-free modules.
2. ℵ1-free abelian groups.
3. Compactness results.
V. PURE INJECTIVE MODULES
1. Structure theory.
2. Cotorsion groups.
VI. MORE SET THEORY
1. Prediction Principles.
2. Models of set theory.
3. L, the constructible universe.
4. MA and PFA.
5. PCF theory and I[&lgr;].
VII. ALMOST FREE MODULES REVISISTED (IV, VI)
0. ℵ1-free abelian groups revisited.
1. &kgr;-free modules revisited.
2. &kgr;-free abelian groups.
3. Transversals, &lgr;-systems and NPT.
3A. Reshuffling &lgr;-systems.
4. Hereditarily separable groups.
5. NPT and the construction of almost free groups.
VIII. ℵ1-SEPARABLE GROUPS (VI, VII.0,1)
1. Constructions and definitions.
2. ℵ1-separable groups under Martin's axiom.
3. ℵ1-separable groups under PFA.
IX. QUOTIENTS OF PRODUCTS OF Z
(III, IV, V)
1. Perps and products.
2. Countable products of the integers.
3. Uncountable products of the integers.
4. Radicals and large cardinals.
X. ITERATED SUMS AND PRODUCTS (III)
1. The Reid class.
2. Types in the Reid class.
XI. TOPOLOGICAL METHODS (X, IV)
1. Inverse and direct limits.
3. Density and dual bases.
4. Groups of continuous functions.
5. Sheaves of abelian groups.
XII. AN ANALYSIS OF EXT (VII, VIII.1)
1. Ext and Diamond.
2. Ext, MA and Proper forcing.
3. Baer modules.
4. The structure of Ext.
5. The structure of Ext when Hom=0.
XIII. UNIFORMIZATION (XII)
0. Whitehead groups and uniformization.
1. The basic construction and its applications.
2. The necessity of uniformization.
3. The diversity of Whitehead groups.
4. Monochromatic uniformization and
hereditarily separable groups.
XIV. THE BLACK BOX AND ENDOMORPHISM RINGS(V, VI)
1. Introducing the Black Box.
2. Proof of the Black Box.
3. Endomorphism rings of cotorsion-free groups.
4. Endomorphism rings of separable groups.
5. Weak realizability of endomorphism rings and the Kaplansky Test problems.
XV. SOME CONSTRUCTIONS IN ZFC (VII, VIII, XIV)
1. A rigid ℵ1-free group of cardinality ℵ1.
2. ℵn-separable groups with the Corner pathology.
3. Absolutely indecomposable modules.
4. The existence of &lgr;-separable groups.
XVI. COTORSION THEORIES, COVERS AND SPLITTERS(IX, XII.1, XIV)
1. Orthogonal classes and splitters.
2. Cotorsion theories.
3. Almost free splitters.
4. The Black Box and Ext.
XVII. DUAL GROUPS (IX, XI, XIV)
1. Invariants of dual groups.
2. Tree groups.
3. Criteria for being a dual group.
4. Some non-reflexive groups.
5. Dual groups in L.