Almost free modules. Set-theoretic methods

Description

I. ALGEBRAIC PRELIMINARIES 1. Homomorphisms and extensions. 2. Direct sums and products. 3. Linear topologies. II. SET THEORY 1. Ordinary set theory. 2. Filters and large cardinals. 3. Ultraproducts. 4. Clubs and stationary sets. 5. Games and trees. 6. u-systems and partitions. III. SLENDER MODULES 1.Introduction to slenderness. 2.Examples of slender modules and rings. 3.The Los-Eda theorem. IV. ALMOST FREE MODULES 0. Introduction to 1free abelian groups. 1. -free modules. 2. 1-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[ ]. VII. ALMOST FREE MODULES REVISISTED (IV, VI) 0. 1-free abelian groups revisited. 1. -free modules revisited. 2. -free abelian groups. 3. Transversals, -systems and NPT. 3A. Reshuffling -systems. 4. Hereditarily separable groups. 5. NPT and the construction of almost free groups. VIII. 1-SEPARABLE GROUPS (VI, VII.0,1) 1. Constructions and definitions. 2. 1-separable groups under Martin's axiom. 3. 1-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. 2. Completions. 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 1-free group of cardinality 1. 2. n-separable groups with the Corner pathology. 3. Absolutely indecomposable modules. 4. The existence of -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