Almost Free Modules (eBook)

Set-theoretic Methods

P.C. Eklof, A.H. Mekler (Autoren)

eBook Download: PDF

2002 | 1. Auflage
620 Seiten
Elsevier Science (Verlag)
978-0-08-052705-5 (ISBN)

Infos

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.

Inhalt

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

Textauszug

Chapter I

Algebraic Preliminaries

Paul C. Eklof    Department of Mathematics, University of California, Irvine CA, U.S.A.

Alan H. Mekler    Department of Mathematics and Statistics Simon Fraser University

In the first two sections of this chapter we review the algebraic background which is assumed in the rest of the book; this also gives us the opportunity to fix notation and conventions. In the last section, we discuss linear topologies on modules. We assume the reader is already familiar with most of the material in this chapter, so it is presented informally and largely without proofs; for more on the topics covered we refer the reader to such texts as Fuchs 1970/1973, Anderson-Fuller 1992, Rotman 1979, or Weibel 1994, as well as any standard graduate text in algebra.

All rings in this book will have a multiplicative identity and all modules will be unitary modules. Unless otherwise specified, “module” will mean left R-module. Much of the time we will focus on abelian groups, that is -modules, and we will often refer to these simply as groups.

If φ: A → B is a function, and X ⊆ A, φ[X] = {φ(a): a ∈ X};φ[A] will also be denoted im(φ) or rge(φ). If Y ⊆ B, φ- 1[Y], = {a ∈ A: φ(a) ∈ Y}; if φ is a homomorphism, ker(φ) = φ_1[{0}]. The restriction of φ to X is denoted φ X, i.e., φ X = {(x, φ(x)): x ∈ X}. In an abuse of notation, sometimes we will write M = 0 instead of M = {0} and φ–1[x] instead of φ–1[{x}]. If ψ: B → C, ψ φ is the composition of ψ with φ, a function from A to C.

If M is a module and Y ⊆ M, then (Y) denotes the submodule generated by Y. The notation Y ⊂ X means Y ⊆ X and Y ≠ 1.

1 Homomorphisms and extensions

If M and H are left R-modules, Homr(M,H) denotes the group of R-homomorphisms from M to H, which is an abelian group under the operation defined by: (f + g)(x) = f(x) + g(x). We will sometimes refer to HomR(M, H) as the H-dual of M. Often we will write Hom (M, H), if R is clear from context. If x ∈ M and y ∈ Hom(M,H), we denote by 〈y, x〉 or 〈x, y〉, interchangeably, the element y(x) of H, i.e., the result of applying y to x.

If H is an R-S bimodule (that is, a left R-module and a right S-module such that (ra)s = r(as) for all r ∈ R, a ∈ H, s ∈ S), then Homr(M,H) has a right S-module structure defined by: (fs)(x) = f(x)s. The bimodule structure that we will be interested in arises as follows. If H is an R-module, let Endr(H) = Homr(H, H); then EndR(H) is a ring under composition of homomorphisms, where we define the product f ⋅ g to be g f; H has a right EndR(H)-module structure defined by: af = f(a) for all a ∈ H and f ∈ Endr(H). This makes H into an R-EndR(H)-bimodule. Note that EndR(5) ≅ R.

If M is an abelian group, then M*, without further explanation, will denote ℤ(M,?ℤ), called the dual group of M. A group is called a dual group if and only if it is of the form Hom(M, ) for some group M. The structure and properties of dual groups will be one of the principal subjects of this book. In particular, we will be interested in when a group is (canonically) isomorphic to its double dual. Sometimes it will be convenient to consider this question in the more general context of H-duals.

Let us temporarily fix an R-module H, and let S denote EndR(H), so that H is an R-S-bimodule. For convenience denote Homr(M,H) by M*. Then M* is a right S-module, and HomS(M*, H) is a left R-module in the obvious fashion; we denote the latter by M**. There is a canonical homomorphism

M:M→M**

defined by: 〈σM(x),y〉 = 〈x,y〉 for all x ∈ M and y ∈ M*. We say that M is H-torsionless if σM is one-one, and that M is H-reflexive if σM is one-one and onto M, i.e., an isomorphism. If H = R, we say torsionless or reflexive instead of R-torsionless or R-reflexive, respectively.

For every R-homomorphism φ: M → N there is an induced S-homomorphism φ* : Hom(N,H) → Hom(M,H) defined by: φ*(f) = f φ for all f ∈ Hom(N,H). There is also an induced S-homomorphism φ*: Hom(H, M) → Hom(H, N) defined by: φ*(g) = φ gfor all g ∈ Hom(H, M). If φ is an isomorphism, then so are φ* and φ*. If φ is surjective, then φ* is injective; if φ is injective, then φ* is injective. But φ* may not be surjective when φ is injective; and φ* may not be surjective when φ is surjective. In fact, we have the following situation. A sequence of homomorphisms

→Mn−1→φn−1Mn→φnMn+1→…

is called exact if ker(φn) = im(φn-1) for all n. A short exact sequence (or s.e.s.) is an exact sequence of the form

→L→ψM→φN→0.

Given a short exact sequence of R-homomorphisms as above and given an R-module H, the sequences

→Hom(N,H)→φ*Hom(M,H)→ψ*Hom(L,H)0→Hom(H,L)→ψ*Hom(H,M)→φ*Hom(H,N)

are exact. We have the following fundamental theorem of Cartan-Eilenberg 1956. (In its statement we will ignore the complication that the domain of the function is a proper class.)

1.1

Theorem.

For all n ≥ 1 and R there is a binary functionRn(_,_)from the class of R-modules to the class of abelian groups so that for any short exact sequence

→L→ψM→φN→0

there are exact...

Produkt-Details

• Erscheint lt. Verlag: 29.4.2002
• Sprache: englisch
• Themenwelt: Mathematik / Informatik ► Mathematik ► Algebra
• Themenwelt: Mathematik / Informatik ► Mathematik ► Logik / Mengenlehre
• Themenwelt: Technik
• ISBN-10: 0-08-052705-1 / 0080527051
• ISBN-13: 978-0-08-052705-5 / 9780080527055