Almost Free Modules: Set-theoretic Methods

By P.C. Eklof and A.H. Mekler

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.

• Language: English
• Publisher: Elsevier Science
• Release date: Apr 29, 2002
• ISBN: 9780080527055

Book preview

Almost Free Modules

Set-theoretic Methods

First edition

Eklof Paul C.

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

Mekler Alan H.

Department of Mathematics and Statistics Simon Fraser University

2002

Elsevier

Amsterdam  –  London  –  New York- Oxford  –  Paris  –  Shannon  –  Tokyo

Table of Contents

Cover image

Title page

Copyright page

Dedication

Preface to the revised edition

Preface

Chapter I: Algebraic Preliminaries

1 Homomorphisms and extensions

2 Direct sums and products

3 Linear topologies

Chapter II: Set Theory

1 Ordinary set theory

2 Filters and large cardinals

3 Ultraproducts

4 Clubs and stationary sets

5 Games and trees

6 Δ-systems and partitions

Exercises

Notes

Chapter III: Slender Modules

1 Introduction to slenderness

2 Examples of slender modules and rings

3 The Łoś-Eda theorem

EXERCISES

NOTES

Chapter IV: Almost Free Modules

0 Introduction to ℵ1-free abelian groups

1 κ-free modules

2 ℵ1-free abelian groups

3 Compactness results

EXERCISES

NOTES

Chapter V: Pure-Injective Modules

1 Structure theory

2 Cotorsion groups

Exercises

Notes

Chapter 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[λ]

Exercises

Chapter VII: Almost Free Modules Revisited

0 ℵ1-free abelian groups revisited

1 κ-free modules revisited

2 κ -free abelian groups

3 Transversals, λ-systems and NPT

3A Appendix: Reshuffling λ -systems

4 Hereditarily separable groups

5 NPT and the construction of almost free groups

Exercises

Notes

Chapter VIII: ℵ1-Separable Groups

1 Constructions and definitions

2 ℵ1-separable groups under Martin’s Axiom

3 ℵ1-separable groups under PFA

EXERCISES

NOTES

1 Perps and products

2 Countable products of the integers

3 Uncountable products of the integers

4 Radicals and large cardinals

Exercises

Notes

Chapter X: Iterated Sums and Products

1 The Reid class

2 Types in the Reid class

Exercises

Notes

Chapter XI: Topological Methods

1 Inverse and direct limits

2 Completions

3 Density and dual bases

4 Groups of continuous functions

5 Sheaves of abelian groups

EXERCISES

Notes

Chapter XII: An Analysis of EXT

1 Ext and Diamond

2 Ext, MA and proper forcing

3 Baer modules

4 The structure of Ext

5 The structure of Ext when Horn = 0

EXERCISES

Notes

Chapter XIII: Uniformization

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

Exercises

Notes

Chapter XIV: The Black Box and Endomorphism Rings

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

Exercises

Notes

Chapter XV: Some Constructions in ZFC

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 λ-separable groups

Notes

Chapter XVI: Cotorsion Theories, Covers and Splitters

1 Orthogonal classes and splitters

2 Cotorsion theories

3 Almost free splitters

4 The Black Box and Ext

Exercises

Notes

Chapter XVII: Dual Groups

1 Invariants of dual groups

2 Tree groups

3 Criteria for being a dual group

4 Some non-reflexive dual groups

5 Dual groups in L

Notes

Appendix: Open and Solved Problems

Bibliography

Index

Copyright

Dedication

Preface to the revised edition

Paul C. Eklof    University of California, Irvine

This revised and expanded edition presents some of the developments in the subject in the years since the first edition was finished in late 1989. The content has been expanded by approximately 20%. Two entirely new chapters have been added and one chapter (XII, on the structure of Ext) has been rewritten and expanded into two chapters (XII and XIII in this edition). In addition, new material has been added to eight other chapters, including seven additional sections.

An Appendix to the first edition contained a list of open problems. The Appendix in this edition gives the significant progress which has been made on some of these problems; in many cases, the solution is presented in the body of this edition. (There is also a new section of additional open problems.) Another indication of the growth of the subject since the first edition is the fact that the bibliography has grown by about 50%.

The focus of this edition remains on the four major problems enumerated in the Preface to the first edition (see pp. xii-xiv). It is true, even more than before, that an account cannot be given of all results, even within these four areas. Some of the new methods that are added here are: the use of pcf theory to construct almost free groups; the use of sheaves to realize double duals; a pushout construction of modules which make Ext vanish, with applications to splitters, cotorsion theories and the Flat Cover Conjecture; an extension of the method used to analyze Whitehead modules in L to the analysis of Baer modules in ZFC; expanded uses of uniformization techniques; the use of algebraically closed subrings to construct negative answers to the Kaplansky Test Problems; and the use of λ-systems to construct λ-separable groups and λ-free Whitehead groups.

The following briefly describes the new content in terms of the four main subject areas:

1. Almost free modules: Almost free groups of cardinality ≥ ℵω+1 are constructed in section VII.5, using results from pcf theory discussed in §VI.5. The new Chapter XV gives several new constructions of almost free groups: a rigid ℵ1-free group of cardinality ℵ1; ℵn-separable groups with pathological decompositions; λ-separable groups of cardinality λ whenever λ-free groups of cardinality λ exist. In the new Chapter XVI, almost free splitters are investigated.

2. The structure of Ext: An entire chapter (XIII) is dedicated to the important method of uniformization which provides, under suitable set-theoretic hypotheses, the existence of non-trivial examples of modules M (over any non-perfect ring) which, for a given N, satisfy Ext(M, N) = 0. The existence of non-trivial Whitehead groups is shown equivalent to a purely combinatorial property. Baer modules (over non-hereditary rings) are considered in §XII.3. Chapter XVI deals with splitters, that is modules B such that Ext(B, B) = 0, in the general context of cotorsion theories; one application is a proof of the Flat Cover Conjecture for modules over arbitrary rings.

3. The structure of Hom: Sheaves of abelian groups are used in §XI.5 to settle a question left open in the first edition and prove that every dual group is isomorphic to A**/σ[A] for some dual group A. In §XVII.5 dual groups in L are investigated and it is proved, among other things, that there is a reflexive group A which is not isomorphic to A .

4. Endomorphism rings: The Kaplansky Test Problems are introduced in §XIV.5 and it is shown that a weak form of realizability of a ring as an endomorphism ring leads to negative answers to the Test Problems. More constructions using the Black Box are given in §XVI.4.

As in the first edition, a reference to Theorem 3.5 of Chapter X is given as 3.5 within Chapter X and as X.3.5 in other chapters. Almost all references to items in the first eleven chapters of the first edition will refer to the same item in this edition. (New material in those chapters generally occurs either in new sections or at the end of old sections.) However, this is not true for references to later chapters of the first edition.

After writing the first edition of this book, I often remarked that I would not want to write a book without the help of a computer and a co-author. But now, while I have the assistance of even more sophisticated software, I have had, sadly, to prepare this revision without the assistance of my co-author, colleague, and friend, Alan Mekler, who died of cancer in 1992 at the age of 44. He left a void which has not been filled; the many areas of mathematics to which he contributed, including the subject of this book, are poorer for the absence of his deep insight, broad knowledge and brilliant intellect. Along with his many other friends, I continue to miss his exuberant personality, but was inspired by his courage in facing his illness. (An obituary and photograph appears in Order, vol. 9 (1992), 99–101.)

I would like to thank Charly Bitton, Matt Foreman, Rüdiger Gӧbel, David Rector, Greg Schlitt, Phill Schultz, Saharon Shelah, Lutz Strüngmann, Jan Trlifaj, Pauli Väisanen, Simone Wallutis and Tom Winckler for their help. I owe a special debt of gratitude to Oren Kolman, who provided a long list of errata to the first edition. I am grateful to the Mittag-Leffler Institute and Jouko Väänanen for their hospitality during Fall 2000 while I worked on this revision in Stockholm. Moreover, I received support from NSF DMS 98-03126 and DMS-0101155. Last but not least, I owe more than I can say to the support and encouragement of my wife, Sherry.

November, 2001

Preface

Paul C. Eklof    University of California, Irvine

Alan H. Mekler    Simon Fraser University

The modern era in set-theoretic methods in algebra can be said to have begun on July 11, 1973 when Saharon Shelah borrowed László Fuchs’ Infinite Abelian Groups from the Hebrew University library. Soon thereafter, he showed that Whitehead’s Problem — to which many talented mathematicians had devoted much creative energy — was not solvable in ordinary set theory (ZFC). In the years since, Shelah and others have made a good deal of progress on other natural and important problems in algebra, in some cases showing that the problem is not solvable in ZFC, and in others, proving results in ZFC using powerful techniques from set theory (combined, of course, with algebraic methods). One purpose of this book is to make these set- theoretic methods available to the algebraist through an exposition of their use in solving a few major problems. In addition, the logician will find here non-trivial applications of set-theoretic techniques, a few topics such as λ-systems and the Black Box which are less well- known, and also the construction of many structures of interest to model theorists.

Actually, there has long been an affinity between set theory and abelian group theory, going back at least as far as Los’ discovery that measurable cardinals arise naturally in the study of slender groups; also, constructions by transfinite induction have been common, exemplified by Hill’s work on almost free groups. Modern set-theoretic techniques were used in work of Gregory, Eklof and Mekler which came immediately before Shelah’s work. The modern developments take advantage of the great progress that has been made in set theory in the last two decades. The major methods used can be summarized as follows: (1) stationary sets and their generalization, λ- systems; (2) prediction principles, including the diamond principles and the Black Box(es); (3) combinatorial consequences of the Axiom of Constructibility in addition to the diamond principles; (4) internal forcing axioms, especially Martin’s Axiom and the Proper Forcing Axiom; and (5) large cardinal axioms. These are explained in the text, principally in Chapters II, VI and XIII; little is presumed of the algebraist besides a nodding acquaintance with cardinal and ordinal numbers. (One major set theoretic technique which is not discussed is that of forcing; for this the non-logician might want to consult Dales-Woodin 1987 or Kunen 1980.) The algebraic prerequisites are no more than would be covered in an introductory course on groups, rings and fields.

It is no longer possible to give, in a reasonable amount of space, an account of all the results in abelian group theory and homological algebra which use set-theoretic methods. So, as an organizing principle, we have chosen to restrict attention to the torsion-free case, in particular, to the general area described by the title of the book. This means that we have been forced to leave out such interesting work as, for example, that on Crawley’s Problem, the socles of p-groups, or uncountable Butler groups; however, the set-theoretic methods used there are largely represented here. (We have included some of this work in the Bibliography.) Our title and focus are inspired by the notes, Almost Free Abelian Groups, of a seminar given by George A. Reid at Tulane in 1966-67; this seminar was concerned with giving an account of the information known to date concerning Whitehead’s problem … The investigations of this question naturally lead to the consideration of other classes of groups, and other conditions, which, while not perhaps implying freeness, do in one sense or another get fairly close (from the Introduction).

by a (countable) p.i.d.).

1. Almost free modules: for which cardinals κ are there nonfree abelian groups of cardinality κ which are κ-free, that is, every subgroup of cardinality < κ is free? This question is implicit in Fuchs’ 1960 text and explicit in his 1970 volume. () which are not. The existence of almost free objects makes sense as a question in universal algebra and our treatment also deals with this wider setting, although we do not aim for total generality. Chapter VIII deals with a related question, that of the structure and classification of abelian groups which are ℵ1-separable, that is, every countable subset is contained in a countable free direct summand.

2. The structure of Ext: solve Whitehead's Problem — does ) = 0 imply A is free?; determine the structure of the divisible group ) for any torsion-free A. Whitehead’s Problem appears in a 1955 paper of Ehrenfeucht and is attributed to J. H. C. Whitehead. Call a group A a W-group ) = 0; an alternative characterization is that if F ⊇ K are free groups such that F/K ≅ A, then every homomorphism from K lifts to one on F. It was early observed, independently by several people, that W-groups are ℵ1-free; later Rotman showed that they are even separable (that is, every finite subset is contained in a free direct summand) and slender and Chase showed that the Continuum Hypothesis implies that W-groups are what we call strongly ℵ1-free. As mentioned above, the complete solution to the problem cannot be given in ZFC; this is discussed in Chapter XII. New difficulties arise over other rings, even for uncountable p.i.d.’s, so we restrict ourselves largely to abelian groups.

For these first two problems, the solution depends very strongly on the set theory; in fact, an explicit reduction of the problem to a purely set-theoretic form is given, respectively, in sections VII.3 and XII.3 (although the latter is only for groups of cardinality ℵ1). For the next two problems, the situation is different: though there are some results that are not provable in ZFC, the major results are theorems of ZFC whose proofs use powerful set-theoretic tools.

3. The structure of Hom: characterize abelian groups of the form A* (= Hom(A)) in some group-theoretic way; is every such group reflexive, that is, canonically isomorphic to its double dual is slender is very important here, since it implies that free groups (of non-measurable cardinality) are reflexive (cf. III.3.8); we discuss slenderness in Chapter III, including Eda’s extension of Los’ theorem to the measurable case. Call a group a dual group if it is of the form A*. Another question which occurs in the Reid notes is whether every dual group belongs to the Reid class, and closed under direct sums and products. The first negative answers (in ZFC) to the latter question and to the second question above were given by Eda and Ohta. This is discussed in Chapters X and XI. One may also ask if a dual group is strongly non-reflexive, that is, not isomorphic, in any way, to its double dual. A great diversity of constructions of dual groups, reflexive, non-reflexive, and strongly non-reflexive is presented in Chapter XIV, almost all of which is new; other questions about dual groups are raised there, and some are answered.

4. Endomorphism rings: characterize the rings which can appear as the endomorphism rings of certain classes of groups. is indecomposable; and Corner showed that the realization of certain rings can lead to the existence of groups with weird decomposition properties, e.g., a group A such that A is not isomorphic to A ⊕ A but A ⊕ A is isomorphic to A ⊕ A ⊕ A ⊕ A. There has been much work by Corner, Dugas, Göbel, Shelah and others which uses the diamond and Black Box prediction principles to solve cases of this problem. Due to limitations in space and in our expertise, our discussion of this problem, in Chapter XIII, is less comprehensive than that of the other three and serves mainly as an illustration of the use of a new version of the Black Box, due to Shelah, introduced here (and also applied in section 5 of XIV).

Detailed historical information is given in the Notes at the end of each chapter. We have left most of the attribution of results to those Notes; theorems are, naturally, attributed to their original authors, though the proofs here may be different in some cases. We make personal claim only to those results specifically so claimed in the Notes. (However, all mistakes are due to us!) The Exercises at the end of the chapters are, in many cases, guides to further results (often highly interesting and non-trivial but which we have not had room to include in full); we have attempted to point the reader to their sources, whenever possible. We have tried to be accurate in our attributions, but wish to apologize for the inevitable mistakes and oversights.

The Table of Contents contains a guide to the major dependencies between chapters. We have tried to design the book so that the reader can enter it at many different points, and then easily refer back to earlier results as needed.

We are indebted to Saharon Shelah for his help and for allowing some of his work to appear first in this book. We would also like to thank: Bernhard Thomé for his careful reading of most of the manuscript; Aboulmotalab Ihwil and Ali Sagar for taking the lecture notes at SFU which were the genesis of this book; Paul Cohn, Mark Davis, Mark DeBonis, Manfred Dugas, Katsuya Eda, Martin Gilchrist, Rudiger Göbel, Menachem Magidor and Martin Ziegler for their comments on parts of the book; and Rob Ballantyne, Katy Eklof, Sherry Eklof, Mike Fried, Martin Gilchrist, Roger Hunter and Barbara Kukan for their help in the production of the final camera-ready copy.

This work was partially supported by NSF Grant No. DMS-8400451 and NSERC Grant No. A8948.

December, 1989

First edition dedication:

To Sherry and Barbara

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-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, called the dual group of M. A group is called a dual group ) 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

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

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

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

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 function from the class of R-modules to the class of abelian groups so that for any short exact sequence

there are exact sequences

and

These sequences are called the long exact (or Cartan-Eilenberg) sequences induced by

which we will often write as ExtR or even as Ext, especially when R (M, H) = 0 for all M, H and all n ) in the next section for an explicit definition of the vanishing of Ext.)

2 Direct sums and products

An indexed family of modules is a function from a set I, the index set, to a set of modules. We will write the indexed family as (Mi: i ∈ I), or, more often, abuse notation and write it as {Mt: i ∈ I}, keeping in mind that we allow the possibility that Mi = Mj for i ∈ j. The direct product of the indexed family {Mi: i ∈ Isuch that x(i) ∈ Mi for all i ∈ I; it is given a module structure via coordinate-wise operations: (x1+ x2)(i) = x1(i)+x2(i) and (rx)(i) = rx(i, and if Mi = M for all i, it is denoted MI, or Mκ if I has cardinality κ. Sometimes we will denote an element x by (ai)I or (ai)i∈ I if x(i) = ai for all ilet

The direct sum of the indexed family {Mi: i ∈ Iconsisting of all x such that supp(x, if Mi = M for all i, it is denoted M(I) or M(κ) if I has cardinality κ.

Associated with the direct product or sum we have a couple of canonical homomorphisms. For each j ∈ I we have the canonical surjection to x(j); we sometimes also denote by ρj the restriction of ρj . For each j ∈ I we also have the canonical injection λj: Mj which takes a ∈ Mj to x defined by:

Obviously, λj is an isomorphism of Mj with a submodule of ⊕I Mi, and we sometimes identify Mj with this submodule; also, we sometimes regard λj .

For any indexed family {Mi: i ∈ I} and any module H, the map:

which takes f: ⊕IMi → H to (f λi)I is an isomorphism of groups (and of S-modules, if H has a right S-structure). Moreover, the map:

is an isomorphism. (These facts express the universal mapping properties of the direct sum and direct product, respectively.) We also have natural isomorphisms

and

Since HomR(R, H) ≅ H, Homr(R(κ), H) ≅ Hκ for any cardinal κ. The question of when Homr(Rk,H) ≅ H(κ) is the subject of Chapter III.

2.1

Lemma.

For any R-modules M and H, M is H-torsionless if and only if M is isomorphic to a submodule of H¹ for some I.

Proof

Here we are using the same convention as in section 1 and regarding H as an R-S-bimodule, where S = Endr(H); M* and M** are also defined as in section 1. Now suppose first that M is H-torsionless. Let I = M* and define θ: M → HI as follows: for all x ∈ M, θ(x)(y) = 〈y, x〉 for all y ∈ I. Then θ is clearly a homomorphism, and because M is H-torsionless, θ is injective. So θ embeds M as a submodule of H¹. Conversely, suppose M is a submodule of H¹; to see that σm is one-one, consider a non-zero element x of M. There exists j ∈ I such that x(j) ≠ 0; then if ρj is the canonical surjection, ρj M belongs to M* and σM(x)(ρj M) ≠ 0.

It follows easily from this lemma, or from the definition, that a submodule of an If-torsionless module is H-torsionless.

If M0 and M1 are submodules of M such that M = M0 + M1 (i.e., M is the smallest submodule of M containing M0 and M1), and M0 ∩ M1 = {0}, M is said to be the (internal) direct sum of M0 and M1, written M = M0 ⊕ M1. In that case, M and to ⊕I Mi where I = {0,1}. For any submodule M0 of M, there is another submodule M1 of M such that M = M0 ⊕ M1 if and only if there is a projection of M onto M0, i.e., a homomorphism π: M → M0 such that π M0 = the identity on M0. (Let M1 = ker(π).) In this case we say that M0 is a (direct) summand of M.

Note that if M = M0 ⊕ M1 and A is a submodule of M containing M0, then A = M0 ⊕ (A ∩ M1). So if M0 is a direct summand of M and M0 ⊆ A ⊆ M, then M0 is a direct summand of A.

We say that a short exact sequence

splits if there is a homomorphism p: N → M such that φ p = lN, the identity on N. In this case, p is called a splitting of φ, and there is a homomorphism q: M → L, called a splitting of ψ, such that q ψ = 1L, then the short exact sequence is called split exact. Moreover, M is the direct sum of ker(φ) (= ψ[L]) and im(p). Conversely, if ker(φ) is a direct summand of Mequals 0 if and only if every short exact sequence

is a split exact sequence.

Let H and S be as in section 1, and denote Homr(M, H) by M*, HomS(M*,H) by M**, and HomR(M**,H) by M***. Define a map

by: 〈ρ(z),x〉 = 〈z,σm(x)〉 for all z ∈ M*** and x ∈ M.

2.2

Lemma.

ρ σM*is the identity on M*. Hence σM*is one-one, so M* is H-torsionless.

Proof

Let y ∈ M*. Then 〈ρ(σM*(y)),x〉 = 〈σM*(y),(σM(x)〉 = 〈y,σM(x)〉 = 〈y,x〉 for all x ∈ M. Hence ρ(σM*(y)) = y.

Notice also that if

splits, then

is a split short exact sequence because q ψ = 1L implies ψq* = 1L.

An R-module N is said to be free in case there is a subset, B, of N such that B generates N and every set map f: B → M, into an arbitrary R-module M, extends to a homomorphism: N → M; in this case, B is called a basis of N, and every element of N is uniquely a linear combination of elements of B. An R-module is free if and only if it is isomorphic to R(κ) for some cardinal κ (= |B|). For any index set I and i ∈ I, define ei ∈ RI by ei(j) = 1, if i = j and ei(j) = 0, otherwise (in other words, ei = λi(l)). Then {ei: i ∈ I} is a basis for R(I).

If N is free, then every short exact sequence

splits, for arbitrary L and M; indeed, if B is a basis of N, the set map f:B → M which takes each element, b, of B to a pre-image in M under φ, extends to a homomorphism p: N → M, which is a splitting of φwhenever N is free.

If R is a p.i.d., every submodule of a free R-module is free. (A p.i.d. is a principal ideal domain, that is, an integral domain such that every ideal is principal; we decree that a field is not a p.i.d.)

For any module M, there is a free module F and a surjective homomorphism φ: F → M; if we let K = ker(φ) and ψ be the inclusion of K into F, then we obtain a short exact sequence

(If R is a p.i.d., then K is also free, since it is a submodule of a free module, and the short exact sequence above is called a free resolution of M.) In any case, for every R and every .R-module H, by Theorem 1.1, this short exact sequence induces the exact sequence

(The last term is 0 because F is free.) So Ext(M,H) is isomorphic to the group of homomorphisms from K to H modulo those which extend to a homomorphism from F to H. In particular, we would like to call attention to the following criterion which we will make frequent use of in later chapters, such as VII, XII, and XIII, where the vanishing of Ext is studied:

) If F is a free module and K is a submodule of F, Ext(F/K,H) = 0 if and only if every homomorphism from K to H extends to a homomorphism from F to H.

since the homomorphism ψ : pn which takes pn .

We will have occasion to use a pushout construction. The pushout of the diagram

is the commutative diagram

where D is the quotient of B ⊕ C modulo the submodule

and γ(b) = (b, 0) + K and δ(c) = (0, c) + K. Then D/δ[C] ≅ B/β[A], and if β is an injection, then so is δ. We refer to other sources (e.g., Fuchs 1970, p. 52 or Enochs-Jenda 2000, pp. 22f) for the categorical definition of the pushout and its uniqueness properties.

Now we turn to projective and injective modules. An R-module M is called projective (respectively, injective) if and only if for every short exact sequence

the induced map β* : Hom(M, B) → Hom(M, C) (respectively, the induced map α* : Hom(B, M) → Hom(A, M)) is surjective. It is easy to see that a free module is projective, a direct sum of projectives is projective, and a direct summand of a projective is projective. Moreover, M is projective if and only if every short exact sequence of the form

for all H. Since there is such a short exact sequence with A free, M is isomorphic to a direct summand of a free module if M is projective; thus M is projective if and only if M is isomorphic to a summand of a free module. If R is a p.i.d., M is projective if and only if M is free.

The following are two useful facts about projectives, the first due to Eilenberg, the second to Kaplansky.

2.3

Lemma.

If M is projective, then there is a free module F such that M ⊕ F is free.

PROOF

Since M is projective, there is a module P such that P ⊕ M ≅ K where K is free. Now let F = K(ω). Then M ⊕ F ≅ M ⊕ (P ⊕ M)(ω) ≅ M⊕P⊕M⊕P⊕M⊕… ≅ (M ⊕ P)(ω) ≅ F which is free.

2.4

Theorem.

Every projective R-module is a direct sum of countably-generated projective modules.

Proof

If P is projective, then P ⊕ M is free for some module Mwhere each Rt is isomorphic to Rof submodules of P ⊕ M such that F0 = 0; P ⊕ M , and for all υ

(a) Fυ ⊆ Fυ+ 1;

(b) if υ ;

is countably-generated.

. For all υ < αfor some Cυ because, by (cis a direct summand of Fυ and, by (d), Fυ is a direct summand of P ⊕ M. Moreover, Cυ is a homomorphic image of

, so we are done once the claim is proved.

The chain is constructed by transfinite induction. We can suppose that I is well-ordered. Suppose that Fμ has been constructed for all μ < υ. If υ to satisfy (b). If υ = μ + 1 for some μ, and Fμ ≠ P ⊕ M, let i ∈ I be minimal such that Ri is not contained in Fμ. Let G0 = Fμ, G1 = Fμ ⊕ Ri and define by induction onn ∈ ω submodules Gn of P ⊕ M such that Gn ⊆ (Gn+ 1 ∩ P) ⊕ (Gn+ 1 ∩ Msome Jn ⊆ I, and Gn+i/Gn the desired properties.

An R-module M is called flat if __ ⊗R M preserves the exactness of all sequences 0 → A → B (of right R-modules). Every projective module is flat.

A direct product of injective modules is always injective, and a direct summand of an injective module is injective. A module M is injective if and only if every short exact sequence of the form

for all H.

For every module M there is an injective module Q and a one-one homomorphism ψ: M → Q.

Let R be an integral domain. We say that an R-module M is torsion-free if for all a ∈ M \ {0} and r ∈ R \ {0}, ra ≠ 0; and we call M divisible if M = rM for all r ∈ R \ {0}. We call M reduced if {0} is the only divisible submodule of M. Every injective R-module is divisible, and if R is a p.i.d. (or a Dedekind domain), the converse holds. Moreover, for any integral domain R, every torsion-free divisible .R-module is injective (cf. Cartan-Eilenberg 1956, p.128).

If R is a p.i.d., we say that a submodule B of an R-module A is relatively divisible, or pure, in A if rB = rA ∩ B for every r ∈ R. If A is torsion-free, then B is pure in A if and only if A/B is torsion-free. We will have more to say about pure submodules in Chapters IV and V.

2.5. be the group of rationals, under addition. Let Z(p; Z(p(the non-zero complex numbers under multiplication) consisting of the pnth roots of unity for all n ∈ ω. Then every divisible group is isomorphic to

where p ranges over the primes and the γp and δ are cardinals ≥ 0; moreover, these cardinals are uniquely determined by the divisible group.

Every group can be embedded in a divisible group. Furthermore, if A is torsion-free, A such that D/A is a torsion group; indeed, D . D is called the injective hull of A.

If A is a group, we let tA denote the torsion subgroup of A, i.e., {a ∈ A: na = 0 for some n ≠ 0}; then A/tA is a torsion-free group. For any n ≥ 2, we let Z(n) denote the cyclic group of order nand of the Z(n). Hence, every finitely-generated torsion-free group is free.

A group is said be of bounded order if there exists n ≠ 0 such that nA = {0}. Every group of bounded order is a direct sum of cyclic groups.

3 Linear topologies

A linear topology on a module M , about 0 such that each element, Uis a submodule of M; then for every point a ∈ M, {a + U: U } is a base of neighborhoods about a. In that case, a subset O of M is open if and only if for every a ∈ O, there exists U such that a + U ⊆ O. Moreover, addition + :M × M → M is continuous, and for every r ∈ R, scalar multiplication by r: M → M is continuous; that is, M is a topological R-module. We shall use M to denote a topological module equipped with a given linear topology, and U to denote the associated neighborhood base of 0; in context there will be no ambiguity.

3.1

Example.

The R-topology on M is the linear topology which has a base of neighborhoods of 0 consisting of all finite intersections of submodules of the form rM for r ∈ R \ {0}. If M¹ is defined to be ∩{rM: r ∈ R \ {0}}, then M is Hausdorff in its R-topology if and only if M¹ = 0. Thus, if R is an integral domain and M is torsion-free, M is Hausdorff in its R-topology if and only if M -adic topology.

If S is a subset of R, we can similarly define the S-topology to be the linear topology with base of neighborhoods of 0 consisting of all finite intersections of submodules of the form sM for s ∈ S \ {0}. For any S and homomorphism φ: M → N, φ is continuous with respect to the S-topology on M and N since φ[sM] ⊆ sN.

If R is a p.i.d. and p is a prime of R, the p-adic topology on M is defined to be the S-topology on R where S = {pn: n ∈ ω}; it has a base of neighborhoods of 0 consisting of the submodules of the form pnM. This topology is a metrizable topology with metric given by: d(a, b) = p–n if and only if a – b ∈ pnM \pn+ lM. M is Hausdorff in the p-adic topology if and only if ∩{pnM: n ∈ ω} = 0.

If a linear topology on M has a countable base of neighborhoods of 0, then the topology is metrizable, and therefore describable in terms of convergent sequences, but in general we must use nets to describe the topology.

A (neat) Cauchy net in M is defined to be an indexed family {au: U } of elements of M with the property that for all U ⊆ V , aU – aV ∈ V. We say that a is the limit of the Cauchy net {aU: U } if for all U , a – aU . We say that M is complete if M is Hausdorff and every Cauchy net has a (unique) limit in M. Two Cauchy nets {aU : U } and {bU : U } are equivalent if and only if for every U, aU – bU . It is easy to see that if M is complete, then two Cauchy nets have the same limit if and only if they are equivalent.

It is fairly routine to check the following:

3.2

Proposition.

A direct product of modules is complete in the R-topology (respectively, the p-adic topology) if and only if each Mi is complete in the R-topology (resp., the p-adic topology).

Given modules M with linear topologies, we say that θ: M is the completion of M is complete and θ is an algebraic and topological embedding such that θ[M. The completion of M is unique up to isomorphism over M.

We can construct the completion of M either by working directly with equivalence classes of Cauchy nets or as an inverse limit; we shall describe the latter. (For a discussion of inverse limits, see section XI.1.) If we define V ≤ U to mean U ⊆ V for all U, V becomes a directed set, i.e., for all U, V there exists W(= U ∩ V) such that W ≥ U, V. For any U ≥ V we have a canonical map πu,v : M/U → M/V which takes m + U to m + V.

3.3

Proposition.

Suppose M is a Hausdorff module. Denote by the inverse limit with the topologyinduced by the product topology on ; let M/U have the discrete topology. If is the map induced by the canonical maps: M → M/U, then θ is the completion of M.

Proof

such that for all U ≥ Vis a Cauchy net. As well, two Cauchy nets give rise to the same element of the inverse limit if and only if they are equivalent. The rest of the proof is routine.

3.4

Example.

Let R be a p.i.d. equipped with the p-adic topology for some prime p. Then R where rn belongs to a fixed set of representatives of R/pRis the limit in the p. Indeed, given x = (…, xn + pnR, we define rn inductively so that for all m.

is also the completion of R(p), the localization of R at pR, equipped with the p-adic topology. If R is called the ring of p-adic integers; its additive group is denoted Jp.

Now we turn to the f?-topology on a p.i.d. R. The completion of Ris the obvious surjection if rR ⊆ tR.

3.5

Proposition.

Let R be a p.i.d. equipped with the R-topology and let be its completion.

(i) The topology on is the R-topology;

(ii) is a torsion-free reduced R-module and R is a pure submodule of ;

(iii) is isomorphic (as ring) to where p ranges over a representative set of generators of prime ideals of R.

Proof

, where U which are zero in a fixed finite set of coordinates, say r1,…, rk. Conversely, if x = (…, xr+ rR, then for any i = 1,… ,k; so for any r ∈ Rand hence (for a fixed i) there exists yr ∈ R such that riyr . Thus if y = (…, yr + rR,…), y and satisfies riy = x.

is reduced since it is Hausdorff in the R-topology. R by the diagonal mapping which is a pure embedding because if a ∈ R , then a + rR = 0 + rR so a ∈ rR/R is torsion-free. Suppose that x = (…, xr + rRand sx = a ∈ R for some s ∈ R \ {0}. Then sxs + sR = a + sR, so a = sb for some b ∈ R. Hence, for all r ∈ R, xsr – b ∈ rR; so since xr – xsr ∈ rR, we have xr – b ∈ rR. Therefore x = b ∈ R.

(iii) Define ψ which takes x = (…, xr + rRwhose pth coordinate is (.xp + pR+ pnR,…). It is clear that ψ is a ring homomorphism. Moreover, ψ, is one-one because if ψ(x) = 0, then for any r, if pn divides r, then pn divides xr (mod pn) and pn ; hence xr ∈ rR. Finally, using the Chinese Remainder Theorem we can show that ψ is surjective.

3.6

Example.

has a base of neighborhoods of 0 consisting of the subgroups n(n ∈ ω where an . Indeed, given x = (…,xn + n , we define an inductively so that for all m. (This representation is not unique because it depends on the choice of the xn in the coset xn .)

(ii) Let M . Suppose H is a reduced torsion-free group. If f ∈ Hom(M,H), then f is completely determined by f , because f is continuous when M and H -adic topology. Thus Hom(M, H. If f(eα) ≠ 0, for infinitely many α. So Hom(M.

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