Preface to the revised edition

PREFACE

to the revised edition

This revised and expanded edition presents s o m e 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 m e t h o d s 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 )~-systems to construct A-separable groups and ),-free Whitehead groups. The following briefly describes the new content in terms of the four main subject areas: 1. A l m o s t free m o d u l e s - Almost free groups of cardinality _> R~+I are constructed in section VII.5, using results from pcf theory discussed in w The new Chapter XV gives several new constructions of almost free groups: a rigid ~l-free group of cardinality ~1;

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P R E F A C E to the revised edition

Rn-separable groups with pathological decompositions; A-separable groups of cardinality )~ whenever :k-free groups of cardinality ), exist. In the new Chapter XVI, almost free splitters are investigated. 2. T h e s t r u c t u r e of E x t : 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 E x t ( 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 w 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. T h e s t r u c t u r e of Horn: Sheaves of abelian groups are used in w to settle a question left open in the first edition and prove that every dual group is isomorphic to A**/a[A] for some dual group A. In w 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 ~) Z. 4. E n d o m o r p h i s m rings: The Kaplansky Test Problems are introduced in w 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 w 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

P R E F A C E to the revised edition

ix

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, Riidiger GSbel, David Rector, Greg Schlitt, Phill Schultz, Saharon Shelah, Lutz Striingmann, Jan Trlifaj, Pauli V~iis~inen, 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-Leffier Institute and Jouko V~i~in~inen 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. University of California, Irvine November, 2001

PAUL C. E K L O F