Book Review: Almost free modules. Set-theoretic methods

Abstract

Forty years ago, J. H. C. Whitehead asked whether or not an abelian group A had to be free (i.e., free abelian) if all abelian extensions of the group Z of integers by A were splitting [in other words, if Ext1^, Z) = 0]. This purely group theoretical question was motivated by problems outside the realm of group theory (the second Cousin problem, see Stein [7], and questions raised by Dixmier [2]). For countable abelian groups A, Stein [7] and Ehrenfeucht [4] gave affirmative answers, but the solution for groups of higher cardinalities looked extremely difficult, as witnessed by several publications, which contained only fragmentary results in the general case. The Whitehead problem remained for a while one of the handful of major open problems in the theory of abelian groups.In 1973, a young mathematician, Saharon Shelah, got interested in the problem. He had the bright idea of approaching the problem from a different angle by scrutinizing the underlying sets. He was able to prove that already for groups A of cardinality Nj , the Whitehead problem was undecidable in ZFC (the Zermelo-Fraenkel axioms of set theory plus the Axiom of Choice). More precisely, in the constructible universe L, E\XX(A, Z) = 0 implies that A has to be free, while in models in which the Continuum Hypothesis fails but Martin's Axiom holds, there do exist nonfree groups A with Ext'(^, Z) = 0. This unexpected result was a big surprise and drew immediately the attention to the relevance of set-theoretical techniques in solving purely algebraic problems.Shelah's discovery marked the beginning of the modern era of applications of powerful set-theoretical methods in algebra. Since then a great deal of significant work has been done in the area. The systematic use of additional set-theoretical hypotheses led to new insight into (and sometimes to a solution of) several open problems in algebra-as it did in other fields of mathematics. Shelah has remained the leading force in the developments, providing leadership and continuous stimulus to the subject.The book by Eklof and Mekler under review presents an excellent up-todate and in-depth survey of most of the recent developments in the area. The authors-who themselves have been at the forefront of the developmentshave written a book, which is a fine example of how two different fields of mathematics can interact and create a new, flourishing field. (Actually, the title