The duals of Warfield groups

Abstract

All considered groups in this paper will be abelian groups. By a p-local group we mean a module over Zp, the ring of integers localized at the prime p. Recall that an abelian group is said to be simply presented provided it can be defined in terms of generators and relations in such a way that all of the relations are of the form mx = 0 or mx = y. For instance, totally projective p-groups and completely decomposable torsion-free groups are simply presented. Warfield groups are direct summands of simply presented groups, in the p-local case they are also called Warfield modules. Warfield [W] established some characterizations of Warfield modules and gave a complete set of isomorphism invariants. Using an alternate definition, Hunter, Richman and Walker [HRW1], [HRW2], [HRi] proved existence theorems for p-local and global Warfield groups and were able to classify global Warfield groups. Moore [M] described Warfield modules in terms of quasi-sequentially nice submodules. Introducing the concept of a knice submodule, Hill and Megibben [HM1] characterized Warfield modules using the third axiom of countability. In [HM3], they proved a variety of characterizations of Warfield groups using Axiom 3 and the global definitions of nice and knice subgroups. Some more characterizations of Warfield groups are contained in [L1]. In [F], the following question was formulated as Problem 65: Which are the compact abelian groups whose duals are totally projective p-groups? Kiefer [K] described those groups dualizing the various characterizations of totally projective p-groups. Since Warfield groups are generalizations of totally projective p-groups, it seems to be a natural question to ask for the structure of the duals of Warfield groups. The aim of this paper is, both