A projective characterization of a class of abelian groups. (Q297504)

All groups in this article are abelian. The \(\mathbb Z\)-adic topology on a group \(G\) uses \(\{nG:n\in\mathbb N\}\) as a neighborhood base for 0. In this topology, the closure of 0 is \(\bigcap_{n\in\mathbb N}nG\) and is known as the first Ulm subgroup of \(G\), denoted \(G^1\).NEWLINENEWLINE In the present paper, Keef defines the \(\mathbb Z^2\)-adic topology on \(G\) using \(\{nG^1:n\in\mathbb N\}\) as neighborhood base of 0. He further defines a collection of short exact sequences called \(L^2\)-pure-exact, whose properties with respect to the \(\mathbb Z^2\)-adic topology have parallels with the properties of pure-exact sequences with respect to the \(\mathbb Z\)-adic topology. In particular, it is established in Proposition 2.8 that \(G\) is \(L^2\)-pure-injective iff \(G\) is the direct sum of a divisible group and a group that is complete in the \(\mathbb Z\)-adic topology iff \(G^1\) is a pure-injective group. Corollary 2.9 shows that there are enough \(L^2\)-pure-injectives. Theorem 2.10 establishes that a group is \(L^2\)-projective iff it has a subgroup that is \(\Sigma\)-cyclic such that the corresponding factor group is also \(\Sigma\)-cyclic. The theorem also gives us that there are enough \(L^2\)-pure-projectives.NEWLINENEWLINE Keef argues that Proposition 2.8 and its corollary give a fairly complete classification of \(L^2\)-pure-injectives, but that the \(L^2\)-pure-projectives are at least as complex as the class of all separable \(p\)-groups. In particular, he claims it would be extremely difficult to obtain a satisfactory description of the class of torsion \(L^2\)-pure-projectives.NEWLINENEWLINE However, in Section 3 Keef shows that if \(G\) is a torsion-free \(L^2\)-pure projective group, then it is locally free and that the converse holds if \(G\) has countable torsion-free rank. Also, Section 4 contains a complete description of the countable groups of torsion-free rank one that are \(L^2\)-pure-projective (Theorem 4.1).