Cotorsion theories cogenerated by \(\aleph_1\)-free Abelian groups. (Q1415022)

This paper deals with \(\aleph_1\)-free Abelian groups \(G\). The authors consider the class \(TC(G)\) of all torsion groups \(T\) with \(\text{Ext}(G,T)=0\), that is all torsion groups cogenerated by \(G\). We also refer to \textit{L. Strüngmann} and \textit{S. L. Wallutis} [Contemp. Math. 273, 269-283 (2001; Zbl 0985.20042)]. Recall from the Baer splitting problem (solved by Griffith) that \(TC(G)\) are all torsion groups if and only if \(G\) is free. Thus ``\(TC(G)\)'' is a natural extension of that splitting problem. Assuming CH the authors characterize classes of Abelian groups realized as \(TC(G)\). Moreover, assuming \(V=L\), for any torsion-free group \(G\) there is an \(\aleph_1\)-free group \(H\) with \(|H|=\aleph_1\) and \(TC(G)=TC(H)\). The latter does not hold under CH.