The Baer splitting problem in the twentyfirst century. (Q1409608)

In 1969 the author published the solution to a 33 year old problem of Reinhold Baer on Abelian groups: If \(\text{Ext}_\mathbb{Z}^1(G,T)=0\) for all torsion groups \(T\), then \(G\) is free. In the present paper the author deals with ``Baer modules'' and attempts to extend the results obtained for Abelian groups to modules over regular domains. A module \(C\) over a (regular) domain \(R\) is a `Baer module' if \(\text{Ext}_R^j(C,T)=0\) for all torsion modules \(T\) and all \(j>0\). It is shown quickly (Corollary~2.2) that \(C\) is a Baer module if \(\text{Ext}_R^1(C,T)=0\) for all torsion modules \(T\). The main result is Theorem~4.5. Let \(R\) be a regular domain and \(C\) a countably generated Baer module over \(R\). Then \(C\) is locally free. Another interesting result that is easily stated is Corollary~3.2. Baer modules over Noetherian integral domains are flat. Methods employed include localization and completion in the \(\mathfrak m\)-adic topology for a local ring \((R,{\mathfrak m},k)\). A theorem of Prüfer is generalized to countable modules over a complete regular local ring (Theorem~4.3).