Extensionless modules of finite rank over countable Dedekind domains (Q1203559)

A module \(M\) is extensionless if every extension of \(M\) by \(M\) splits. Let \(D\) be a countable Dedekind domain with quotient field, \(K\). Let \(M\) be a \(D\)-module with \(k\otimes_ D M\) an \(n\)-dimensional \(K\)-vector space, i.e. \(M\) is of finite rank, \(n\). We show that \(M\) is extensionless if and only if \(M\) is isomorphic to \(\bigoplus_ nR\), where \(R\) is a rank one \(D\)-module with \(R/D\) a direct sum of Prüfer modules (possibly none) and a finitely-generated torsion \(D\)-module. In order to prove this result we first prove that if \(M_ 1\) and \(M_ 2\) are torsion-free \(D\)- modules of finite rank, then \(\text{Ext}(M_ 2,M_ 1)\) is either 0 or has cardinality \(2^{\aleph_ 0}\). Analogous results are also proved when \(D\) is replaced by \(S\), \(S\) a countable finite-dimensional hereditary tame algebra over an algebraically closed field.