Homogeneity and complete decomposability of torsion free knot modules (Q1174756)

Let \(\Lambda\) be the integral group ring of the infinite cyclic group \(\langle t\rangle\). A \(\Lambda\)-module \(\mathcal A\) is called a knot module if \(\mathcal A\) is finitely generated over \(\Lambda\) and \(t-1\) induces an automorphism of \(\mathcal A\). In this paper, all groups considered are torsion free abelian groups, and all \(\Lambda\)-modules are \(\mathbb{Z}\)-torsion free knot modules. Generalizing the results of \textit{R. H. Crowell} [Duke Math. J. 30, 349--354 (1963; Zbl 0118.39304)] and of \textit{D. W. Sumners} [Invent. Math. 15, 78--90 (1972; Zbl 0229.55001)], the author obtains a necessary and sufficient condition for a cyclic \(\Lambda\)-module to be homogeneous of type \(\tau\). Secondly, he gives a necessary and sufficient condition for a cyclic \(\Lambda\)-module to be completely decomposable, from which he derives the result that there exists a \(\Lambda\)-module \(\mathcal A\) whose model is not \(\mathbb{Z}\)-isomorphic to \(\mathcal A\). This is a negative answer to a conjecture of D. W. Sumners in the above paper.