Tilting modules over tame hereditary algebras. (Q2856653)

Let \(R\) be a hereditary tame algebra over a field. The paper under review gives a complete classification of the infinite dimensional tilting modules over \(R\). One of the key ingredients used to do this, is the so called tilting module arising from a universal localization, that is the tilting modules which are a direct sum of the form \(R_{\mathcal U}\oplus R_{\mathcal U}/R\), where \(\mathcal U\) is a set of quasi-simple modules and by \(R_{\mathcal U}\) is denoted the universal localization at \(\mathcal U\). A large tilting module \(T\) over \(R\), is shown to have a direct sum decomposition \(T=Y\oplus M\), where \(Y\) is finite dimensional and \(M\) has no finite dimensional indecomposable direct summand. The finite dimensional part \(Y\) is explicitly described, and it is called a branch module. For every such branch module and every subset of the set of tubes in the Auslander-Reiten quiver of \(R\), it is constructed a tilting module, using a suitable universal localization, and it is shown that every large tilting module is of this form. Further this classification leads also to a direct decomposition of every large tilting module \(T=\bigoplus t_\lambda(T)\oplus\overline T\), where \(\bigoplus t_\lambda(T)\) is indexed over the set of all tubes and is a torsion module (hence a direct sum of Prüfer modules and finite dimensional regular modules) and \(\overline T\) is a torsion-free module.