Torsion Pairs and Rigid Objects in Tubes

We classify the torsion pairs in a tube category and show that they are in bijection with maximal rigid objects in the extension of the tube category containing the Prüfer and adic modules. We show that the annulus geometric model for the tube category can be extended to the larger category and interpret torsion pairs, maximal rigid objects and the bijection between them geometrically. We also give a similar geometric description in the case of the linear orientation of a Dynkin quiver of type A.