  Torsion pairs in the category of finitely presented modules over a noetherian
ring can be parametrised by the class of cosilting modules. In this paper, we
characterise such modules in terms of their indecomposable summands, providing
a new approach to the classification of torsion pairs. In particular, we
classify cosilting modules over cluster-tilted algebras of type $\tilde{A}$. We
do this by using a geometric model for finite- and infinite-dimensional modules
over such algebras.
