Induced and Coinduced Modules over Cluster-Tilted Algebras

We propose a new approach to study the relation between the module categories of a tilted algebra C and the corresponding cluster-tilted algebra B. This new approach consists of using the induction functor as well as the coinduction functor. We give an explicit construction of injective resolutions of projective B-modules, and as a consequence, we obtain a new proof of the 1-Gorenstein property for cluster-tilted algebras. We show that the relation extension bimodule is a partial tilting and a tau-rigid C-module and that the corresponding induced module is a partial tilting and a tau-rigid B-module. Furthermore, if C tilted from a hereditary algebra A, we compare the induction and coinduction functors to the Buan-Marsh-Reiten functor from the cluster-category of A to the module category of B. We also study the question which B-modules are actually induced or coinduced from a module over a tilted algebra