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=C⋉EB=C⋉E. This new approach consists of using the induction functor −⊗CB−⊗CB as well as the coinduction functor D(B⊗CD−)D(B⊗CD−). We show that DE is a partial tilting and a τ-rigid C  -module and that the induced module DE⊗CBDE⊗CB is a partial tilting and a τ-rigid B  -module. Furthermore, if C=EndATC=EndAT for a tilting module T over a hereditary algebra A  , we compare the induction and coinduction functors to the Buan–Marsh–Reiten functor HomCA(T,−)HomCA(T,−) from the cluster-category of A to the module category of B. We also study the question as to which B-modules are actually induced or coinduced from a module over a tilted algebra.