INDUCED AND COINDUCED MODULES IN CLUSTER-TILTED ALGEBRAS

Abstract. 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 n E. This new approach consists of using the induction functor −⊗C B as well as the coinduction functor D(B ⊗C D−). 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 DE is a partial tilting and a τ-rigid C-module and that the induced module DE⊗CB is a partial tilting and a τ-rigid B-module. Furthermore, if C = 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,−) from the cluster