Injective Presentations of Induced Modules over Cluster-Tilted Algebras

Every cluster-tilted algebra B is the relation extension C⋉ExtC2(DC,C) of a tilted algebra C. A B-module is called induced if it is of the form M⊗_CB for some C-module M. We study the relation between the injective presentations of a C-module and the injective presentations of the induced B-module. Our main result is an explicit construction of the modules and morphisms in an injective presentation of any induced B-module. In the case where the C-module, and hence the B-module, is projective, our construction yields an injective resolution. In particular, it gives a module theoretic proof of the well-known 1-Gorenstein property of cluster-tilted algebras.