Induced and Coinduced Modules in 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 $B=C\ltimes E$. This new approach consists of using the induction functor $-\otimes_C B$ as well as the coinduction functor $D(B\otimes_C D-)$. 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 $\tau$-rigid $C$-module and that the induced module $DE\otimes_C B$ is a partial tilting and a $\tau$-rigid $B$-module. Furthermore, if $C=\text{End}_A T$ for a tilting module $T$ over a hereditary algebra $A$, we compare the induction and coinduction functors to the Buan-Marsh-Reiten functor $\text{Hom}_{\mathcal{C}_A}(T,-)$ 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.