Injective Presentations of Induced Modules over Cluster-Tilted Algebras

Every cluster-tilted algebra $B$ is the relation extension $C\ltimes $\text{Ext}$^2_C(DC,C)$ofatiltedalgebra$C$\mathrm{.}A$B$-module is called induced if it is of the form $M{\otimes }_{C}B$ 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.