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 $B=C⋉E$. This new approach consists of using the induction functor $-{\otimes }_{C}B$ as well as the coinduction functor $D(B{\otimes }_{C}D-)$. We 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.

27 pages, 1 figure, v2: Following a referee's suggestion, the section on injective resolutions has been removed and is now part of our paper `Injective Presentations of Induced Modules over Cluster-Tilted Algebras'

References in corpus (1)

• Hochschild cohomology of the cluster-tilted algebras of finite representation type

Cited by in corpus (2)

• $\tau$-Tilting Theory and $\tau$-Slices
• Projective Dimensions and Extensions of Modules from Titled to Cluster-Tilted Algebras