$\tau$-Rigid Modules from Tilted to Cluster-Tilted Algebras

TL;DR

This paper investigates the relationship between $ au$-rigid modules in tilted algebras and their corresponding cluster-tilted algebras, providing insights into module category structures.

Contribution

It characterizes which $ au$-rigid modules in tilted algebras remain $ au$-rigid after transitioning to cluster-tilted algebras.

Findings

Identifies conditions for $ au$-rigid modules to persist across algebra types

Establishes connections between module categories of tilted and cluster-tilted algebras

Enhances understanding of module rigidity in algebra transformations

Abstract

We study the module categories of a tilted algebra C and the corresponding cluster-tilted algebra B. In particular, we study which $\tau$-rigid C-modules are also $\tau$-rigid B-modules.