Projective Dimensions and Extensions of Modules from Titled to Cluster-Tilted Algebras

TL;DR

This paper explores how module properties, especially projective dimension and rigidity, transfer from tilted algebra C to cluster-tilted algebra B, providing classifications and conditions for rigidity preservation.

Contribution

It offers a complete classification of projective dimensions of C-modules in B and establishes conditions under which rigidity is preserved from C to B.

Findings

Complete classification of projective dimensions in B

Conditions for C-module rigidity to imply B-module rigidity

Indecomposable rigid C-modules are always rigid in B

Abstract

We study the module categories of a tilted algebra C and the corresponding cluster-tilted algebra B. We investigate how various properties of a C-module are affected when considered in the module category of B. We give a complete classification of the projective dimension of a C-module inside the module category of B. If a C-module M is rigid, we show two sufficient conditions for M to be a rigid B-module. In particular, if M is an indecomposable and rigid C-module, we prove M is always a rigid B-module.