Modules of infinite projective dimension

TL;DR

This paper characterizes modules with infinite projective dimension over certain endomorphism algebras in $(n+2)$-angulated categories, linking infinite projective dimension to non-zero factorization ideals, and generalizes a recent result in cluster-tilted algebras.

Contribution

It provides a new characterization of infinite projective dimension modules in the context of $(n+2)$-angulated categories and extends existing results to a broader class of algebras.

Findings

Modules have infinite projective dimension iff associated ideals are non-zero.

Introduces the ideal $I_M$ of endomorphisms factoring through an object M.

Generalizes a recent result for cluster-tilted algebras.

Abstract

We characterize the modules of infinite projective dimension over the endomorphism algebras of Opperman-Thomas cluster tilting objects $X$ in $(n+2)$-angulated categories $(\mathcal C,\Sigma^n,\Theta)$. For an indecomposable object $M$ of $\mathcal C$, we define in this article the ideal $I_M$ of ${\rm End}_{\mathcal C}(\Sigma^nX)$ given by all endomorphisms that factor through ${\rm add} M$, and show that the ${\rm End}_{\mathcal C}(X)$-module ${\rm Hom}_{\mathcal C}(X,M)$ has infinite projective dimension precisely when $I_M$ is non-zero. As an application, we generalize a recent result by Beaudet-Br\"{u}stle-Todorov for cluster-tilted algebras.