Modules of infinite projective dimension

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 ${\mathrm{End}}_{\mathcal{C}}({\Sigma }^{n}X)$ given by all endomorphisms that factor through $\mathrm{add}M$, and show that the ${\mathrm{End}}_{\mathcal{C}}(X)$-module ${\mathrm{Hom}}_{\mathcal{C}}(X,M)$ has infinite projective dimension precisely when $I_M$ is non-zero. As an application, we generalize a recent result by Beaudet-Brüstle-Todorov for cluster-tilted algebras.

We have added Corollary 2.5