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 ${\rm End}_{\mathcal C}(\Sigma^nX)$ 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.