Modules of infinite projective dimension

We characterize the modules of infinite projective dimension over the endomorphism algebras of Opperman-Thomas cluster tilting objects $X$ in $(n+2)$-angulated categories $(\mathcal{C},{\mathrm{\Sigma }}^n,\mathrm{\Theta })$. For an indecomposable object $M$ of $\mathcal{C}$, we define in this article the ideal $I_M$ of ${\mathrm{E}\mathrm{n}\mathrm{d}}_{\mathcal{C}}({\mathrm{\Sigma }}^{n}X)$ given by all endomorphisms that factor through $\mathrm{a}\mathrm{d}\mathrm{d}M$, 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.