Xingjia Zhou | Searcharxiv

arXiv2022

Pre-$(n+2)$-angulated categories

In this article, we introduce the notion of pre-$(n+2)$-angulated categories as higher dimensional analogues of pre-triangulated categories defined by Beligiannis-Reiten. We first show that the idempotent completion of a pre-$(n+2)$-angulated category admits a unique structure of pre-$(n+2)$-angulated category. Let $(\mathscr{C},\mathbb{E},\mathfrak{s})$ be an $n$-exangulated category and $\mathscr{X}$ be a strongly functorially finite subcategory of $\mathscr{C}$. We then show that the quotient category $\mathscr{C}/\mathscr{X}$ is a pre-$(n+2)$-angulated category.These results allow to construct several examples of pre-$(n+2)$-angulated categories. Moreover, we also give a necessary and sufficient condition for the quotient $\mathscr{C}/\mathscr{X}$ to be an $(n+2)$-angulated category.

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arXiv2020

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 $(\mathcal C,Σ^n,Θ)$. For an indecomposable object $M$ of $\mathcal C$, we define in this article the ideal $I_M$ of ${\rm End}_{\mathcal C}(Σ^nX)$ 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-Brüstle-Todorov for cluster-tilted algebras.

math.RT↗