$n$-abelian quotient categories

TL;DR

This paper demonstrates that quotient categories derived from $(n+2)$-angulated categories with cluster-tilting subcategories are $n$-abelian, linking them to $n$-cluster tilting subcategories of abelian categories and establishing their Gorenstein properties.

Contribution

It establishes that such quotient categories are $n$-abelian and connects them to $n$-cluster tilting subcategories in abelian categories, extending recent results.

Findings

Quotient categories are $n$-abelian.

When $ ext{C}$ has a Serre functor, $ ext{C}/ ext{X}$ is equivalent to an $n$-cluster tilting subcategory.

The module category $ ext{mod}( ext{Σ}^{-1} ext{X})$ is Gorenstein with dimension at most $n$.

Abstract

Let $\C$ be an $(n+2)$-angulated category with shift functor $\Sigma$ and $\X$ be a cluster-tilting subcategory of $\C$. Then we show that the quotient category $\C/\X$ is an $n$-abelian category. If $\C$ has a Serre functor, then $\C/\X$ is equivalent to an $n$-cluster tilting subcategory of an abelian category $\textrm{mod}(\Sigma^{-1}\X)$. Moreover, we also prove that $\textrm{mod}(\Sigma^{-1}\X)$ is Gorenstein of Gorenstein dimension at most $n$. As an application, we generalize recent results of Jacobsen-J{\o}rgensen and Koenig-Zhu.