$d$-abelian quotients of $(d+2)$-angulated categories

Abstract

Let $\mathcal{T}$ be a triangulated category. If $T$ is a cluster tilting object and $I=[\mathrm{add}T]$ is the ideal of morphisms factoring through an object of $\mathrm{add}T$, then the quotient category $\mathcal{T}/I$ is abelian. This is an important result of cluster theory, due to Keller-Reiten and König-Zhu. More general conditions which imply that $\mathcal{T}/I$ is abelian were determined by Grimeland and the first author. Now let $\mathcal{T}$ be a suitable $(d+2)$-angulated category for an integer $d⩾1$. If $T$ is a cluster tilting object in the sense of Oppermann-Thomas and $I=[\mathrm{add}T]$ is the ideal of morphisms factoring through an object of $\mathrm{add}T$, then we show that $\mathcal{T}/I$ is $d$-abelian. The notions of $(d+2)$-angulated and $d$-abelian categories are due to Geiss-Keller-Oppermann and Jasso. They are higher homological generalisations of triangulated and abelian categories, which are recovered in the special case $d=1$. We actually show that if $\Gamma ={\mathrm{End}}_{\mathcal{T}}T$ is the endomorphism algebra of $T$, then $\mathcal{T}/I$ is equivalent to a $d$-cluster tilting subcategory of $\mathrm{mod}\Gamma$ in the sense of Iyama; this implies that $\mathcal{T}/I$ is $d$-abelian. Moreover, we show that $\Gamma$ is a $d$-Gorenstein algebra. More general conditions which imply that $\mathcal{T}/I$ is $d$-abelian will also be determined, generalising the triangulated results of Grimeland and the first author.

19 pages. This is the final accepted version, which has been accepted for publication in the Journal of Algebra

References in corpus (1)

• Abelian quotients of triangulated categories

Cited by in corpus (5)

• Maximal ${\tau }_d$-rigid pairs
• $d$-Auslander-Reiten sequences in subcategories
• The index with respect to a rigid subcategory of a triangulated category
• Ideal approximation in $n$-angulated categories
• $n$-abelian quotient categories