Browse by author
Lookup NU author(s): Dr Karin Jacobsen, Professor Peter Jorgensen
This work is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License (CC BY-NC-ND).
Let ${\mathscr T}$ be a triangulated category. If $T$ is a cluster tilting object and $I = [ \operatorname{add} T ]$ is the ideal of morphisms factoring through an object of $\operatorname{add} T$, then the quotient category ${\mathscr T} / I$ is abelian. This is an important result of cluster theory, due to Keller--Reiten and K\"{o}nig--Zhu. More general conditions which imply that ${\mathscr T} / I$ is abelian were determined by Grimeland and the first author.Now let ${\mathscr T}$ be a suitable $( d+2 )$-angulated category for an integer $d \geqslant 1$. If $T$ is a cluster tilting object in the sense of Oppermann--Thomas and $I = [ \operatorname{add} T ]$ is the ideal of morphisms factoring through an object of $\operatorname{add} T$, then we show that ${\mathscr T} / I$ 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 $\Gamma = \operatorname{End}_{ \mathscr T }T$ is the endomorphism algebra of $T$, then ${\mathscr T} / I$ is equivalent to a $d$-cluster tilting subcategory of $\operatorname{mod} \Gamma$ in the sense of Iyama; this implies that ${\mathscr T} / I$ is $d$-abelian. Moreover, we show that $\Gamma$ is a $d$-Gorenstein algebra.More general conditions which imply that ${\mathscr T} / I$ is $d$-abelian will also be determined, generalising the triangulated results of Grimeland and the first author.
Author(s): Jacobsen KM, Jorgensen P
Publication type: Article
Publication status: Published
Journal: Journal of Algebra
Year: 2018
Volume: 521
Pages: 114-136
Print publication date: 01/03/2019
Online publication date: 29/11/2018
Acceptance date: 28/11/2018
Date deposited: 03/12/2018
ISSN (print): 0021-8693
ISSN (electronic): 1090-266X
Publisher: Academic Press Inc.
URL: https://doi.org/10.1016/j.jalgebra.2018.11.019
DOI: 10.1016/j.jalgebra.2018.11.019
Altmetrics provided by Altmetric
