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

Abstract

Let T be a triangulated category. If T is a cluster tilting object and I=[addT] is the ideal of morphisms factoring through an object of add T, then the quotient category 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 T/I is abelian were determined by Grimeland and the first author.Now let 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=[addT] is the ideal of morphisms factoring through an object of add T, then we show that 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 Γ=EndTT is the endomorphism algebra of T, then T/I is equivalent to a d-cluster tilting subcategory of mod Γ in the sense of Iyama; this implies that T/I is d-abelian. Moreover, we show that Γ is a d-Gorenstein algebra.More general conditions which imply that T/I is d-abelian will also be determined, generalising the triangulated results of Grimeland and the first author.