d-abelian quotients of (d+2)-angulated categories
Journal article, Peer reviewed
Accepted version
Åpne
Permanent lenke
http://hdl.handle.net/11250/2633604Utgivelsesdato
2019Metadata
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- Institutt for matematiske fag [2531]
- Publikasjoner fra CRIStin - NTNU [38525]
Sammendrag
Let T be a triangulated category. If t is a cluster tilting object and I=add t 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>=0. If t is a cluster tilting object in the sense of Oppermann–Thomas and I=add t 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 A is the endomorphism algebra of t, then T/I is equivalent to a d-cluster tilting subcategory of mod A 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. d-abelian quotients of (d+2)-angulated categories