Maximal ${\tau }_d$-rigid pairs

Abstract

Let $\mathcal{T}$ be a $2$-Calabi--Yau triangulated category, $T$ a cluster tilting object with endomorphism algebra $\Gamma$. Consider the functor $\mathcal{T}(T,-):\mathcal{T}\to \mathrm{mod}\Gamma$. It induces a bijection from the isomorphism classes of cluster tilting objects to the isomorphism classes of support $\tau$-tilting pairs. This is due to Adachi, Iyama, and Reiten. The notion of $(d+2)$-angulated categories is a higher analogue of triangulated categories. We show a higher analogue of the above result, based on the notion of maximal ${\tau }_d$-rigid pairs.

arXiv comment13 pages. This is the final version, accepted for publication in the Journal of Algebra

References in corpus (13)

• $n$-angulated categories
• $n$-abelian and $n$-exact categories
• n-representation-finite algebras and twisted fractionally Calabi-Yau algebras
• The axioms for n-angulated categories
• Selfinjective quivers with potential and 2-representation-finite algebras
• Higher Nakayama algebras I: Construction
• An introduction to higher Auslander-Reiten theory
• Auslander-Reiten $(d+2)$-angles in subcategories and a $(d+2)$-angulated generalisation of a theorem by Brüning
• $d$-abelian quotients of $(d+2)$-angulated categories
• $d$-Auslander-Reiten sequences in subcategories
• A Gabriel-type theorem for cluster tilting
• Higher support tilting I: higher Auslander algebras of linearly oriented type $A$
• Nakayama-type phenomena in higher Auslander--Reiten theory