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.

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