Support ${\tau }_n$-tilting pairs

Abstract

We introduce the higher version of the notion of Adachi-Iyama-Reiten's support $\tau$-tilting pairs, which is a generalization of maximal ${\tau }_n$-rigid pairs in the sense of Jacobsen-Jørgensen. Let $\mathcal{C}$ be an $(n+2)$-angulated category with an $n$-suspension functor ${\Sigma }^n$ and an Opperman-Thomas cluster tilting object. We show that relative $n$-rigid objects in $\mathcal{C}$ are in bijection with ${\tau }_n$-rigid pairs in the $n$-abelian category $\mathcal{C}/\mathrm{add}{\Sigma }^{n}T$, and relative maximal $n$-rigid objects in $\mathcal{C}$ are in bijection with support ${\tau }_n$-tilting pairs. We also show that relative $n$-self-perpendicular objects are in bijection with maximal ${\tau }_n$-rigid pairs. These results generalise the work for $\mathcal{C}$ being $2n$-Calabi-Yau by Jacobsen-Jørgensen and the work for $n=1$ by Yang-Zhu.

arXiv comment17 pages, comments are welcome