Support $\tau_n$-tilting pairs

TL;DR

This paper generalizes support $ au$-tilting theory to higher dimensions within $(n+2)$-angulated categories, establishing bijections between various classes of $n$-rigid objects and support $ au_n$-tilting pairs, extending prior work.

Contribution

It introduces higher support $ au_n$-tilting pairs and establishes fundamental bijections in $(n+2)$-angulated categories, extending existing theories to a broader higher-dimensional context.

Findings

Bijection between relative $n$-rigid objects and $ au_n$-rigid pairs.

Bijection between relative maximal $n$-rigid objects and support $ au_n$-tilting pairs.

Extension of previous results from $2n$-Calabi-Yau categories and $n=1$ case.

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-J{\o}rgensen. 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 $\mathcal C/{\rm add}\Sigma^n T$, 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-J{\o}rgensen and the work for $n=1$ by Yang-Zhu.