# Relative rigid objects in triangulated category

Research paper by Changjian Fu, Shengfei Geng, Pin Liu

Indexed on: 13 Aug '18Published on: 13 Aug '18Published in: arXiv - Mathematics - Rings and Algebras

#### Abstract

Let $\mathcal{T}$ be a Krull-Schmidt, Hom-finite triangulated category with suspension functor $[1]$ and $R\in \mathcal{T}$ a basic rigid object with endomorphism algebra $\Gamma$. We investigate the $R[1]$-rigid objects in the finitely presented subcategory $\operatorname{pr} R$ of $\mathcal{T}$. Among others, it is proved that there is a bijection between the set of basic $R[1]$-rigid objects in $\operatorname{pr} R$ and the set of basic $\tau$-rigid pairs of $\Gamma$-modules, which induces a one-to-one correspondence between the set of basic maximal $R[1]$-rigid objects with respect to $\operatorname{pr} R$ and the set of basic support $\tau$-tilting $\Gamma$-modules. As applications, we recover various previously known bijections provided that $R$ and $\mathcal{T}$ satisfy corresponding assumptions.