Relative rigid objects in triangulated categories

TL;DR

This paper explores the relationship between relative rigid objects in triangulated categories and support τ-tilting modules over endomorphism algebras, establishing bijections that unify several known results.

Contribution

It introduces the concept of $R[1]$-rigid objects in triangulated categories and proves they correspond to support τ-tilting modules, extending existing bijections.

Findings

$R[1]$-rigid objects in $ ext{pr}(R)$ correspond to $ au$-rigid $ ext{End}(R)$-modules.

Maximal $R[1]$-rigid objects correspond to support $ au$-tilting modules.

Various known bijections are recovered under specific assumptions.

Abstract

Let $\mathcal{T}$ be a Krull-Schmidt, Hom-finite triangulated category with suspension functor $[1]$. Let $R$ be a basic rigid object, $\Gamma$ the endomorphism algebra of $R$, and $\operatorname{\mathsf{pr}}(R)\subseteq \mathcal{T}$ the subcategory of objects finitely presented by $R$. We investigate the relative rigid objects, \ie $R[1]$-rigid objects of $\mathcal{T}$. Our main results show that the $R[1]$-rigid objects in $\operatorname{\mathsf{pr}}(R)$ are in bijection with $\tau$-rigid $\Gamma$-modules, and the maximal $R[1]$-rigid objects with respect to $\operatorname{\mathsf{pr}}(R)$ are in bijection with support $\tau$-tilting $\Gamma$-modules. We also show that various previously known bijections involving support $\tau$-tilting modules are recovered under respective assumptions.