Relative cluster tilting objects in triangulated categories

TL;DR

This paper introduces and studies relative cluster tilting objects in triangulated categories, establishing a bijection with support τ-tilting modules and generalizing cluster-tilting mutation concepts.

Contribution

It generalizes cluster-tilting objects to relative versions, develops a theory including partial orders and mutations, and links these to support τ-tilting modules.

Findings

Bijection between $T[1]$-cluster tilting objects and support τ-tilting modules.

Introduction of partial order and mutation for $T[1]$-cluster tilting objects.

Provides partial answers to existing questions in the field.

Abstract

Assume that $\D$ is a Krull-Schmidt, Hom-finite triangulated category with a Serre functor and a cluster-tilting object $T$. We introduce the notion of relative cluster tilting objects, and $T[1]$-cluster tilting objects in $\D$, which are a generalization of cluster-tilting objects. When $\D$ is $2$-Calabi-Yau, the relative cluster tilting objects are cluster-tilting. Let $\la={\rm End}^{op}_{\D}(T)$ be the opposite algebra of the endomorphism algebra of $T$. We show that there exists a bijection between $T[1]$-cluster tilting objects in $\D$ and support $\tau$-tilting $\la$-modules, which generalizes a result of Adachi-Iyama-Reiten \cite{AIR}. We develop a basic theory on $T[1]$-cluster tilting objects. In particular, we introduce a partial order on the set of $T[1]$-cluster tilting objects and mutation of $T[1]$-cluster tilting objects, which can be regarded as a generalization of `cluster-tilting mutation'. As an application, we give a partial answer to a question posed in \cite{AIR}.