Relative cluster tilting objects in triangulated categories

Abstract:

Assume that $\mathcal {D}$ 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 $\mathcal {D}$, which are a generalization of cluster-tilting objects. When $\mathcal {D}$ is $2$-Calabi–Yau, the relative cluster tilting objects are cluster-tilting. Let $\Lambda =\textrm {End}^{op}_{\mathcal {D}}(T)$ be the opposite algebra of the endomorphism algebra of $T$. We show that there exists a bijection between $T[1]$-cluster tilting objects in $\mathcal {D}$ and support $\tau$-tilting $\Lambda$-modules, which generalizes a result of Adachi–Iyama–Reiten [$\tau$-tilting theory, Compos. Math. 150 (2014), no. 3, 415–452]. 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 Adachi–Iyama–Reiten, loc. cit.