Relative cluster tilting objects in triangulated categories

Abstract

© 2018 American Mathematical Society. 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 Λ = 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 τ-tilting Λ-modules, which generalizes a result of Adachi–Iyama–Reiten [τ-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.