On cluster-tilting objects in a triangulated category with Serre duality

Let D be a Krull–Schmidt, Hom-finite triangulated category with a Serre functor and a cluster-tilting object T. We introduce the notion of an F_Λ-stable support τ-tilting module, induced by the shift functor and the Auslander–Reiten translation, in the cluster-tilted algebra (Formula presented.). We show that there exists a bijection between basic cluster-tilting objects in D and basic F_Λ-stable support τ-tilting Λ-modules. This generalizes a result of Adachi–Iyama–Reiten [1]. As a consequence, we obtain that all cluster-tilting objects in D have the same number of nonisomorphic indecomposable direct summands.