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If two cluster-tilting objects of an acyclic cluster category are related bya mutation, then their endomorphism algebras are nearly-Morita equivalent[Buan-Marsh-Reiten], i.e. their module categories are equivalent "up to asimple module". This result has been generalised by D. Yang, using a result ofP-G. Plamondon, to any simple mutation of maximal rigid objects in a2-Calabi--Yau triangulated category. In this paper, we investigate the moregeneral case of any mutation of a (non-necessarily maximal) rigid object in atriangulated category with a Serre functor. In that setup, the endomorphismalgebras might not be nearly-Morita equivalent and we obtain a weaker propertythat we call pseudo-Morita equivalence. Inspired by results of Buan-Marsh, wealso describe our result in terms of localisations.

Nearly Morita equivalences and rigid objects