Nearly Morita equivalences and rigid objects

TL;DR

This paper extends the concept of nearly-Morita equivalence to mutations of rigid objects in triangulated categories with Serre functors, introducing pseudo-Morita equivalence and exploring localisations.

Contribution

It generalizes the nearly-Morita equivalence concept from cluster-tilting objects to arbitrary rigid objects, defining pseudo-Morita equivalence and analyzing localisations.

Findings

Endomorphism algebras may not be nearly-Morita equivalent after mutation.

Introduces pseudo-Morita equivalence as a weaker property.

Describes results in terms of localisations.

Abstract

If two cluster-tilting objects of an acyclic cluster category are related by a mutation, then their endomorphism algebras are nearly-Morita equivalent [Buan-Marsh-Reiten], i.e. their module categories are equivalent "up to a simple module". This result has been generalised by D. Yang, using a result of P-G. Plamondon, to any simple mutation of maximal rigid objects in a 2-Calabi--Yau triangulated category. In this paper, we investigate the more general case of any mutation of a (non-necessarily maximal) rigid object in a triangulated category with a Serre functor. In that setup, the endomorphism algebras might not be nearly-Morita equivalent and we obtain a weaker property that we call pseudo-Morita equivalence. Inspired by results of Buan-Marsh, we also describe our result in terms of localisations.