Morita equivalence of deformations of Kleinian singularities

TL;DR

This paper classifies Morita equivalences among deformations of Kleinian singularities and generalized Weyl algebras, resolving a long-standing open problem for generic parameters in noncommutative algebra.

Contribution

It provides a complete classification of Morita equivalence classes for these algebraic structures under generic conditions, advancing understanding in noncommutative geometry.

Findings

Classified all Morita equivalent pairs of generalized Weyl algebras for generic parameters.

Extended classification results to noncommutative deformations of Kleinian singularities.

Resolved a 30-year-old open question in the field.

Abstract

In this paper we classify all Morita equivalent pairs of (classical) generalized Weyl algebras for generic values of the parameters, thus positively settling a 30 year old question posed by T.Hodges. We also prove a similar result for noncommutative deformations of arbitrary Kleinian singularities provided the corresponding parameters are very generic.