Non-Commutative Deformations of Derived McKay Correspondence for A(n) singularities

TL;DR

This paper proves that the derived McKay correspondence extends to semi-universal non-commutative deformations of toric surface singularities, linking commutative and non-commutative resolutions in a broader deformation context.

Contribution

It establishes the extension of derived equivalences to semi-universal non-commutative deformations for toric surface singularities, advancing the understanding of McKay correspondence.

Findings

Derived equivalence extends to non-commutative deformations.

Links between commutative and non-commutative resolutions are established.

Enhances the framework of McKay correspondence in deformation settings.

Abstract

The derived McKay correspondence conjecture says that there is an equivalence of triangulated categories between the bounded derived categories of commutative and non-commutative crepant resolutions of a Gorenstein singularity. We will prove that this derived equivalence extends between the semi-universal non-commutative deformations of the commutative and the non-commutative crepant resolutions of a toric surface singularity.