McKay correspondence for canonical orders

TL;DR

This paper extends the McKay correspondence to canonical orders, establishing derived equivalences between their minimal resolutions and skew group rings, and explicitly relating exceptional curves to indecomposable modules.

Contribution

It constructs minimal resolutions of canonical orders using non-commutative cyclic covers and skew group rings, and explicitly formulates the McKay correspondence in this context.

Findings

Derived equivalence between minimal resolutions and skew group rings for canonical orders

Explicit numerical McKay correspondence relating curves and modules

Construction of minimal resolutions via non-commutative cyclic covers

Abstract

Canonical orders, introduced in the minimal model program for orders, are simultaneous generalisations of Kleinian singularities and their associated skew group rings. In this paper, we construct minimal resolutions of canonical orders via non-commutative cyclic covers and skew group rings. This allows us to exhibit a derived equivalence between minimal resolutions of canonical orders and the skew group ring form of the canonical order in all but one case. The Fourier-Mukai transform used to construct this equivalence allows us to make explicit, the numerical version of the McKay correspondence for canonical orders which, relates the exceptional curves of the minimal resolution to the indecomposable reflexive modules of the canonical order.