Resolutions of type A quantum surface singularities

Let B = q[u,v]^{Cₙ₊₁} be a type Aₙ quantum Kleinian singularity, which is an example of a noncommutative surface singularity. This singularity is known to have a noncommutative quasi-crepant resolution Λ , which is an “algebraic” resolution of B . We construct a category X which serves as a “geometric” resolution of B by adapting techniques from quiver GIT and show that X and {mod-}Λ are derived equivalent. Furthermore, we show that the intersection arrangement of lines in the exceptional locus of X corresponds to a type Aₙ Dynkin diagram. This generalises the geometric McKay correspondence for classical Kleinian singularities.