Resolutions of type $\mathbb{A}$ quantum surface singularities

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