Singularity categories of deformations of Kleinian singularities

Abstract

Let $G$ be a finite subgroup of $SL\left(2,k\right)$ and let $R=k{\left[x,y\right]}^G$ be the coordinate ring of the corresponding Kleinian singularity. In 1998, Crawley-Boevey and Holland defined deformations ${\mathcal{𝒪}}^{\lambda }$ of $R$ parametrised by weights $\lambda$. In this paper, we determine the singularity categories ${\mathcal{𝒟}}_{sg}\left({\mathcal{𝒪}}^{\lambda }\right)$ of these deformations, and show that they correspond to subgraphs of the Dynkin graph associated to $R$. This generalises known results on the structure of ${\mathcal{𝒟}}_{sg}\left(R\right)$. We also provide a generalisation of the intersection theory appearing in the geometric McKay correspondence to a noncommutative setting.