Vol. 14, No. 2, 2020

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Singularity categories of deformations of Kleinian singularities

Simon Crawford

Vol. 14 (2020), No. 2, 349–382

Let G be a finite subgroup of SL(2,k) and let R = k[x,y]G be the coordinate ring of the corresponding Kleinian singularity. In 1998, Crawley-Boevey and Holland defined deformations 𝒪λ of R parametrised by weights λ. In this paper, we determine the singularity categories 𝒟sg(𝒪λ) 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 𝒟sg(R). We also provide a generalisation of the intersection theory appearing in the geometric McKay correspondence to a noncommutative setting.

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singularity categories, Kleinian singularities, preprojective algebras
Mathematical Subject Classification 2010
Primary: 14J17
Secondary: 16G20, 16G50, 18E30
Received: 24 February 2017
Revised: 26 July 2019
Accepted: 14 September 2019
Published: 29 May 2020
Simon Crawford
Department of Pure Mathematics
University of Waterloo