Singularity categories of deformations of Kleinian singularities

Crawford, Simon

doi:10.2140/ant.2020.14.349

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Singularity categories of deformations of Kleinian singularities

Simon Crawford

Vol. 14 (2020), No. 2, 349–382

DOI: 10.2140/ant.2020.14.349

Abstract

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 $𝒪^{λ}$ of $R$ parametrised by weights $λ$ . In this paper, we determine the singularity categories $𝒟_{sg} (𝒪^{λ})$ 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 $𝒟_{sg} (R)$ . We also provide a generalisation of the intersection theory appearing in the geometric McKay correspondence to a noncommutative setting.

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Keywords

singularity categories, Kleinian singularities, preprojective algebras

Mathematical Subject Classification 2010

Primary: 14J17

Secondary: 16G20, 16G50, 18E30

Milestones

Received: 24 February 2017

Revised: 26 July 2019

Accepted: 14 September 2019

Published: 29 May 2020

Authors

Simon Crawford
Department of Pure Mathematics University of Waterloo ON Canada

Vol. 14, No. 2, 2020