Singularity categories of rational double points in arbitrary characteristic

TL;DR

This paper establishes a correspondence between singularity categories of rational double points and Dynkin graphs in any characteristic, revealing new phenomena in positive characteristic and providing counterexamples to existing theorems.

Contribution

It extends the known correspondence between rational double points and Dynkin graphs to arbitrary characteristic, including positive characteristic, and constructs counterexamples to a theorem relating dg singularity categories and Tyurina algebras.

Findings

Correspondence holds in arbitrary characteristic.

Existence of non-isomorphic rational double points with equivalent singularity categories.

Counterexample to Hua and Keller's theorem in positive characteristic.

Abstract

We establish a one-to-one correspondence between the singularity categories of rational double points and the simply-laced Dynkin graphs in arbitrary characteristic. This correspondence is well-known in characteristic zero since the rational double points are quotient singularities in characteristic zero whereas not necessarily in positive characteristic. Considering some rational double points are not taut in characteristic two, three or five, we can see there exist two rational double points which are not analytically isomorphic but whose singularity categories are triangulated equivalent. As an application, we construct a counter-example in positive characteristic of a theorem of Hua and Keller: the dg singularity category of a hypersurface singularity determines its Tyurina algebra.