The K-theory of (compound) Du Val singularities

TL;DR

This thesis provides an explicit description of the Grothendieck and divisor class groups for a broad class of Du Val and compound Du Val singularities, including their deformations and related quotient singularities.

Contribution

It establishes an explicit isomorphism between the Grothendieck group and the direct sum of integers and the class group for various singularities, extending known results to new cases.

Findings

Explicit descriptions of G_0(R) and Cl(R) for cDV singularities

Isomorphism between G_0(R) and Z ⊕ Cl(R) established in multiple settings

Conjecture on the size of the Grothendieck group for symmetric groups

Abstract

This thesis gives a complete description of the Grothendieck group and divisor class group for large families of two and three dimensional singularities. The main results presented throughout, and summarised in Theorem 8.1.1, give an explicit description of the Grothendieck group and class group of Kleinian singularities, their deformations, and compound Du Val (cDV) singularities in a variety of settings. For such rings R, the main results assert that there exists an isomorphism between $G_0(R)$ and $\mathbb{Z} \oplus \mathrm{Cl}(R)$, and the class group is explicitly presented. More precisely, we establish these results for 2-dimensional deformations of global type A Kleinian singularities, 3-dimensional isolated complete local cDV singularities admitting a noncommutative crepant resolution, any 3-dimensional type A complete local cDV singularity, polyhedral quotient singularities (which are non-isolated), and any isolated cDV singularity admitting a minimal model with only type cAn cDV singularities. We also study various complex reflection groups in the setting of symplectic quotient singularities, for which this isomorphism does not hold, and conjecture based on computer evidence that the reduced Grothendieck group in the case of the symmetric group has size $n!$.