The K-theory of (compound) Du Val singularities

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 Z + 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 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!. This work requires a range of tools including, but not limited to, Nagata’s theo- rem, knitting techniques, Knörrer periodicity, the singularity category, and the computer-algebra system MAGMA. Of particular note is the application of knitting techniques which leads to independently interesting results on the symmetry of the quivers underlying the modifying algebras of Kleinian and cDV singularities