K-theory and the singularity category of quotient singularities

TL;DR

This paper investigates the algebraic K-theory of the singularity category for quotient singularities, revealing finiteness and vanishing results, and applying these to compute Grothendieck groups and duality properties.

Contribution

It establishes new finiteness and vanishing results for K-theory groups of quotient singularities and applies these to compute Grothendieck groups and duality in algebraic geometry.

Findings

K_0 of the singularity category is finite torsion for isolated quotient singularities

K_1 of the singularity category vanishes for these singularities

Varieties with isolated quotient singularities satisfy rational Poincare duality

Abstract

In this paper we study Schlichting's K-theory groups of the Buchweitz-Orlov singularity category $\mathcal{D}^{sg}(X)$ of a quasi-projective algebraic scheme $X/k$ with applications to Algebraic K-theory. We prove that for isolated quotient singularities $\mathrm{K}_0(\mathcal{D}^{sg}(X))$ is finite torsion, and that $\mathrm{K}_1(\mathcal{D}^{sg}(X)) = 0$. One of the main applications is that algebraic varieties with isolated quotient singularities satisfy rational Poincare duality on the level of the Grothendieck group; this allows computing the Grothendieck group of such varieties in terms of their resolution of singularities. Other applications concern the Grothendieck group of perfect complexes supported at a singular point and topological filtration on the Grothendieck groups.