Vanishing of the negative homotopy K-theory of quotient singularities

TL;DR

This paper improves the understanding of negative homotopy K-theory groups for quotient singularities, showing they vanish below -1 and providing explicit bounds for cyclic cases, using noncommutative algebraic geometry.

Contribution

It extends previous results by proving vanishing of negative homotopy K-theory below -1 for quotient singularities, with explicit bounds in cyclic cases, employing noncommutative algebraic geometry techniques.

Findings

Negative homotopy K-theory groups vanish below -1 for quotient singularities.

Explicit bounds are provided for the first negative homotopy K-theory group in cyclic quotient singularities.

The results are achieved using noncommutative algebraic geometry methods.

Abstract

Making use of Gruson-Raynaud's technique of "platification par eclatement", Kerz and Strunk proved that the negative homotopy K-theory groups of a Noetherian scheme X of Krull dimension d vanish below -d. In this note, making use of noncommutative algebraic geometry, we improve this result in the case of quotient singularities by proving that the negative homotopy K-theory groups vanish below -1. Furthermore, in the case of cyclic quotient singularities, we provide an explicit "upper bound" for the first negative homotopy K-theory group.