A negative answer to a question of Bass

TL;DR

This paper provides a counterexample in algebraic K-theory showing that the equality of K_0 groups does not extend from one polynomial variable to two, answering a question posed by Bass.

Contribution

It constructs a specific isolated surface singularity over a number field where K_0(R) equals K_0(R[t]) but not K_0(R[t_1,t_2]), demonstrating a negative answer to Bass's question.

Findings

K_0(R) = K_0(R[t]) for the constructed singularity

K_0(R) ≠ K_0(R[t_1,t_2]) for the same singularity

Counterexample to Bass's conjecture in algebraic K-theory

Abstract

In this companion paper to arXiv:0802.1928 we provide an example of an isolated surface singularity $R$ over a number field such that $K_0(R) = K_0(R[t])$ but $K_0(R) \neq K_0(R[t_1,t_2])$. This answers, negatively, a question of Bass.