Homotopy Invariance of $K$-groups using Grayson's Technique

TL;DR

This paper proves the homotopy invariance of higher $K$-groups for Pr"ufer domains by demonstrating the vanishing of Nil$K$-groups using Grayson's technique, providing a new proof of a known result.

Contribution

It offers a new proof of homotopy invariance of $K$-theory for Pr"ufer domains by showing Nil$K$-groups vanish, differing from previous methods.

Findings

$K_n(R) \,\cong\, K_n(R[s])$ for all $n>0$ when $R$ is a Pr"ufer domain

Nil$K$-groups vanish for these rings

New proof technique using Grayson's approach

Abstract

Homotopy invariance of $K$-theory has always been a point of interest. In this article, with the help of the generators of Nil$K$-groups using Grayson's technique, it is shown that if $R$ is a Pr\"{u}fer domain, then $K_n(R) \cong K_n(R[s])$ for all $n>0.$ This is a specific case of the already published work of the author and Vivek Sadhu. However, contrary to the method used before, we specifically prove the isomorphism by showing that Nil$K$-groups vanish.