On the vanishing of Relative Negative K-theory

Abstract

In this article, we study the relative negative K-groups $K_{-n}(f)$ of a map $f:X\to S$ of schemes. We prove a relative version of the Weibel conjecture i.e. if $f:X\to S$ is a smooth affine map of noetherian schemes with $\dim S=d$ then $K_{-n}(f)=0$ for $n>d+1$ and the natural map $K_{-n}(f)\to K_{-n}(f\times {\mathbb{A}}^r)$ is an isomorphism for all $r>0$ and $n>d.$ We also prove a vanishing result for relative negative K-groups of a subintegral map.

arXiv comment15 pages, Final Version, Some changes made due to referee report