# On the vanishing of Relative Negative K-theory

## TL;DR

This paper proves a relative version of the Weibel conjecture, showing vanishing of certain negative K-groups for smooth affine maps of schemes and establishing stability under affine extensions.

## Contribution

It establishes a vanishing theorem for relative negative K-groups in the context of smooth affine maps of noetherian schemes, extending the Weibel conjecture.

## Key findings

- Proves $K_{-n}(f)=0$ for $n > d+1$ in smooth affine maps.
- Shows the natural map $K_{-n}(f) o K_{-n}(f imes A^r)$ is an isomorphism for $n > d$.
- Provides a vanishing result for relative negative K-groups of subintegral maps.

## 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.