On the vanishing of negative K-groups

TL;DR

This paper proves the vanishing of negative K-groups for certain schemes over an infinite perfect field of positive characteristic, assuming strong resolution of singularities, thus supporting Weibel's conjecture.

Contribution

It establishes the vanishing of negative K-groups for schemes over specific fields under the assumption of strong resolution of singularities, partially confirming Weibel's conjecture.

Findings

Negative K-groups vanish for schemes over infinite perfect fields of positive characteristic.

Supports Weibel's conjecture in the context of schemes with finite type over such fields.

Results depend on the assumption of strong resolution of singularities.

Abstract

Let k be an infinite perfect field of positive characteristic p and assume that strong resolution of singularities holds over k. We prove that, if X is a d-dimensional noetherian scheme whose underlying reduced scheme is essentially of finite type over the field k, then the negative K-group K_q(X) vanishes for every q < -d. This partially affirms a conjecture of Weibel.