Vanishing of negative $K$-theory in positive characteristic

TL;DR

This paper proves that negative algebraic K-theory vanishes in positive characteristic for certain schemes, using Gabber's alterations and recent advances in motivic homotopy theory, partially answering a question by Weibel.

Contribution

It establishes the vanishing of negative K-theory in positive characteristic for quasi-excellent schemes, extending previous results through new geometric and homotopical methods.

Findings

Negative K-theory vanishes for n < -dim X in positive characteristic.

Uses Gabber's alterations to relate K-theory to motivic homotopy theory.

Provides partial confirmation of Weibel's conjecture in this setting.

Abstract

We show how a theorem of Gabber on alterations can be used to apply work of Cisinski, Suslin, Voevodsky, and Weibel to prove that $K_n(X)[1/p] = 0$ for $n < - \dim X$ where $X$ is a quasi-excellent noetherian scheme, $p$ is a prime that is nilpotent on $X$, and $K_n$ is the $K$-theory of Bass-Thomason-Trobaugh. This gives a partial answer to a question of Weibel.