# Negative $K$-theory and Chow group of monoid algebras

## TL;DR

This paper extends Weibel's K-dimension conjecture to monoid algebras, showing isomorphisms in negative K-theory for certain monoid algebras over quasi-excellent Q-algebras and analyzing Chow groups.

## Contribution

It generalizes Weibel's conjecture to monoid algebras and investigates the behavior of K-theory and Chow groups in this context.

## Key findings

- K_i(R) ≅ K_i(R[M]) for i ≤ -d when R is a quasi-excellent Q-algebra
- K_i(R[M]) = 0 for i < -d under specified conditions
- Chow group of 0-cycles vanishes for finitely generated monoid algebras over algebraically closed fields

## Abstract

We show, for a finitely generated partially cancellative torsion-free commutative monoid $M$, that $K_i(R) \cong K_i(R[M])$ whenever $i \le -d$ and $R$ is a quasi-excellent $\Q$-algebra of Krull dimension $d \ge 1$. In particular, $K_i(R[M]) = 0$ for $i < -d$. This is a generalization of Weibel's $K$-dimension conjecture to monoid algebras. We show that this generalization fails for $X[M]$ if $X$ is not an affine scheme. We also show that the Levine-Weibel Chow group of 0-cycles $\CH^{LW}_0(k[M])$ vanishes for any finitely generated commutative partially cancellative monoid $M$ if $k$ is an algebraically closed field.