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

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.

Final version, 24 pages, To appear in Contemporary Math. (volume title: K-theory in algebra, analysis and topology)