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We show, for a finitely generated partially cancellative torsion-freecommutative 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 $d \ge 1$. Inparticular, $K_i(R[M]) = 0$ for $i < -d$. This is a generalization of Weibel's$K$-dimension conjecture to monoid algebras. We show that this generalizationfails for $X[M]$ if $X$ is not an affine scheme. We also show that theLevine-Weibel Chow group of 0-cycles $\CH^{LW}_0(k[M])$ vanishes for anyfinitely generated commutative partially cancellative monoid $M$ if $k$ is analgebraically closed field.

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