K0-stability over monoid algebras

Abstract

(1) Let $R$ be a commutative Noetherian ring of dimension $d$ and $M$ a commutative partially cancellative torsion-free seminormal monoid. Then ${\text{Vec}}_r(R[M])$ is injective stable at $d+1$. This settles a conjecture of Gubeladze for the mentioned class of monoids.

(2) Take the same $R$ as in (1) with an additional assumption that $R$ has a positive characteristic which is prime to $(r-1)!$. Let $M$ be a commutative cancellative torsion-free seminormal positive monoid with a radical ideal $I\subset M$. Then the map ${\text{Um}}_r(R[M])\to {\text{Um}}_r(R[M]∕IR[M])$ is surjective for $r\ge d∕2+2$. This answers a question of Wiemers in some special cases.