Abstract

(1)  Let $R$ be a commutative Noetherian ring of dimension $d$ and $M$ a commutative partially cancellative torsion-free seminormal monoid. Then ${Vec}_r\left(R\left[M\right]\right)$ 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 $\left(r-1\right)!$. Let $M$ be a commutative cancellative torsion-free seminormal positive monoid with a radical ideal $I\subset M$. Then the map ${Um}_r\left(R\left[M\right]\right)\to {Um}_r\left(R\left[M\right]∕IR\left[M\right]\right)$ is surjective for $r\ge d∕2+2$. This answers a question of Wiemers in some special cases.

Keywords

Mathematical Subject Classification 2010

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