Unimodular rows over monoid extensions of overrings of polynomial rings

Abstract

Let $R$ be a commutative Noetherian ring of dimension $d$ and $M$ a commutative cancellative torsion-free seminormal monoid. Then: (1) Let $A$ be a ring of type $R[d,m,n]$ and $P$ be a projective $A[M]$-module of rank $r\ge \text{max }\{2,d+1\}$. Then the action of $E(A[M]\oplus P)$ on $(A[M]\oplus P)$ is transitive and (2) Assume $(R,m,K)$ is a regular local ring containing a field $k$ such that either $\text{char }k=0$ or $\text{char }k=p$ and $\text{tr}$-$\text{deg }K∕{\mathbb{𝔽}}_p\ge 1$. Let $A$ be a ring of type $R{[d,m,n]}^{\ast }$ and $f\in R$ be a regular parameter. Then all finitely generated projective modules over $A[M]$, $A{[M]}_f$ and $A[M]{\otimes }_{R}R(T)$ are free. When $M$ is free both results are due to Keshari and Lokhande (2014).