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 $r \geq max\{2,d+1\}$. Then the action of $E(A[M] \oplus P)$ on $Um(A[M] \oplus P)$ is transitive and (2) Assume $(R, m, K)$ is a regular local ring containing a field $k$ such that either $char$ $k=0$ or $ char$ $k = p$ and $tr$-$deg$ $K/\mathbb{F}_p \geq 1$. Let $A$ be a ring of type $R[d,m,n]^*$ 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.