Unimodular rows over monoid extensions of overrings of polynomial rings

TL;DR

This paper proves transitivity of elementary group actions on unimodular rows and freeness of projective modules over certain monoid-extended polynomial rings, extending known results to broader classes of rings and monoids.

Contribution

It establishes new transitivity and freeness results for projective modules over monoid extensions of polynomial rings, generalizing prior work to seminormal monoids and specific ring types.

Findings

Transitivity of $E(A[M] igoplus P)$ on $Um(A[M] igoplus P)$ for certain rings and modules.

Freeness of projective modules over $A[M], A[M]_f$, and $A[M] ensor_R R(T)$ under specified conditions.

Extension of known results to seminormal monoids and rings of type $R[d,m,n]$ and $R[d,m,n]^*$.

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.