Projective modules over Rees-like algebras and its monoid extensions

TL;DR

This paper extends classical results on projective modules over Rees-like algebras and their monoid extensions, establishing conditions for the existence of unimodular elements and transitivity of elementary group actions.

Contribution

It generalizes Serre and Bass's results by proving unimodularity and transitivity for projective modules over Rees-like algebras and monoid extensions under broader conditions.

Findings

Projective modules of rank ≥ dimension have unimodular elements.

Elementary group actions are transitive on unimodular elements.

Results improve classical theorems of Serre and Bass.

Abstract

Let $A$ be a Rees-like algebra of dimension $d$ and $N$ a commutative partially cancellative torsion-free seminormal monoid. We prove the following results. \begin{enumerate} \item Let $P$ be a finitely generated projective $A$-module of $\rank\geq d$. Then $(i)$ $P$ has a unimodular element; $(ii)$ The action of $\EL(A\oplus P)$ on $\Um(A\oplus P)$ is transitive. \item Let $P$ be a finitely generated projective $A[N]$-module of $\rank~r$. Then $(i)$ $P$ has a unimodular element for $r\geq\max\{3,d\}$; $(ii)$ The action of $\EL(A[N]\oplus P)$ on $\Um(A[N]\oplus P)$ is transitive for $r\geq\max\{2,d\}$. \end{enumerate} These improve the classical results of Serre \cite{Se58} and Bass \cite{Ba64}.