Equivalent generating vectors of finitely generated modules over commutative rings

TL;DR

This paper investigates the action of special linear groups on generating vectors of finitely generated modules over certain commutative rings, extending previous results to broader classes of rings.

Contribution

It provides a description of the quotient of generating vectors by group action for modules over elementary divisor or almost local-global coherent Prüfer rings, generalizing earlier work.

Findings

Describes the structure of generating vector orbits under group action.

Extends previous results to broader classes of rings.

Provides a classification of generating vectors for finitely presented modules.

Abstract

Let $R$ be a commutative ring with identity and let $M$ be an $R$-module which is generated by $\mu$ elements but not fewer. We denote by $\operatorname{SL}_n(R)$ the group of the $n \times n$ matrices over $R$ with determinant $1$. We denote by $\operatorname{E}_n(R)$ the subgroup of $\operatorname{SL}_n(R)$ generated by the the matrices which differ from the identity by a single off-diagonal coefficient. Given $n \ge \mu$ and $G \in \left\{\operatorname{SL}_n(R),\operatorname{E}_n(R)\right\}$, we study the action of $G$ by matrix right-multiplication on $\operatorname{V}_n(M)$, the set of elements of $M^n$ whose components generate $M$. Assuming that $M$ is finitely presented and that $R$ is an elementary divisor ring or an almost local-global coherent Pr\"ufer ring, we obtain a description of $\operatorname{V}_n(M)/G$ which extends the author's earlier result on finitely generated modules over quasi-Euclidean rings.