# Unimodular rows over monoid rings

## TL;DR

This paper proves that the elementary group acts transitively on unimodular rows over monoid rings for sufficiently large n, extending classical results from polynomial rings and advancing K-theory of monoid rings.

## Contribution

It generalizes Suslin's polynomial ring results to monoid rings, completing a long-standing project and opening new avenues in K-theory research.

## Key findings

- Elementary action is transitive on unimodular rows over monoid rings for n >= max(d+2,3)
- Extends classical polynomial ring results to more general monoid rings
- Provides new insights into the structure of K-theory of monoid rings

## Abstract

For a commutative Noetherian ring R of dimension d and a commutative cancellative monoid M, the elementary action on unimodular n-rows over the monoid ring R[M] is transitive for n>=max(d+2,3). The starting point is the case of polynomial rings, considered by A. Suslin in the 1970s. The main result completes a project, initiated in the early 1990s, and suggests a new direction in the study of K-theory of monoid rings.