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Elementary Action of Classical Groups on Unimodular Rows Over Monoid Rings

Rabeya Basu, Maria Ann Mathew

Abstract

The elementary action of symplectic and orthogonal groups on unimodular rows of length $2n$ is transitive for $2n \geq \max(4, d+2)$ in the symplectic case, and $2n \geq \max(6, 2d+4)$ in the orthogonal case, over monoid rings $R[M]$, where $R$ is a commutative noetherian ring of dimension $d$, and $M$ is commutative cancellative torsion free monoid. As a consequence, one gets the surjective stabilization bound for the $K_1$ for classical groups. This is an extension of J. Gubeladze's results for linear groups.

Elementary Action of Classical Groups on Unimodular Rows Over Monoid Rings

Abstract

The elementary action of symplectic and orthogonal groups on unimodular rows of length is transitive for in the symplectic case, and in the orthogonal case, over monoid rings , where is a commutative noetherian ring of dimension , and is commutative cancellative torsion free monoid. As a consequence, one gets the surjective stabilization bound for the for classical groups. This is an extension of J. Gubeladze's results for linear groups.

Paper Structure

This paper contains 6 sections, 23 theorems, 62 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Classical Groups
  3. Monoids
  4. Prologue: Simplification of hypothesis
  5. Transitive action of ${\rm E}(2n,R[M])$ on ${\rm Um}(2n,R[M])$
  6. Surjective Stabilization of $\rm K_1$-group

Key Result

Theorem 1.1

Let $R$ be a ring and $M$ a monoid. Assume $2n \geq \hbox{$\mathbb{D}$}(R)$. Then ${\rm E}(2n,R[M])$ acts transitively on ${\rm Um}(2n,R[M]),$ in other words,

Figures (4)

  • Figure :
  • Figure :
  • Figure :
  • Figure :

Theorems & Definitions (46)

  • Theorem 1.1
  • Theorem 1.2
  • Remark 2.1
  • Definition 2.2
  • Proposition 2.3
  • Lemma 2.4
  • Proposition 3.1
  • Remark 3.2
  • Definition 3.3
  • Definition 3.4
  • ...and 36 more