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On completion of unimodular rows over polynomial extension of finitely generated rings over $\mathbb{Z}$

Sampat Sharma

Abstract

In this article, we prove that if $R$ is a finitely generated ring over $\mathbb{Z}$ of dimension $d, d\geq2, \frac{1}{d!}\in R$, then any unimodular row over $R[X]$ of length $d+1$ can be mapped to a factorial row by elementary transformations.

On completion of unimodular rows over polynomial extension of finitely generated rings over $\mathbb{Z}$

Abstract

In this article, we prove that if is a finitely generated ring over of dimension , then any unimodular row over of length can be mapped to a factorial row by elementary transformations.

Paper Structure

This paper contains 7 sections, 22 theorems, 31 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. The elementary symplectic Witt group $W_{E}(R)$
  4. The Vaserstein Rule
  5. Coordinate power in Witt group
  6. Roitman's degree reduction technique
  7. The main results

Key Result

Theorem 1.1

Let $R$ be a finitely generated ring of dimension $d,$$d\geq 2,$$\frac{1}{d!}\in R$ and $v\in Um_{d+1}(R[X]).$ Then for some $(u_{0},\ldots, u_{d})\in Um_{d+1}(R[X]).$

Theorems & Definitions (25)

  • Theorem 1.1
  • Definition 2.1
  • Lemma 2.2
  • Theorem 3.1
  • Lemma 3.2
  • Proposition 3.3
  • Proposition 3.4
  • Lemma 3.5
  • Lemma 3.6
  • Lemma 4.1
  • ...and 15 more