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Absence of torsion in orbit space

Sampat Sharma

Abstract

In this paper, we prove that if $R$ is a local ring of dimension $d,$ $d\geq 2$ and $\frac{1}{d!}\in R$ then the group $\frac{Um_{d+1}(R[X])}{E_{d+1}(R[X])}$ has no $k$-torsion, provided $k\in GL_{1}(R).$ We also prove that if $R$ is a regular ring of dimension $d,$ $d\geq 2$ and $\frac{1}{d!}\in R$ such that $E_{d+1}(R)$ acts transitively on $Um_{d+1}(R)$ then $E_{d+1}(R[X])$ acts transitively on $Um_{d+1}(R[X]).$

Absence of torsion in orbit space

Abstract

In this paper, we prove that if is a local ring of dimension and then the group has no -torsion, provided We also prove that if is a regular ring of dimension and such that acts transitively on then acts transitively on

Paper Structure

This paper contains 11 sections, 24 theorems, 18 equations.

Table of Contents

  1. Introduction
  2. Preliminary Results
  3. Abelian group structure on $\frac{Um_{n}(R)}{E_{n}(R)}$ for $\hbox{sdim}(R)\leq 2n-4$
  4. Mennicke--Newman Lemma
  5. L.N Vaserstein's power operation $\chi_{k}$ on $\frac{Um_{r}(R)}{E_{r}(R)},$ for $k\in \mathbb{Z}, r\geq 3$
  6. On the equality of $\chi_{k}([v])$ with $[v]^{k}, k\in \mathbb{N}$
  7. The elementary symplectic Witt group $W_{E}(R)$
  8. The Suslin matrices
  9. Relative elementary groups
  10. Absence of torsion in $\frac{Um_{d+1}(R[X])}{E_{d+1}(R[X])}$
  11. An application to completion of unimodular rows

Key Result

Theorem 1.0.1

Let $R$ be a local ring of dimension $d$ and let $\frac{1}{d!} \in R$ , then the group $\frac{Um_{d+1}(R[X])}{E_{d+1}(R[X])}$ has no $k$-torsion, provided $kR = R.$

Theorems & Definitions (35)

  • Theorem 1.0.1
  • Corollary 1.0.2
  • Corollary 1.0.3
  • Definition 2.0.1
  • Example 2.0.2
  • Definition 2.1.1
  • Definition 2.1.2
  • Definition 2.1.3
  • Theorem 2.1.4
  • Lemma 2.2.1
  • ...and 25 more