Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Symplectic orbits of unimodular rows

Tariq Syed

Abstract

For a smooth affine algebra $R$ of dimension $d \geq 3$ over a field $k$ and an invertible alternating matrix $χ$ of rank $2n$, the group $Sp(χ)$ of invertible matrices of rank $2n$ over $R$ which are symplectic with respect to $χ$ acts on the right on the set $Um_{2n}(R)$ of unimodular rows of length $2n$ over $R$. In this paper, we prove that $Sp(χ)$ acts transitively on $Um_{2n}(R)$ if $k$ is algebraically closed, $d! \in k^{\times}$ and $2n \geq d$.

Symplectic orbits of unimodular rows

Abstract

For a smooth affine algebra of dimension over a field and an invertible alternating matrix of rank , the group of invertible matrices of rank over which are symplectic with respect to acts on the right on the set of unimodular rows of length over . In this paper, we prove that acts transitively on if is algebraically closed, and .

Paper Structure

This paper contains 6 sections, 12 theorems.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. The groups $W_E (R)$ and $W_{SL}(R)$
  4. Motivic homotopy theory and Suslin matrices
  5. The generalized Vaserstein symbol
  6. Symplectic orbits of unimodular rows

Key Result

Lemma 2.1

Let $R$ be a commutative ring and let $\chi \in A_{2n} (R)$. If $\varphi \in GL_{2n} (R)$ such that its class $[\varphi] \in K_{1} (R)$ lies in $\ker (H)$, then there are $m \in \mathbb{N}$ and $\varphi' \in SL_{2n+2m} (R)$ such that $[\varphi] = [\varphi'] \in K_1 (R)$ and $\varphi'$ is symplectic

Theorems & Definitions (21)

  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Proposition 2.3
  • Theorem 3.1
  • Theorem 3.2
  • Theorem 3.3
  • Theorem 3.4
  • Lemma 3.5
  • ...and 11 more