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Completion of skew completable unimodular rows

Sampat Sharma

Abstract

Skew completable unimodular rows of odd length are completable over polynomial extension of a local ring if dimension of local ring and length of unimodular rows are same.

Completion of skew completable unimodular rows

Abstract

Skew completable unimodular rows of odd length are completable over polynomial extension of a local ring if dimension of local ring and length of unimodular rows are same.

Paper Structure

This paper contains 8 sections, 15 theorems, 25 equations.

Table of Contents

  1. Introduction
  2. Preliminary Remarks
  3. The elementary symplectic Witt group $W_{E}(R)$
  4. W. Van der Kallen's group structure on ${Um_{d+1}(R)}/{E_{d+1}(R)}$
  5. Krusemeyer's completion of the square of a skew completable row
  6. Completion of skew-completable unimodular rows of length ${d}$
  7. Completion of unimodular 3-vectors
  8. Completion over regular local rings

Key Result

Theorem 1.1

Let $R$ be a local ring of Krull dimension $d\geq 3$ with $d$ odd and $\frac{1}{(d-1)!} \in R.$ Let $v = (v_{0}, v_{1},\dots,v_{d-1})\in Um_{d}(R[X])$ be skew-completable unimodular row over $R[X].$ Then there exists $\rho \in SL_{d}(R[X])\cap E_{d+2}(R[X])$ and an invertible alternating matrix $W \

Theorems & Definitions (19)

  • Theorem 1.1
  • Theorem 1.2
  • Definition 2.1
  • Lemma 2.2
  • Proposition 2.3
  • Definition 3.1
  • Example 3.2: Kaplansky
  • Theorem 3.3
  • Remark 3.5
  • Lemma 3.6
  • ...and 9 more