In this article, an analog of the Quillen’s local-global principle was established for elementary subgroup of Infinite Unimodular Vector Group <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$SUm_{\ge r}^\infty (R)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>S</mi> <mi>U</mi> <msubsup> <mi>m</mi> <mrow> <mo>≥</mo> <mi>r</mi> </mrow> <mi>∞</mi> </msubsup> <mrow> <mo stretchy="false">(</mo> <mi>R</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> obtained from the Special Unimodular Vector Group <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$SUm_r(R)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>S</mi> <mi>U</mi> <msub> <mi>m</mi> <mi>r</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>R</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>, a subgroup of <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$SL_{2^r}(R)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>S</mi> <msub> <mi>L</mi> <msup> <mn>2</mn> <mi>r</mi> </msup> </msub> <mrow> <mo stretchy="false">(</mo> <mi>R</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>, generated by the Suslin matrices <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$S_r(v,w)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>S</mi> <mi>r</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>v</mi> <mo>,</mo> <mi>w</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>, corresponding to the pairs (v, w), with <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$v \cdot w^T = 1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>v</mi> <mo>·</mo> <msup> <mi>w</mi> <mi>T</mi> </msup> <mo>=</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation>, by the placement of “circle operation”.

A local global principle for the infinite elementary unimodular vector group

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发表日期：

2026年2月16日

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Selby Jose,
Dipali A. Savant

摘要

In this article, an analog of the Quillen’s local-global principle was established for elementary subgroup of Infinite Unimodular Vector Group $SUm_{\ge r}^\infty (R)$ $S U m_{\geq r}^{\infty} (R)$ obtained from the Special Unimodular Vector Group $SUm_r(R)$ $S U m_{r} (R)$ , a subgroup of $SL_{2^r}(R)$ $S L_{2^{r}} (R)$ , generated by the Suslin matrices $S_r(v,w)$ $S_{r} (v, w)$ , corresponding to the pairs (v, w), with $v \cdot w^T = 1$ $v \cdot w^{T} = 1$ , by the placement of “circle operation”.