Abstract

Let R be a Dedekind domain and G a split reductive group, i.e. a Chevalley-Demazure group scheme, of rank ≥ 2 {geq 2}. We prove that G ? (R ? [ x 1, ..., x n ]) = G ? (R) ? E ? (R ? [ x 1, ..., x n ]) ? for any ? n ≥ 1. G(R[x-{1},ldots,x-{n}])=G(R)E(R[x-{1},ldots,x-{n}])quadtext{for any} n% geq 1. In particular, this extends to orthogonal groups the corresponding results of A. Suslin and F. Grunewald, J. Mennicke and L. Vaserstein for G = SL N, Sp 2 ? N {G=mathrm{SL}-{N},mathrm{Sp}-{2N}}. We also deduce some corollaries of the above result for regular rings R of higher dimension and discrete Hodge algebras over R.

Original languageEnglish
JournalJournal of Group Theory
DOIs
Publication statusPublished - 1 Jan 2019

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Dedekind Domain
Chevalley Groups
Polynomial ring
Regular Ring
Group Scheme
Orthogonal Group
Reductive Group
Higher Dimensions
Deduce
Corollary
Algebra

Scopus subject areas

  • Algebra and Number Theory

Cite this

@article{e0c54381d69a44a9b51f426ca70e3f47,
title = "Chevalley groups of polynomial rings over Dedekind domains",
abstract = "Let R be a Dedekind domain and G a split reductive group, i.e. a Chevalley-Demazure group scheme, of rank ≥ 2 {geq 2}. We prove that G ? (R ? [ x 1, ..., x n ]) = G ? (R) ? E ? (R ? [ x 1, ..., x n ]) ? for any ? n ≥ 1. G(R[x-{1},ldots,x-{n}])=G(R)E(R[x-{1},ldots,x-{n}])quadtext{for any} n{\%} geq 1. In particular, this extends to orthogonal groups the corresponding results of A. Suslin and F. Grunewald, J. Mennicke and L. Vaserstein for G = SL N, Sp 2 ? N {G=mathrm{SL}-{N},mathrm{Sp}-{2N}}. We also deduce some corollaries of the above result for regular rings R of higher dimension and discrete Hodge algebras over R.",
author = "Anastasia Stavrova",
year = "2019",
month = "1",
day = "1",
doi = "10.1515/jgth-2019-0100",
language = "English",
journal = "Journal of Group Theory",
issn = "1433-5883",
publisher = "De Gruyter",

}

TY - JOUR

T1 - Chevalley groups of polynomial rings over Dedekind domains

AU - Stavrova, Anastasia

PY - 2019/1/1

Y1 - 2019/1/1

N2 - Let R be a Dedekind domain and G a split reductive group, i.e. a Chevalley-Demazure group scheme, of rank ≥ 2 {geq 2}. We prove that G ? (R ? [ x 1, ..., x n ]) = G ? (R) ? E ? (R ? [ x 1, ..., x n ]) ? for any ? n ≥ 1. G(R[x-{1},ldots,x-{n}])=G(R)E(R[x-{1},ldots,x-{n}])quadtext{for any} n% geq 1. In particular, this extends to orthogonal groups the corresponding results of A. Suslin and F. Grunewald, J. Mennicke and L. Vaserstein for G = SL N, Sp 2 ? N {G=mathrm{SL}-{N},mathrm{Sp}-{2N}}. We also deduce some corollaries of the above result for regular rings R of higher dimension and discrete Hodge algebras over R.

AB - Let R be a Dedekind domain and G a split reductive group, i.e. a Chevalley-Demazure group scheme, of rank ≥ 2 {geq 2}. We prove that G ? (R ? [ x 1, ..., x n ]) = G ? (R) ? E ? (R ? [ x 1, ..., x n ]) ? for any ? n ≥ 1. G(R[x-{1},ldots,x-{n}])=G(R)E(R[x-{1},ldots,x-{n}])quadtext{for any} n% geq 1. In particular, this extends to orthogonal groups the corresponding results of A. Suslin and F. Grunewald, J. Mennicke and L. Vaserstein for G = SL N, Sp 2 ? N {G=mathrm{SL}-{N},mathrm{Sp}-{2N}}. We also deduce some corollaries of the above result for regular rings R of higher dimension and discrete Hodge algebras over R.

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U2 - 10.1515/jgth-2019-0100

DO - 10.1515/jgth-2019-0100

M3 - Article

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JO - Journal of Group Theory

JF - Journal of Group Theory

SN - 1433-5883

ER -