An elementary description of K1(R) without elementary matrices

Thomas Hüttemann; Zuhong Zhang

doi:10.12958/adm1568

An elementary description of K₁(R) without elementary matrices

Thomas Hüttemann, Zuhong Zhang^*

^*Corresponding author for this work

School of Mathematics and Physics

Research output: Contribution to journal › Article › peer-review

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Abstract

Let R be a ring with unit. Passing to the colimit with respect to the standard inclusions GL(n,R)→GL(n+1,R) (which add a unit vector as new last row and column) yields, by definition, the stable linear group GL(R); the same result is obtained, up to isomorphism, when using the `opposite' inclusions (which add a unit vector as new first row and column). In this note it is shown that passing to the colimit along both these families of inclusions simultaneously recovers the algebraic K-group K1(R)=GL(R)/E(R) of R, giving an elementary description that does not involve elementary matrices explicitly.

Original language	English
Pages (from-to)	79
Number of pages	4
Journal	Algebra and Discrete Mathematics
Volume	30
Issue number	1
Early online date	01 Jun 2020
DOIs	https://doi.org/10.12958/adm1568
Publication status	Early online date - 01 Jun 2020

ASJC Scopus subject areas

Algebra and Number Theory

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An elementary description of K1(R) without elementary matrices
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2 Invited talk
1 Oral presentation

Some remarks on the “fundamental theorem” in algebraic K-theory
Huettemann, T. (Invited speaker)
01 Sept 2023
Activity: Talk or presentation types › Invited talk
Fun with K-theory
Huettemann, T. (Speaker)
13 May 2022
Activity: Talk or presentation types › Invited talk
K-theory: from linear equations to the fundamental theorem
Huettemann, T. (Speaker)
19 Feb 2021
Activity: Talk or presentation types › Oral presentation

Cite this

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title = "An elementary description of K1(R) without elementary matrices",

abstract = "Let R be a ring with unit. Passing to the colimit with respect to the standard inclusions GL(n,R)→GL(n+1,R) (which add a unit vector as new last row and column) yields, by definition, the stable linear group GL(R); the same result is obtained, up to isomorphism, when using the `opposite' inclusions (which add a unit vector as new first row and column). In this note it is shown that passing to the colimit along both these families of inclusions simultaneously recovers the algebraic K-group K1(R)=GL(R)/E(R) of R, giving an elementary description that does not involve elementary matrices explicitly.",

author = "Thomas H{\"u}ttemann and Zuhong Zhang",

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doi = "10.12958/adm1568",

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AU - Zhang, Zuhong

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N2 - Let R be a ring with unit. Passing to the colimit with respect to the standard inclusions GL(n,R)→GL(n+1,R) (which add a unit vector as new last row and column) yields, by definition, the stable linear group GL(R); the same result is obtained, up to isomorphism, when using the `opposite' inclusions (which add a unit vector as new first row and column). In this note it is shown that passing to the colimit along both these families of inclusions simultaneously recovers the algebraic K-group K1(R)=GL(R)/E(R) of R, giving an elementary description that does not involve elementary matrices explicitly.

AB - Let R be a ring with unit. Passing to the colimit with respect to the standard inclusions GL(n,R)→GL(n+1,R) (which add a unit vector as new last row and column) yields, by definition, the stable linear group GL(R); the same result is obtained, up to isomorphism, when using the `opposite' inclusions (which add a unit vector as new first row and column). In this note it is shown that passing to the colimit along both these families of inclusions simultaneously recovers the algebraic K-group K1(R)=GL(R)/E(R) of R, giving an elementary description that does not involve elementary matrices explicitly.

U2 - 10.12958/adm1568

DO - 10.12958/adm1568

M3 - Article

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SP - 79

JO - Algebra and Discrete Mathematics

JF - Algebra and Discrete Mathematics

IS - 1

ER -

An elementary description of K1(R) without elementary matrices

Abstract

ASJC Scopus subject areas

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Activities

Some remarks on the “fundamental theorem” in algebraic K-theory

Fun with K-theory

K-theory: from linear equations to the fundamental theorem

Cite this

An elementary description of K₁(R) without elementary matrices