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RESEARCH ARTICLE
An elementary description of $K_1(R)$ without elementary matrices
T. Hüttemanna, Z. Zhangb a Queen's University Belfast, School of Mathematics and Physics, Mathematical Sciences Research Centre, Belfast BT7 1NN, UK
b School of Mathematics, Beijing Institute of Technology, 5 South Zhongguancun Street, Haidian District, 100081 Beijing, P. R. China
Abstract:
Let $R$ be a ring with unit. Passing to the colimit with respect to the standard inclusions $\mathrm{GL}(n,R) \to \mathrm{GL}(n+1,R)$ (which add a unit vector as new last row and column) yields, by definition, the stable linear group $\mathrm{GL}(R)$; 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 $K_1(R) = \mathrm{GL}(R)/E(R)$ of $R$, giving an elementary description that does not involve elementary matrices explicitly.
Keywords:
$K$-theory, invertible matrix, elementary matrix.
Received: 18.03.2020
Citation:
T. Hüttemann, Z. Zhang, “An elementary description of $K_1(R)$ without elementary matrices”, Algebra Discrete Math., 30:1 (2020), 79–82
Linking options:
https://www.mathnet.ru/eng/adm766 https://www.mathnet.ru/eng/adm/v30/i1/p79
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Abstract page: | 46 | Full-text PDF : | 24 | References: | 11 |
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