An elementary description of \(K_1(R)\) without elementary matrices
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 isomorp...
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| Date: | 2020 |
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| Main Authors: | , |
| Format: | Article |
| Language: | English |
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Lugansk National Taras Shevchenko University
2020
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| Online Access: | https://admjournal.luguniv.edu.ua/index.php/adm/article/view/1568 |
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| Journal Title: | Algebra and Discrete Mathematics |
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Algebra and Discrete Mathematics| _version_ | 1856543381095710720 |
|---|---|
| author | Hüttemann, T. Zhang, Z. |
| author_facet | Hüttemann, T. Zhang, Z. |
| author_sort | Hüttemann, T. |
| baseUrl_str | |
| collection | OJS |
| datestamp_date | 2021-01-04T06:22:04Z |
| description | 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. |
| first_indexed | 2025-12-02T15:42:13Z |
| format | Article |
| id | admjournalluguniveduua-article-1568 |
| institution | Algebra and Discrete Mathematics |
| language | English |
| last_indexed | 2025-12-02T15:42:13Z |
| publishDate | 2020 |
| publisher | Lugansk National Taras Shevchenko University |
| record_format | ojs |
| spelling | admjournalluguniveduua-article-15682021-01-04T06:22:04Z An elementary description of \(K_1(R)\) without elementary matrices Hüttemann, T. Zhang, Z. \(K\)-theory, invertible matrix, elementary matrix Primary 19B99; Secondary 16E20 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. Lugansk National Taras Shevchenko University 2020-12-30 Article Article Peer-reviewed Article application/pdf https://admjournal.luguniv.edu.ua/index.php/adm/article/view/1568 10.12958/adm1568 Algebra and Discrete Mathematics; Vol 30, No 1 (2020) 2415-721X 1726-3255 en https://admjournal.luguniv.edu.ua/index.php/adm/article/view/1568/pdf https://admjournal.luguniv.edu.ua/index.php/adm/article/downloadSuppFile/1568/684 https://admjournal.luguniv.edu.ua/index.php/adm/article/downloadSuppFile/1568/726 https://admjournal.luguniv.edu.ua/index.php/adm/article/downloadSuppFile/1568/727 https://admjournal.luguniv.edu.ua/index.php/adm/article/downloadSuppFile/1568/728 https://admjournal.luguniv.edu.ua/index.php/adm/article/downloadSuppFile/1568/729 Copyright (c) 2020 Algebra and Discrete Mathematics |
| spellingShingle | \(K\)-theory invertible matrix elementary matrix Primary 19B99 Secondary 16E20 Hüttemann, T. Zhang, Z. An elementary description of \(K_1(R)\) without elementary matrices |
| title | An elementary description of \(K_1(R)\) without elementary matrices |
| title_full | An elementary description of \(K_1(R)\) without elementary matrices |
| title_fullStr | An elementary description of \(K_1(R)\) without elementary matrices |
| title_full_unstemmed | An elementary description of \(K_1(R)\) without elementary matrices |
| title_short | An elementary description of \(K_1(R)\) without elementary matrices |
| title_sort | elementary description of \(k_1(r)\) without elementary matrices |
| topic | \(K\)-theory invertible matrix elementary matrix Primary 19B99 Secondary 16E20 |
| topic_facet | \(K\)-theory invertible matrix elementary matrix Primary 19B99 Secondary 16E20 |
| url | https://admjournal.luguniv.edu.ua/index.php/adm/article/view/1568 |
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