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...
Збережено в:
| Дата: | 2020 |
|---|---|
| Автори: | , |
| Формат: | Стаття |
| Мова: | English |
| Опубліковано: |
Lugansk National Taras Shevchenko University
2020
|
| Теми: | |
| Онлайн доступ: | https://admjournal.luguniv.edu.ua/index.php/adm/article/view/1568 |
| Теги: |
Додати тег
Немає тегів, Будьте першим, хто поставить тег для цього запису!
|
| Назва журналу: | Algebra and Discrete Mathematics |
Репозитарії
Algebra and Discrete Mathematics| id |
oai:ojs.admjournal.luguniv.edu.ua:article-1568 |
|---|---|
| record_format |
ojs |
| spelling |
oai:ojs.admjournal.luguniv.edu.ua: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 |
| institution |
Algebra and Discrete Mathematics |
| baseUrl_str |
|
| datestamp_date |
2021-01-04T06:22:04Z |
| collection |
OJS |
| language |
English |
| topic |
\(K\)-theory invertible matrix elementary matrix Primary 19B99 Secondary 16E20 |
| 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 |
| topic_facet |
\(K\)-theory invertible matrix elementary matrix Primary 19B99 Secondary 16E20 |
| format |
Article |
| author |
Hüttemann, T. Zhang, Z. |
| author_facet |
Hüttemann, T. Zhang, Z. |
| author_sort |
Hüttemann, T. |
| title |
An elementary description of \(K_1(R)\) without elementary matrices |
| title_short |
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_sort |
elementary description of \(k_1(r)\) without elementary matrices |
| 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. |
| publisher |
Lugansk National Taras Shevchenko University |
| publishDate |
2020 |
| url |
https://admjournal.luguniv.edu.ua/index.php/adm/article/view/1568 |
| work_keys_str_mv |
AT huttemannt anelementarydescriptionofk1rwithoutelementarymatrices AT zhangz anelementarydescriptionofk1rwithoutelementarymatrices AT huttemannt elementarydescriptionofk1rwithoutelementarymatrices AT zhangz elementarydescriptionofk1rwithoutelementarymatrices |
| first_indexed |
2025-07-17T10:33:27Z |
| last_indexed |
2025-07-17T10:33:27Z |
| _version_ |
1837889914770817024 |