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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| Дата: | 2020 |
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| Автори: | , |
| Формат: | Стаття |
| Мова: | English |
| Опубліковано: |
Lugansk National Taras Shevchenko University
2020
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| Теми: | |
| Онлайн доступ: | https://admjournal.luguniv.edu.ua/index.php/adm/article/view/1568 |
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| Назва журналу: | Algebra and Discrete Mathematics |
Репозитарії
Algebra and Discrete Mathematics| Резюме: | 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. |
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