An elementary description of K₁(R) without elementary matrices
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” inclusio...
Збережено в:
| Дата: | 2020 |
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| Автори: | , |
| Формат: | Стаття |
| Мова: | English |
| Опубліковано: |
Інститут прикладної математики і механіки НАН України
2020
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| Назва видання: | Algebra and Discrete Mathematics |
| Онлайн доступ: | http://dspace.nbuv.gov.ua/handle/123456789/188554 |
| Теги: |
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| Назва журналу: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Цитувати: | An elementary description of K₁(R) without elementary matrices / T. Hüttemann, Z. Zhang // Algebra and Discrete Mathematics. — 2020. — Vol. 30, № 1. — С. 79–82. — Бібліогр.: 1 назв. — англ. |
Репозитарії
Digital Library of Periodicals of National Academy of Sciences of Ukraine| Резюме: | 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. |
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