An elementary description of K1(R) without elementary matrices

T. Huettemann, Zuhong Zhang
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Abstract

Let $R$ be a ring with unit. Passing to the colimit with respect to the standard inclusions $GL(n,R) \to GL(n+1,R)$ (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 $K_1(R) = GL(R)/E(R)$ of~$R$, giving an elementary description that does not involve elementary matrices explicitly.
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没有初等矩阵的K1(R)的初等描述
设$R$是一个带单位的环。将标准包含项$GL(n,R) \传递给GL(n+1,R)$(其中添加了一个单位向量作为新的最后一行和最后一列),根据定义,得到稳定的线性群$GL(R)$;当使用“相反”包含(添加一个单位向量作为新的第一行和第一列)时,可以获得相同的结果,直至同构。本文证明了沿这两个包体族同时传递到极限可以恢复~$R$的代数$K$-群$K_1(R) = GL(R)/E(R)$,给出了一个不显式涉及初等矩阵的初等描述。
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