多项式代数的单侧逆代数的群${\rm K}_1(\mS_n)$
The group ${\rm K}_1(\mS_n)$ of the algebra of one-sided inverses of a polynomial algebra
摘要：
多项式代数$P_n$在$n$变量中的单侧逆的代数$\mS_n$由$P_n$通过添加交换，代数$P_n$的正则生成元的{\em left}（但不是双侧）逆而获得。代数$\mS_n$是一个经典Krull维数$2n$和整体维数$n$的非交换非Noether代数，它不是一个域。如果基域$K$具有特征零，则代数$\mS_n$与标量积分微分算子代数$K< \frac{\der}{\der x_1}, ..., frac{\der}{\der x_n}, \int_1, ..., \int_n>$正则同构。忽略非Noether性质，代数$\mS_n$属于类似于$n$ th Weyl代数$A_n$和多项式代数$P_{2n}$的代数族。证明了$ {\rm K}_1(\mS_n)\simeq K^*$.其主要思想是证明群$\GL_\infty (\mS_n)$是由$K^*$、初等矩阵群$E_\infty (\mS_n)$和$(n-2)2^{n-1}+1$显式矩阵生成的，然后证明所有矩阵都是初等矩阵。对于代数$\mS_n$的每个高度为$m$的非零幂等素理想$\gp$，证明了$$ \rm K}_1(\mS_n, \gp)\simeq K^*, & \text{if}m=1, \Z^{\frac{m(m-1)}{2}}\times K^{*m}& \text{if}m> 1.$$
英文摘要：
The algebra $\mS_n$ of one-sided inverses of a polynomial algebra $P_n$ in $n$ variables is obtained from $P_n$ by adding commuting, {\em left} (but not two-sided) inverses of the canonical generators of the algebra $P_n$. The algebra $\mS_n$ is a noncommutative, non-Noetherian algebra of classical Krull dimension $2n$ and of global dimension $n$ which is not a domain. If the ground field $K$ has characteristic zero then the algebra $\mS_n$ is canonically isomorphic to the algebra $K< \frac{\der}{\der x_1}, ..., frac{\der}{\der x_n}, \int_1, ..., \int_n>$ of scalar integro-differential operators. %Ignoring non-Noetherian % property, the algebra $\mS_n$ belongs to a family of algebras % like the $n$th Weyl algebra $A_n$ and the polynomial algebra %$P_{2n}$. It is proved that $ {\rm K}_1(\mS_n)\simeq K^*$. The main idea is to show that the group $\GL_\infty (\mS_n)$ is generated by $K^*$, the group of elementary matrices $E_\infty (\mS_n)$ and $(n-2)2^{n-1}+1$ explicit (tricky) matrices and then to prove that all the matrices are elementary. For each nonzero idempotent prime ideal $\gp$ of height $m$ of the algebra $\mS_n$, it is proved that $$ \rm K}_1(\mS_n, \gp)\simeq K^*, & \text{if}m=1, \Z^{\frac{m(m-1)}{2}}\times K^{*m}& \text{if}m> 1.$$
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
