In this paper, we prove that if $R$ is a local ring of dimension $d,$$d\ge 2$ and $\frac{1}{d!}\in R$ then the group $\frac{{\mathrm{\text{Um}}}_{d+1}(R[X])}{E_{d+1}(R[X])}$ has no $k$-torsion, provided $\mathrm{\text{kR}}=R.$ We also prove that if $R$ is a regular ring of dimension $d,$$d\ge 2$ and $\frac{1}{d!}\in R$ such that $E_{d+1}(R)$ acts transitively on ${\mathrm{\text{Um}}}_{d+1}(R)$ then $E_{d+1}(R[X])$ acts transitively on ${\mathrm{\text{Um}}}_{d+1}(R[X]).$

中文翻译：

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在本文中，我们证明如果$R$是维度的局部环$d,$$d\ge 2$和$\frac{1}{d！}\in R$然后组$\frac{{\mathrm{\text{嗯}}}_{d+1}(R[X])}{{乙}_{d+1}(R[X])}$没有$ķ$-扭力，提供$\mathrm{\text{kR}}=R.$我们还证明如果$R$是一个规则的维度环$d,$$d\ge 2$和$\frac{1}{d！}\in R$这样${乙}_{d+1}(R)$传递地作用于${\mathrm{\text{嗯}}}_{d+1}(R)$然后${乙}_{d+1}(R[X])$传递地作用于${\mathrm{\text{嗯}}}_{d+1}(R[X]).$