论文标题
商单二型矢量组是nilpotent
The quotient Unimodular Vector group is nilpotent
论文作者
论文摘要
Jose-rao介绍并研究了特殊的单模型矢量组$ sum_r（r）$和$ eum_r（r）$，即其基本的单模型矢量亚组。他们证明，对于$ r \ geq 2 $，$ eum_r（r）$是$ sum_r（r）$的普通子组。 Jose-rao定理说，商$ sum_r（r）/eum_r（r）$，对于$ r \ geq 2 $，是正交商组$ so_ so_ {2（r + 1）}（r + 1）}（r）}（r）/eo__ {2（r + 1）$。已知后一组被Hazrat-vavilov的工作赋予了Nilpotent，遵循A. bak的方法；前者也是如此。 在本文中，我们将直接证明A. Bak的想法，以证明商的单模型矢量群是$ \ leq d = \ dim（r）$的类nilpotent。我们还使用A. Bak方法启发的Quillen-Suslin理论，以证明如果$ r = a [x] $，则使用$ a $ a a $ a a local戒指，那么商的单型向量群是Abelian。
Jose-Rao introduced and studied the Special Unimodular Vector group $SUm_r(R)$ and $EUm_r(R)$, its Elementary Unimodular Vector subgroup. They proved that for $r \geq 2$, $EUm_r(R)$ is a normal subgroup of $SUm_r(R)$. The Jose-Rao theorem says that the quotient Unimodular Vector group, $SUm_r(R)/EUm_r(R)$, for $r \geq 2$, is a subgroup of the orthogonal quotient group $SO_{2(r+1)}(R)/EO_{2(r + 1)}(R)$. The latter group is known to be nilpotent by the work of Hazrat-Vavilov, following methods of A. Bak; and so is the former. In this article we give a direct proof, following ideas of A. Bak, to show that the quotient Unimodular Vector group is nilpotent of class $\leq d = \dim(R)$. We also use the Quillen-Suslin theory, inspired by A. Bak's method, to prove that if $R = A[X]$, with $A$ a local ring, then the quotient Unimodular Vector group is abelian.
