关于$2\ × 2$线性群的亲$2$恒等式

David el-Chai Ben-Ezra, E. Zelmanov
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引用次数: 0

摘要

设$\hat{F}$为自由亲$p$非阿贝尔群,设$\Delta$为具有极大理想$I$的局部可交换完全环,使得$\textrm{char}(\Delta/I)=p$。在[Zu]中,Zubkov证明当$p\neq2$时,亲$p$同余子群$GL_{2}^{1}(\Delta)=\ker(GL_{2}(\Delta)\overset{\Delta\to\Delta/I}{\longrightarrow}GL_{2}(\Delta/I))$承认一个亲$p$同一性。即存在一个元素$1\neq w\in\hat{F}$,它在任何连续同态$\hat{F}\to GL_{2}^{1}(\Delta)$下消失。在本文中,我们调查的情况$p=2$。主要的结果是,当$\textrm{char}(\Delta)=2$,亲$2$组$GL_{2}^{1}(\Delta)$承认一个亲$2$的身份。这个结果是通过使用起源于pi理论的迹恒等式得到的。
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On pro-$2$ identities of $2\times 2$ linear groups
Let $\hat{F}$ be a free pro-$p$ non-abelian group, and let $\Delta$ be a local commutative complete ring with a maximal ideal $I$ such that $\textrm{char}(\Delta/I)=p$. In [Zu], Zubkov showed that when $p\neq2$, the pro-$p$ congruence subgroup $GL_{2}^{1}(\Delta)=\ker(GL_{2}(\Delta)\overset{\Delta\to\Delta/I}{\longrightarrow}GL_{2}(\Delta/I))$ admits a pro-$p$ identity. I.e. there exists an element $1\neq w\in\hat{F}$ that vanishes under any continuous homomorphism $\hat{F}\to GL_{2}^{1}(\Delta)$. In this paper we investigate the case $p=2$. The main result is that when $\textrm{char}(\Delta)=2$, the pro-$2$ group $GL_{2}^{1}(\Delta)$ admits a pro-$2$ identity. This result was obtained by the use of trace identities that are originated in PI-theory.
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