In the present paper, which is a direct sequel of our paper [14] joint with Roozbeh Hazrat, we prove an unrelativized version of the standard commutator formula in the setting of Chevalley groups. Namely, let Φ be a reduced irreducible root system of rank ≥ 2, let R be a commutative ring and let I,J be two ideals of R. We consider subgroups of the Chevalley group G(Φ, R) of type Φ over R. The unrelativized elementary subgroup E(Φ, I) of level I is generated (as a group) by the elementary unipotents xα(ξ), α ∈ Φ, ξ ∈ I, of level I. Obviously, in general, E(Φ, I) has no chance to be normal in E(Φ, R); its normal closure in the absolute elementary subgroup E(Φ, R) is denoted by E(Φ, R, I). The main results of [14] implied that the commutator [E(Φ, I), E(Φ, J)] is in fact normal in E(Φ, R). In the present paper we prove an unexpected result, that in fact [E(Φ, I), E(Φ, J)] = [E(Φ, R, I), E(Φ, R, J)]. It follows that the standard commutator formula also holds in the unrelativized form, namely [E(Φ, I), C(Φ, R, J)] = [E(Φ, I), E(Φ, J)], where C(Φ, R, I) is the full congruence subgroup of level I. In particular, E(Φ, I) is normal in C(Φ, R, I).

中文翻译：

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Chevalley 群中相对换向子群的生成。二

在本文中，这是我们与 Roozbeh Hazrat 联合的论文 [14] 的直接续篇，我们在 Chevalley 群的设置中证明了标准换向器公式的非相对化版本。即，让Φ是秩≥ 2 的约化不可约根系统，令R是一个交换环并且让一世,Ĵ成为两个理想R. 我们考虑 Chevalley 群的子群G(Φ,R) 类型Φ超过R. 未相对化的基本子群乙(Φ,一世) 水平一世由基本单能者（作为一个组）生成Xα(ξ), α ∈ Φ, ξ ∈一世, 水平一世. 显然，一般来说，乙(Φ,一世) 没有机会正常乙(Φ,R); 它在绝对初等子群中的正常闭包乙(Φ,R) 表示为乙(Φ,R,一世）。[14] 的主要结果暗示了换向器 [乙(Φ,一世),乙(Φ,Ĵ)] 实际上是正常的乙(Φ,R）。在本文中，我们证明了一个意想不到的结果，事实上 [乙(Φ,一世),乙(Φ,Ĵ)] = [乙(Φ,R,一世),乙(Φ,R,Ĵ)]。由此可见，标准交换子公式也以非相对化形式成立，即 [乙(Φ,一世),C(Φ,R,Ĵ)] = [乙(Φ,一世),乙(Φ,Ĵ）]， 在哪里C(Φ,R,一世) 是水平的全同子群一世. 特别是，乙(Φ,一世) 是正常的C(Φ,R,一世）。