Local-global principle for congruence subgroups of Chevalley groups

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We prove Suslins local-global principle for principal congruence subgroups of Chevalley groups. Let $G$ be a Chevalley-Demazure group scheme with a root system $\Phi e A 1$ and $E$ its elementary subgroup. Let $R$ be a ring and $I$ an ideal of $R$. Assume additionally that $R$ has no residue fields of 2 elements if $\Phi=C 2$ or $G 2$. Theorem. Let $g\in GRX,XRX$. Suppose that for every maximal ideal $\m$ of $R$ the image of $g$ under the localization homomorphism at $\m$ belongs to $ER \mX,IR \mX$. Then, $g\in ERX,IRX$. The theorem is a common generalization of the result of E.Abe for the absolute case $I=R$ and H.Apte-P.Chattopadhyay-R.Rao for classical groups. It is worth mentioning that for the absolute case the local-global principle was obtained by V.Petrov and A.Stavrova in more general settings of isotropic reductive groups.

Autor: Himanee Apte; Alexei Stepanov

Fuente: https://archive.org/