Centrality of the congruence kernel for elementary subgroups of Chevalley groups of rank $> 1$ over noetherian rings
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Abstract:
Let $G$ be a universal Chevalley-Demazure group scheme associated to a reduced irreducible root system of rank $>1.$ For a commutative ring $R$, we let $\Gamma = E(R)$ denote the elementary subgroup of the group of $R$-points $G(R).$ The congruence kernel $C(\Gamma )$ is then defined to be the kernel of the natural homomorphism $\widehat {\Gamma } \to \overline {\Gamma },$ where $\widehat {\Gamma }$ is the profinite completion of $\Gamma$ and $\overline {\Gamma }$ is the congruence completion defined by ideals of finite index. The purpose of this paper is to show that for an arbitrary noetherian ring $R$ (with some minor restrictions if $G$ is of type $C_n$ or $G_2$), the congruence kernel $C(\Gamma )$ is central in $\widehat {\Gamma }.$References
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Additional Information
- Andrei S. Rapinchuk
- Affiliation: Department of Mathematics, University of Virginia, Charlottesville, Virginia 22904
- MR Author ID: 206801
- Email: asr3x@virginia.edu
- Igor A. Rapinchuk
- Affiliation: Department of Mathematics, Yale University, New Haven, Connecticut 06502
- Email: igor.rapinchuk@yale.edu
- Received by editor(s): July 22, 2010
- Received by editor(s) in revised form: August 12, 2010
- Published electronically: January 20, 2011
- Communicated by: Jonathan I. Hall
- © Copyright 2011
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc. 139 (2011), 3099-3113
- MSC (2010): Primary 19B37; Secondary 20G35
- DOI: https://doi.org/10.1090/S0002-9939-2011-10736-6
- MathSciNet review: 2811265