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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Centrality of the congruence kernel for elementary subgroups of Chevalley groups of rank $ > 1$ over noetherian rings


Authors: Andrei S. Rapinchuk and Igor A. Rapinchuk
Journal: Proc. Amer. Math. Soc. 139 (2011), 3099-3113
MSC (2010): Primary 19B37; Secondary 20G35
Published electronically: January 20, 2011
MathSciNet review: 2811265
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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}.$


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Additional Information

Andrei S. Rapinchuk
Affiliation: Department of Mathematics, University of Virginia, Charlottesville, Virginia 22904
Email: asr3x@virginia.edu

Igor A. Rapinchuk
Affiliation: Department of Mathematics, Yale University, New Haven, Connecticut 06502
Email: igor.rapinchuk@yale.edu

DOI: http://dx.doi.org/10.1090/S0002-9939-2011-10736-6
PII: S 0002-9939(2011)10736-6
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
Article copyright: © Copyright 2011 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.