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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



On the smoothness of centralizers in reductive groups

Author: Sebastian Herpel
Journal: Trans. Amer. Math. Soc. 365 (2013), 3753-3774
MSC (2010): Primary 20G15
Published electronically: December 12, 2012
MathSciNet review: 3042602
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Abstract: Let $ G$ be a connected reductive algebraic group over an algebraically closed field $ k$. In a recent paper, Bate, Martin, Röhrle and Tange show that every (smooth) subgroup of $ G$ is separable provided that the characteristic of $ k$ is very good for $ G$. Here separability of a subgroup means that its scheme-theoretic centralizer in $ G$ is smooth. Serre suggested extending this result to arbitrary, possibly non-smooth, subgroup schemes of $ G$. The aim of this paper is to prove this more general result. Moreover, we provide a condition on the characteristic of $ k$ that is necessary and sufficient for the smoothness of all centralizers in $ G$. We finally relate this condition to other standard hypotheses on connected reductive groups.

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Sebastian Herpel
Affiliation: Fakultät für Mathematik, Ruhr-Universität Bochum, 44780 Bochum, Germany
Address at time of publication: Fachbereich Mathematik, TU Kaiserslautern, Postfach 3049, 67653 Kaiserslautern, Germany

Received by editor(s): March 29, 2011
Received by editor(s) in revised form: November 4, 2011
Published electronically: December 12, 2012
Article copyright: © Copyright 2012 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.