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Transactions of the American Mathematical Society

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On the congruence kernel of isotropic groups over rings


Author: A. Stavrova
Journal: Trans. Amer. Math. Soc. 373 (2020), 4585-4626
MSC (2010): Primary 19B37, 20H05, 20G35, 19C09
DOI: https://doi.org/10.1090/tran/8091
Published electronically: March 31, 2020
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Abstract: Let $ R$ be a connected noetherian commutative ring, and let $ G$ be a simply connected reductive group over $ R$ of isotropic rank $ \ge 2$. The elementary subgroup $ E(R)$ of $ G(R)$ is the subgroup generated by $ U_{P^+}(R)$ and $ U_{P^-}(R)$, where $ U_{P^\pm }$ are the unipotent radicals of two opposite parabolic subgroups $ P^\pm $ of $ G$. Assume that $ 2\in R^\times $ if $ G$ is of type $ B_n,C_n,F_4,G_2$ and $ 3\in R^\times $ if $ G$ is of type $ G_2$. We prove that the congruence kernel of $ E(R)$, defined as the kernel of the natural homomorphism $ \widehat {E(R)}\to \overline {E(R)}$ between the profinite completion of $ E(R)$ and the congruence completion of $ E(R)$ with respect to congruence subgroups of finite index, is central in $ \widehat {E(R)}$. In the course of the proof, we construct Steinberg groups associated to isotropic reducive groups and show that they are central extensions of $ E(R)$ if $ R$ is a local ring.


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Additional Information

A. Stavrova
Affiliation: Department of Mathematics and Computer Science, St. Petersburg State University, 14th Line V.O. 29B, 199178 Saint Petersburg, Russia
Email: anastasia.stavrova@gmail.com

DOI: https://doi.org/10.1090/tran/8091
Keywords: Isotropic reductive group, congruence kernel, congruence subgroup problem, elementary subgroup, Steinberg group
Received by editor(s): February 5, 2019
Received by editor(s) in revised form: August 14, 2019
Published electronically: March 31, 2020
Additional Notes: The author is a winner of the contest “Young Russian Mathematics”, and was supported at different stages of her work by the postdoctoral grant 6.50.22.2014 “Structure theory, representation theory and geometry of algebraic groups” at St. Petersburg State University, by the J. E. Marsden postdoctoral fellowship of the Fields Institute, the Government of Russian Federation megagrant 14.W03.31.0030, by the RFBR grants 18-31-20044, 14-01-31515, 13-01-00709, 12-01-33057, 12-01-31100, and the research program 6.38.74.2011 “Structure theory and geometry of algebraic groups and their applications in representation theory and algebraic K-theory” at St. Petersburg State University.
Article copyright: © Copyright 2020 American Mathematical Society