Local-global principle for transvection groups
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- by A. Bak, Rabeya Basu and Ravi A. Rao PDF
- Proc. Amer. Math. Soc. 138 (2010), 1191-1204
Abstract:
In this article we extend the validity of Suslinās Local-Global Principle for the elementary transvection subgroup of the general linear group GL$_n(R)$, the symplectic group Sp$_{2n}(R)$, and the orthogonal group O$_{2n}(R)$, where $n > 2$, to a Local-Global Principle for the elementary transvection subgroup of the automorphism group Aut$(P)$ of either a projective module $P$ of global rank $> 0$ and constant local rank $> 2$, or of a nonsingular symplectic or orthogonal module $P$ of global hyperbolic rank $> 0$ and constant local hyperbolic rank $> 2$. In Suslinās results, the local and global ranks are the same, because he is concerned only with free modules. Our assumption that the global (hyperbolic) rank $> 0$ is used to define the elementary transvection subgroups. We show further that the elementary transvection subgroup ET$(P)$ is normal in Aut$(P)$, that ET$(P) =$ T$(P)$, where the latter denotes the full transvection subgroup of Aut$(P)$, and that the unstable K$_1$-group K$_1($Aut$(P)) =$ Aut$(P)/$ET$(P) =$ Aut$(P)/$T$(P)$ is nilpotent by abelian, provided $R$ has finite stable dimension. The last result extends previous ones of Bak and Hazrat for GL$_n(R)$, Sp$_{2n}(R)$, and O$_{2n}(R)$.
An important application to the results in the current paper can be found in a preprint of Basu and Rao in which the last two named authors studied the decrease in the injective stabilization of classical modules over a nonsingular affine algebra over perfect C$_1$-fields. We refer the reader to that article for more details.
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Additional Information
- A. Bak
- Affiliation: Department of Mathematics, University of Bielefeld, Bielefeld, Germany
- Email: bak@mathematik.uni-bielefeld.de
- Rabeya Basu
- Affiliation: Indian Institute of Science Education and Research, Kolkata, India
- Email: rabeya.basu@gmail.com, rbasu@iiserkol.ac.in
- Ravi A. Rao
- Affiliation: Tata Institute of Fundamental Research, Mumbai, India
- Email: ravi@math.tifr.res.in
- Received by editor(s): July 2, 2009
- Published electronically: November 20, 2009
- Communicated by: Martin Lorenz
- © Copyright 2009 By the authors
- Journal: Proc. Amer. Math. Soc. 138 (2010), 1191-1204
- MSC (2000): Primary 13C10, 15A63, 19B10, 19B14
- DOI: https://doi.org/10.1090/S0002-9939-09-10198-3
- MathSciNet review: 2578513