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Local-global principle for transvection groups

Authors: A. Bak, Rabeya Basu and Ravi A. Rao
Journal: Proc. Amer. Math. Soc. 138 (2010), 1191-1204
MSC (2000): Primary 13C10, 15A63, 19B10, 19B14
Published electronically: November 20, 2009
MathSciNet review: 2578513
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Abstract: In this article we extend the validity of Suslin's 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 Suslin's 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

Rabeya Basu
Affiliation: Indian Institute of Science Education and Research, Kolkata, India

Ravi A. Rao
Affiliation: Tata Institute of Fundamental Research, Mumbai, India
Email: email:

Keywords: Projective, symplectic, orthogonal modules, nilpotent groups, $ {K}_1$.
Received by editor(s): July 2, 2009
Published electronically: November 20, 2009
Communicated by: Martin Lorenz
Article copyright: © Copyright 2009 By the authors