Local-global principle for transvection groups

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.