PUB - Publikationen an der Universität Bielefeld

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 SP2n(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 CLn(R), SP2n(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.