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Abstract: In this article we extend the validity Suslin-s Local-Global Principle forthe elementary transvection subgroup of the general linear group, thesymplectic group, and the orthogonal group, where n > 2, to a Local-GlobalPrinciple for the elementary transvection subgroup of the automorphism groupAutP of either a projective module P of global rank > 0 and constant localrank > 2, or of a nonsingular symplectic or orthogonal module P of globalhyperbolic rank > 0 and constant local hyperbolic rank > 2. In Suslin-sresults, the local and global ranks are the same, because he is concerned onlywith free modules. Our assumption that the global hyperbolic rank > 0 is usedto define the elementary transvection subgroups. We show further that theelementary transvection subgroup ETP is normal in AutP, that ETP = TPwhere the latter denotes the full transvection subgroup of AutP, and that theunstable K 1-group K 1AutP = AutP-ETP = AutP-TP is nilpotent byabelian, provided R has finite stable dimension. The last result extendsprevious ones of Bak and Hazrat for the above mentioned classical groups.An important application to the results in the current paper can be found inthe work of last two named authors where they have studied the decrease in theinjective stabilization of classical modules over a non-singular affine algebraover perfect C 1-fields. We refer the reader to that article for more details.

Author: A. Bak, Rabeya Basu, Ravi A. Rao

Source: https://arxiv.org/

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