2009/08/21 by Anthony Bak, A. Bak, Rabeya Basu +4
Mathematics · #13C10 #15A63 #19B10 #19B14 #Algebraic Geometry and Number Theory #Commutative Algebra (math.AC) #FOS: Mathematics #Geometric and Algebraic Topology #Homotopy and Cohomology in Algebraic Topology #K-Theory and Homology (math.KT) #math.AC #math.KT #msc:13C10 #msc:15A63 #msc:19B10 #msc:19B14
paper · pdf · doi:10.48550/arxiv.0908.3094
15 pages
openalex publication_date 2009/08/21 · arxiv created 2009/08/24 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In this article we extend the validity Suslin's Local-Global Principle for the elementary transvection subgroup of the general linear group, the symplectic group, and the orthogonal group, 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 K1-group K1(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 the above mentioned classical groups. An important application to the results in the current paper can be found in the work of last two named authors where they have studied the decrease in the injective stabilization of classical modules over a non-singular affine algebra over perfect C1-fields. We refer the reader to that article for more details.