Local-Global Principle for Transvection Groups

TL;DR

This paper extends the Local-Global Principle to automorphism groups of modules with certain rank conditions, showing normality of elementary transvections and properties of associated K-theory groups.

Contribution

It generalizes Suslin's Local-Global Principle to broader module automorphism groups with rank conditions, establishing normality and nilpotency results.

Findings

Elementary transvection subgroup is normal in automorphism group.

The elementary transvection subgroup equals the full transvection subgroup.

The unstable K_1-group is nilpotent by abelian under finite stable dimension.

Abstract

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 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 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 C_1-fields. We refer the reader to that article for more details.