Quillen-Suslin theory for a structure theorem for the Elementary Symplectic Group

A new set of elementary symplectic elements is described, It is shown that these also generate the elementary symplectic group {\rm ESp}$_{2n}(R)$. These generators are more symmetrical than the usual ones, and are useful to study the action of the elementary symplectic group on unimodular rows. Also, an alternate proof of, {\rm ESp}$_{2n}(R)$ is a normal subgroup of {\rm Sp}$_{2n}(R)$, is shown using the Local Global Principle of D. Quillen for the new set of generators.Comment: 14 pages, few typos corrected. To appear in Ramanujan Math. Soc. Lect. Notes Se

Extendability of quadratic modules over a polynomial extension of an equicharacteristic regular local ring

We prove that a quadratic $A[T]$-module $Q$ with Witt index ($Q/TQ$)$\geq d$, where $d$ is the dimension of the equicharacteristic regular local ring $A$, is extended from $A$. This improves a theorem of the second named author who showed it when $A$ is the local ring at a smooth point of an affine variety over an infinite field. To establish our result, we need to establish a Local-Global Principle (of Quillen) for the Dickson--Siegel--Eichler--Roy (DSER) elementary orthogonal transformations.Comment: 19 page

Local-Global Principle for Transvection Groups

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.Comment: 15 page