Symplectic orbits of unimodular rows

TL;DR

This paper proves that the symplectic group associated with an invertible alternating matrix acts transitively on unimodular rows over a smooth affine algebra, under certain conditions on the field, dimension, and size.

Contribution

It establishes the transitivity of the symplectic group action on unimodular rows for smooth affine algebras over algebraically closed fields with specific dimension and size constraints.

Findings

Transitive action of $Sp( ext{ extchi})$ on $Um_{2n}(R)$ under given conditions.

Conditions include $k$ algebraically closed, $d!$ invertible in $k$, and $2n ext{ } ext{geq} ext{ } d$.

Results extend understanding of symplectic orbits in algebraic K-theory.

Abstract

For a smooth affine algebra $R$ of dimension $d \geq 3$ over a field $k$ and an invertible alternating matrix $\chi$ of rank $2n$, the group $Sp(\chi)$ of invertible matrices of rank $2n$ over $R$ which are symplectic with respect to $\chi$ acts on the right on the set $Um_{2n}(R)$ of unimodular rows of length $2n$ over $R$. In this paper, we prove that $Sp(\chi)$ acts transitively on $Um_{2n}(R)$ if $k$ is algebraically closed, $d! \in k^{\times}$ and $2n \geq d$.