Optimal injective stability for the symplectic $K_1Sp$ group

TL;DR

This paper investigates the stability properties of the symplectic K_1 group over certain rings and algebras, establishing conditions under which elementary and symplectic orbits coincide and providing examples of the limits of these stability results.

Contribution

It proves new stability results for the symplectic K_1 group over affine algebras, including orbit equivalences and optimal stability estimates under specific algebraic conditions.

Findings

Elementary and symplectic orbits coincide under certain conditions.

Established stability of the symplectic K_1 group for specific algebraic dimensions.

Provided examples showing the limits and optimality of the stability results.

Abstract

If $R$ is a commutative ring, $I$ an ideal of $R$ and $v, w \in Um_{2n}(R, I)$ then we show that $v, w$ are in the same orbit of elementary action if and only if they are in the same orbit of elementary symplectic action. We also show that if $A$ is a non-singular affine algebra of dimension $d$ over an algebraically closed field $k$ such that $d! A = A$, $d \equiv 2 \pmod 4$ and $I$ an ideal of $A$, then $Um_d(A, I) = e_1{Sp}_d(A, I)$. As a consequence it is proved that if $A$ is a non-singular affine algebra of dimension $d$ over an algebraically closed field $k$ such that $(d + 1)!A = A$, $d \equiv 1 \pmod 4$ and $I$ a principal ideal then $Sp_{d-1}(A, I) \cap {ESp}_{d+1}(A, I) = {ESp}_{d -1}(A, I)$. We give an example to show that the above result does not hold true for an affine algebra over a $C_2$ field and also show by an example that the above stability estimate is optimal.