Optimal injective stability for the symplectic $K_1Sp$ group

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.