Optimal injective stability for the symplectic $K_{1}Sp$ group

Abstract

If $R$ is a commutative ring, $I$ an ideal of $R$ and $v,w\in U{m}_{2n}(R,I)$ 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 $d!A=A$, $d\equiv 2(\mathrm{mod}4)$ and $I$ an ideal of $A$, then $U{m}_d(A,I)=e_1{Sp}_d(A,I)$. 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 $(d+1)!A=A$, $d\equiv 1(\mathrm{mod}4)$ and $I$ a principal ideal then $S{p}_{d-1}(A,I)\cap {ESp}_{d+1}(A,I)={ESp}_{d-1}(A,I)$. 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.

16 pages

Cited by in corpus (4)

• The cancellation of projective modules of rank 2 with a trivial determinant
• Nice group structure on the elementary orbit space of unimodular rows
• Zero cycles, Mennicke symbols and ${\mathrm{K}}_1$-stability of certain real affine algebras
• Results on ${\mathrm{K}}_1$ of general quadratic groups