Optimal injective stability for the symplectic K1Sp group

If R is a commutative ring, I an ideal of R   and v,w∈Um2n(R,I)v,w∈Um2n(R,I) then we show that v,wv,w are in the same orbit of elementary action if and only if they are in that of elementary symplectic action extending a result in [4]. 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=Ad!A=A, d≡2(mod4) and I an ideal of A  , then Umd(A,I)=e1Spd(A,I)Umd(A,I)=e1Spd(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+1)!A=A, d≡1(mod4) and I   a principal ideal then Spd−1(A,I)∩ESpd+1(A,I)=ESpd−1(A,I)Spd−1(A,I)∩ESpd+1(A,I)=ESpd−1(A,I). We give an example to show that the above result does not hold true for an affine algebra over a C2C2 field and also show by an example that the above stability estimate is optimal.