Optimal injective stability for the symplectic K₁Sp group

Original · EN

If R is a commutative ring, I an ideal of R and v, w ∈ Um₂ₙ(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 ≡ 2 4 and I an ideal of A, then 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 4 and I a principal ideal then 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 C₂ field and also show by an example that the above stability estimate is optimal.