  In 1994, the second author and W. van der Kallen showed that the injective
stabilization bound for K_1 of general linear group is d+1 over a regular
affine algebra over a perfect C_1-field, where d is the krull dimension of the
base ring and it is finite and at least 2. In this article we prove that the
injective stabilization bound for K_1 of the symplectic group is d+1 over a
geometrically regular ring containing a field, where d is the stable dimension
of the base ring and it is finite and at least 2. Then using the Local-Global
Principle for the transvection subgroup of the automorphism group of projective
and symplectic modules we show that the injective stabilization bound is d+1
for k_1 of projective and symplectic modules of global rank at least 1 and
local rank at least 3 respectively in each of the two cases above.
