Injective stability for K1 of classical modules

In Rao (1994) [14], the second author and W. van der Kallen showed that the injective stabilization bound for K1 of the general linear group is d+1 over a regular affine algebra over a perfect C1-field, where d is the Krull dimension of the base ring which is finite and at least 2. In this article we prove that the injective stabilization bound for K1 of the symplectic group is d+1 over a geometrically regular ring containing a field, where d is the dimension of the base ring which is finite and at least 2. 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 K1 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.