Injective Stability for K_1 of Classical Modules

TL;DR

This paper establishes the injective stabilization bound for K_1 of symplectic groups over certain rings, extending previous results for general linear groups and applying local-global principles to projective and symplectic modules.

Contribution

It proves the stabilization bound for K_1 of symplectic groups over geometrically regular rings and extends results to projective and symplectic modules using local-global techniques.

Findings

The stabilization bound for K_1 of symplectic groups is d+1 over geometrically regular rings.

The bound applies to projective and symplectic modules with specified ranks.

The results extend previous bounds for general linear groups to symplectic groups.

Abstract

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.