K1-Stability of symplectic modules over monoid algebras

Abstract

Let R be a regular ring of dimension d and L be a c-divisible monoid. If $K_1\mathrm{Sp}\left(R\right)$ is trivial and $k\ge d+2$, then we prove that the symplectic group ${\mathrm{Sp}}_{2k}\left(R\right[L\left]\right)$ is generated by elementary symplectic matrices over $R\left[L\right]$. When $d\le 1$ or R is a geometrically regular ring containing a field, then improved bounds have been established. We also discuss the linear case, extending the work of [14].