On the structure of the symplectic group over polynomial rings with regular coefficients

In this note we prove the following result. Let $A$ be a ring of the geometric type or $A=C\bigl[[T_1,\ldots,T_{m}]\bigr]$, where $C$ is a regular ring and $\dim C\leq1$. Then the group $\operatorname{Sp}_{2r}\left(A[X_1,\ldots,X_{n}]\right)$ ($r\geq2$) is generated by elementary symplectic matrices.