Let $R$ be a Noetherian regular local ring of dimension $n$ with maximal ideal $\mathfrak{m}$. Given two systems of regular parameters $\vec{u}=\left<u\_1, \dots, u\_n\right>$ and $\vec{v}=\left<v\_1, \dots, v\_n\right>$ of $R$, does there exist an element $M$ in $\mathrm{GL}\_n(R)$ which takes $\vec{u}$ to $\vec{v}$?


Any comments on this question are most appreciated!



Systems of regular parameters of a regular local ring
Since $\vec{u},\vec{v}$ both generate $\mathfrak{m}$, there exist matrices $U,V \in \mathrm{Mat}\_{n \times n}(R)$ such that $\vec{u} = U\vec{v}$ and $\vec{v} = V\vec{u}$. Then $(UV-I\_{n})\vec{u} = \vec{0}$ so $UV-I\_{n}$ has entries contained in $\mathfrak{m}$ since $\vec{u}$ is a regular sequence; thus $UV$ is invertible (because it is $I\_{n}$ modulo $\mathfrak{m}$), hence both $U,V$ are invertible.


