Example 2

Let $k$ be a field and let $n$ be a positive integer. The ring of formal power series $k[[X_1, ..., X_n]]$ is a regular noetherian local ring of dimension $n$ (§ 3, No. 4, Cor. 3 to Prop. 8). Let $F_1, ..., F_n$ be formal power series without constant term in $k[[X_1, ..., X_n]]$; in order that the sequence $(F_1, ..., F_n)$ be a system of coordinates of $k[[X_1, ..., X_n]]$, it is necessary and sufficient that the matrix $\left( \frac{\partial F_i}{\partial X_j}(0, ..., 0) \right)$ be invertible.

More generally, let $A$ be a regular noetherian local ring of dimension $r$; then $A[[X_1, ..., X_n]]$ is a regular noetherian local ring of dimension $r + n$. If $(a_1, ..., a_r)$ is a system of coordinates of $A$, then $(a_1, ..., a_r, X_1, ..., X_n)$ is a system of coordinates of $A[[X_1, ..., X_n]]$.