Definition 1

The noetherian local ring $A$ is said to be regular if one has $\dim(A) = [\mathfrak{m}_A/\mathfrak{m}_A^2 : \kappa_A]$. A system of coordinates of $A$ is then any family of elements of $\mathfrak{m}_A$ whose classes modulo $\mathfrak{m}_A^2$ form a basis of the $\kappa_A$-vector space $\mathfrak{m}_A/\mathfrak{m}_A^2$.

A system of coordinates in a regular noetherian local ring $A$ is therefore a finite family $(x_i)_{i \in I}$ generating the ideal $\mathfrak{m}_A$ of $A$ and such that $\mathrm{Card}(I) = \dim(A)$. Conversely, if the ideal $\mathfrak{m}_A$ of a noetherian local ring $A$ is generated by $d$ elements with $d \leq \dim(A)$, the ring $A$ is regular.