Example

Let $k$ be a field, $A = k[[X_1, ..., X_n]]$ and $q$ an ideal of $A$, distinct from $A$. In order that $A/q$ be regular, it is necessary and sufficient that one can find an integer $r \geqslant 0$ and elements $F_1, ..., F_r$ of $A$, generating $q$, and such that the matrix $\left( \frac{\partial F_i}{\partial X_j}(0, ..., 0) \right)$ be of rank $r$ ("jacobian criterion"). One then has $\dim(A/q) = n - r$.