regular local ambient cover minimal dimension

For a surjection $R\to A=R\mathrm{/}I$ of local rings the maximal ideal of $A$ is ${\mathfrak{m}}_R\mathrm{/}I$ and the residue fields agree. Thus its cotangent space is the quotient ${\mathfrak{m}}_R\mathrm{/}(I+{\mathfrak{m}}_R^2)$. If $R$ is regular of dimension $d$, this gives $\mathrm{edim}A\le d$.

For a supplied regular cover put $c={\dim }_k((I+{\mathfrak{m}}_R^2)\mathrm{/}{\mathfrak{m}}_R^2)$. Lift a basis to $x_1,\dots ,x_c\in I$ and extend their cotangent classes to a basis. The construction in the parameter-generated quotient lemma and repeated parameter reduction show that $R\mathrm{/}(x_1,\dots ,x_c)$ is regular of dimension $d-c=\mathrm{edim}A$. It still surjects onto $A$, so the lower bound is attained. If $c=0$ retain the original cover; if $c=d$ the new cover is the residue field.