Example 5

Let $k$ be a field, $A$ an integral $k$-algebra of finite type, and $m$ a maximal ideal of $A$. The local noetherian ring $A_m$ is regular if and only if one has $\dim(A) = [m/m^2 : A/m]$; in fact, one has $\dim(A_m) = \dim(A)$ (§ 2, No. 4, Cor. 2 to Theorem 3) and the vector spaces $m/m^2$ and $mA_m/(mA_m)^2$ over the field $A/m$ are isomorphic (II, § 3, No. 3, Prop. 9). In particular, if $k$ is algebraically closed, the stated condition is equivalent to $\dim(A) = [m/m^2 : k]$ (V, § 3, No. 3, Prop. 1).