Example 3

Let $A$ be a regular local $k$-algebra. If the extension $\kappa_A$ of $k$ is separable, the algebra $A$ is absolutely regular. Let indeed $k'$ be a finite-degree extension of $k$. The $A$-module $A_{(k')}$ is free. Since it is of finite type, every maximal ideal of $A_{(k')}$ lies over $\mathfrak{m}_A$ (V, § 2, No. 1, prop. 1). The ring $\kappa_A \otimes_A A_{(k')}$, isomorphic to $\kappa_A \otimes_k k'$, is regular (No. 3, lemma 2). By the corollary to prop. 9 of § 4, No. 5, the ring $A_{(k')}$ is regular, which proves that the $k$-algebra $A$ is absolutely regular.

$^1$ Some authors use the terminology “geometrically regular” (resp. “geometrically normal”.