Corollary 1

Let $k$ be a field and $A$ a $k$-algebra essentially of finite type. The set of elements $\mathfrak{p}$ of $\operatorname{Spec}(A)$ such that the $k$-algebra $A_{\mathfrak{p}}$ is absolutely regular is open in $\operatorname{Spec}(A)$.

We may suppose that the $k$-algebra $A$ is of finite type. The set considered is then formed of the prime ideals $\mathfrak{p}$ such that $[\kappa(\mathfrak{p}) \otimes_k \Omega_k(A) : \kappa(\mathfrak{p})] \leq \dim_{\mathfrak{p}}(A)$. Now the mapping $\mathfrak{p} \mapsto \dim_{\mathfrak{p}}(A)$ is lower semi-continuous by definition, and the mapping $\mathfrak{p} \mapsto [\kappa(\mathfrak{p}) \otimes_k \Omega_k(A) : \kappa(\mathfrak{p})]$ is upper semi-continuous (Nakayama's lemma and II, § 5, No. 1, Prop. 2).

We shall see later (§ 7, No. 9, Cor. 4 of Th. 3) that under the hypotheses of Cor. 1, the set of prime ideals $\mathfrak{p}$ of $A$ such that the ring $A_{\mathfrak{p}}$ is regular is open in $\operatorname{Spec}(A)$.