Theorem 3 (Zariski-Nagata)

Let $A$ be a semi-local noetherian ring. The following conditions are equivalent: (i) $A$ is a Nagata ring; (ii) for every prime ideal $p$ of $A$, the $\kappa(p)$-algebra $\kappa(p) \otimes_A \hat{A}$ is separable; (iii) for every reduced $A$-algebra $R$, the ring $R \otimes_A \hat{A}$ is reduced. Let us first prove the equivalence of conditions (ii) and (iii). The implication (iii) $\Rightarrow$ (ii) is trivial; conversely, suppose that $A$ satisfies condition (ii). Then, for every $A$-algebra $K$ which is a field, the ring $K \otimes_A \hat{A}$ is reduced. Let now $C$ be a reduced $A$-algebra of finite type; the ring $C$, being noetherian, is isomorphic to a subring of a finite product $K_1 \times \cdots \times K_n$ of fields (IV, § 2, No. 5, Prop. 10); since $\hat{A}$ is flat over $A$, the ring $C \otimes_A \hat{A}$ is isomorphic to a subring of the reduced ring $\prod_i (K_i \otimes_A \hat{A})$, hence is reduced. Finally, let $R$ be any reduced $A$-algebra; then $R$ is the union of the filtered family $(C_\alpha)$ of its subalgebras of finite type, and $R \otimes_A \hat{A}$ is the direct limit of the filtered family $(C_\alpha \otimes_A \hat{A})$ of reduced rings, hence is reduced. Let us show that (ii) implies (i). Let $p$ be a prime ideal of $A$; the field of fractions $K$ of the ring $A/p$ is identified with $\kappa(p)$, and the $K$-algebra $K \otimes_{A/p} (\widehat{A/p})$ is identified with $\kappa(p) \otimes_{A/p} \hat{A}/p\hat{A}$, hence with $\kappa(p) \otimes_A \hat{A}$. If $\kappa(p) \otimes_A \hat{A}$ is a separable $\kappa(p)$-algebra, the ring $A/p$ is Japanese (No. 2, Prop. 3). Let us prove the implication (i) $\Rightarrow$ (ii) by induction on $\dim(A)$. It is obvious if $\dim(A) = 0$ since then $A$ is artinian, hence complete. Let $n$ be an integer $> 0$; consider the following assumption:

(R$_n$) $\left\{ \begin{array}{l} \text{for every local noetherian Nagata ring } C \text{ of dimension } < n \text{ and every prime ideal } r \text{ of } C, \text{ the ring } \kappa(r) \otimes_C \hat{C} \text{ is reduced.} \end{array} \right.$

Let $A$ be a semi-local noetherian Nagata ring of dimension $n$, let $p$ be a prime ideal of $A$ and $L$ a finite extension of the field $\kappa(p)$; it is enough to prove,