Corollary 3

Let $k$ be a field of characteristic 0, and $A$ a local noetherian $k$-algebra. In order that $A$ be a Nagata ring, it is necessary and sufficient that, for every prime ideal $p$ of $A$, the ring $(\widehat{A/p})$ be reduced. In fact, since the fields $\kappa(p)$ are of characteristic 0, it is equivalent to say that the algebras $\kappa(p) \otimes_A \hat{A} = \kappa(p) \otimes_{A/p} (\widehat{A/p})$ are reduced or that they are separable (A, V, p. 117, Theorem 1), which shows that the stated condition is sufficient (Theorem 3, (ii) ⇒ (i)); moreover it is necessary (Theorem 3, (i) ⇒ (iii) with R = A/p).