Exercise 19, § 4

Let $A$ be a noetherian ring, $a$ a nonzero ideal of $A$. Suppose that $A$ is separated and complete for the $a$-adic topology and that $A/a$ is a Nagata ring. Prove that $A$ is a Nagata ring. One may reason as follows: a) Suppose that $\overline{A}$ is an integral domain and that for every nonzero prime ideal $p$ of $A$, the ring $A/p$ is Japanese. Let $\overline{A}$ be the integral closure of $A$ in a finite extension of its field of fractions. Then, for every nonzero prime ideal $\mathfrak{P}$ of $\overline{A}$, the ring $\overline{A}/\mathfrak{P}$ is a finite algebra over the ring $A/(\mathfrak{P} \cap A)$. (Use Exercise 11, a), p. 80.) b) Under the same hypotheses and notations as in a), prove that $\overline{A}$ is noetherian. (Use Exercises 7, p. 79 and 14, p. 81.) Then, arguing as in the preceding exercise, prove that $\overline{A}/a\overline{A}$ is a finitely generated $A$-module, that $\overline{A}$ is separated for the $a\overline{A}$-adic topology, and that $\overline{A}$ is a finitely generated $A$-module.