Exercise 31, § 4

Let $A$ be a noetherian integral ring and $\overline{A}$ the integral closure of $A$ in a finite extension of its field of fractions. If $A$ is of dimension at most 2, then $\overline{A}$ is a noetherian ring. (Use Exercise 7, p. 79, Exercise 14, p. 81 and Krull-Akizuki's theorem (cf. VII, § 2, No. 5).)