Exercise 25, § 4

Let A be a noetherian integral domain, $X$ the topological space $\operatorname{Spec}(A)$, $\operatorname{Nor}(X)$ the set of points $p$ of $A$ such that the local ring $A_p$ is integrally closed. (If the integral closure of $A$ is a finite $A$-algebra, then $\operatorname{Nor}(X)$ is open in $X$, cf. V, § 1, No. 5, Corollary 5.) Let $f$ be a nonzero element of $A$ such that the ring $A[f^{-1}]$ is integrally closed. a) If a prime ideal $p$ of $A$ does not contain $f$, then $p$ belong to $\operatorname{Nor}(X)$. b) Let $E$ be the set of prime ideals $p$ of $A$, associated to $A/f A$, of height $> 1$ or else of height $1$ and such that $A_p$ is not regular. Then one has

(Use Exercises 4 and 5, p. 79.) c) $\operatorname{Nor}(X)$ is open in $X$.