Exercise 7, § 2

Let $A$ be a Noetherian integral domain and $A'$ its integral closure. Show that for every prime ideal $p$ of $A$ there exists only a finite number of prime ideals of $A'$ lying above $p$. (Reduce it first to the case where $A$ is a local ring with maximal ideal $p$; consider its completion $\hat{A}$ and the total ring of fractions $B$ of $\hat{A}$ and note that the field of fractions $K$ of $A$ is identified with a subring of $B$ (Chapter III, § 3, no. 4, Corollary 2 to Theorem 3). Note that the spectrum of $B$ is finite and therefore that in $B$ there is only a finite number of idempotents which are orthogonal to one another. Then let $C \supseteq A$ be a subring of $K$ which is a finitely generated $A$-module and therefore semi-local. Note that the completion $\hat{C}$ of $C$ is identified with a subring of $B$; conclude by noting that $\hat{C}$ is the product of a finite number of local rings, this number being equal to the number of maximal ideals of $C$). Generalize to the integral closure of a Noetherian ring in its total ring of fractions (reduce it to the case of a reduced ring).