Exercise 9, § 4

Let A be an integral noetherian ring, K its field of fractions, E a finite extension of K, and $\overline{A}$ the integral closure of A in E. a) If A is a local ring with completion $\hat{A}$, then $(\hat{A} \otimes_A \overline{A})_{\text{red}}$ is a finite $\hat{A}$-algebra $^1$. (Reduce to the case where $K = E$. Using III, § 3, No. 4, Cor. 2 of Theorem 3, prove that the non-zero elements of A are not zero-divisors in $\hat{A}_{\text{red}}$. Then prove that there exists an injective $\hat{A}_{\text{red}}$-homomorphism of $(\hat{A} \otimes_A \overline{A})_{\text{red}}$ into the total ring of fractions of $\hat{A}_{\text{red}}$. Conclude.) b) For every prime ideal $p$ of A, the $\kappa(p)$-algebra $(\overline{A} \otimes_A \kappa(p))_{\text{red}}$ is finite. (Reduce to the case where A is local, with maximal ideal $p$. Then observe that $(\overline{A} \otimes_A \kappa(p))_{\text{red}}$ is a quotient of $(\overline{A} \otimes_A \hat{A})_{\text{red}} \otimes \kappa(p)$.)