Exercise 10, § 4

Let A be an integral noetherian local ring, $\hat{A}$ its completion, $p_1, ..., p_n$ the minimal prime ideals of $\hat{A}$, and for $1 \leq i \leq n$, put $B_i = \hat{A}/p_i$. a) The ring $B_i$ is Japanese. b) The integral closure $\overline{B}_i$ of $B_i$ is a Krull ring. c) The integral closure $\overline{A}$ of A is a Krull ring. (Use Exercise 8 and VII, § 1, No. 3, Examples 3 and 4.) d) $\overline{A}$ has only a finite number of maximal ideals and their residue fields are finite extensions of that of A. (Use Exercise 9, b).) e) For every prime ideal $p$ of A, there are only a finite number of prime ideals $\mathfrak{P}$ of $\overline{A}$ lying over $p$, and the fields $\kappa(\mathfrak{P})$ are finite extensions of $\kappa(p)$. (Reduce to the case where A is local with maximal ideal $p$.)