Exercise 11, § 2

(a) Let $A$ be a local integrally closed domain, $m$ its maximal ideal, $K$ its field of fractions, $K'$ a finite Galois extension of $K$, $A'$ the integral closure of $A$ in $K'$, $m'$ a maximal ideal of $A'$ lying above $m$, $B = A^z(m')$ its decomposition ring and $n = m' \cap B$. For every integer $k > 0$, $B = A + n^k$ (argue as in Proposition 4). Moreover there exists a primitive element $z$ of $K^z$ (field of fractions of $B$) such that $z \in n$; deduce that $n \cap A[z] = mA[z]$.

(b) Show that, for every integer $k$, $m^kB_n \cap A = m^k$. (If $x \in m^kB_n \cap A$, note that there exists $t \in A[z]$ (where $z$ is defined in (a)) such that $t \notin n$ and $tx \in m^kA[z]$; writing $t = t_0 + t_1z + \cdots + t_{r-1}z^{r-1}$ (where $r$ is the degree $[K^z : K]$), deduce that $t_0x \in m^k$ and $t_0 \notin m$.)

(c) Suppose now that $K$ is any Galois extension of $K$. Keeping the same notation as above, show that $B = A + n^k$ and $m^kB_n \cap A = m^k$ still (note that every element of $A^z(m')$ is contained in the decomposition ring of $m' \cap C$, where $C$ is the integral closure of $A$ in a finite Galois sub-extension of $K'$).

(d) Under the hypotheses of (c), show that, if $A$ is Noetherian, so is $B$. (Observe that the Hausdorff completion of $B_n$ with respect to the $m$-adic topology is identified with the completion $\hat{A}$ of $A$ by (c); if $\phi : B_n \to \hat{A}$ is the canonical homomorphism, show that, for every ideal $a$ of $B_n$, $\phi(a\hat{A}) = a$, using the fact that in $\hat{A}$ every ideal is finitely generated and hence that $a\hat{A}$ is generated by a finite number of elements of $a$ and thus reduce it to the case where $K'$ is a finite extension of $K$.)