Exercise 21, § 1

(a) Let $K_0$ be a perfect field of characteristic $p > 0$, $L$ the field of rational functions $K_0(X_n)_{n \in \mathbf{N}}$, $A'$ the ring of formal power series $L[[Y, Z]]$, where $Y$ and $Z$ are two indeterminates, and $A$ the tensor product ring $K[[Y, Z]] \otimes_K L$, where we write $K = L^p \subset L$; $A$ is a Noetherian local ring, which is not complete and whose completion is identified with $A'$ (Chapter III, § 3, Exercise 17) and $A$ is integral over $A$. Let $c_n$ denote the element

of $A'$. Let $B$ be the subring of $A'$ generated by $A$ and the elements $c_n$ ($n \geq 0$). If $C = A[c_0]$, $C$ is Noetherian and the ring $B$ (which is not integrally closed) is contained in the integral closure of $C$ and contains $C$; but show that $B$ is not Noetherian (observe that $c_n$ does not belong to the ideal of $B$ generated by the $c_i$ of index $i < n$).

(b) Suppose $p = 2$; $K_0, K$ and $L$ being used with the same meaning as in (a), consider this time the ring of formal power series $A' = L[[Y, Z, T]]$ in three indeterminates, the subring

and write

Show that the ring $A[b]$ is Noetherian but that its integral closure B is not a Noetherian ring. (As $b^2 \in A$, every element of the field of fractions of $A[b]$ is of the form $P + Q b$, where P and Q are linear combinations of a finite number of $X_k$ with coefficients in the field of fractions of $K[[Y, Z, T]]$; for such an element to be integral over $A[b]$, show that it is necessary and sufficient that it belong to $A'$. Show that B is generated by the elements and conclude the argument as in (a).)