Exercise 8, § 3

Let $A$ be an integral domain and $x, y, z$ three non-zero elements of $A$, where $x$ is extremal and $Ax \cap Ay = Axy$. Let $S$ be the multiplicative set of the $x^n$ ($n \geq 0$) and let $B = S^{-1}A$; consider the domains of formal power series $A[[T]]$ and $B[[T]]$. (a) Let $i, j, k$ be three integers $\geq 0$ such that $ijk - ij - jk - ki \geq 0$ and $z^i \in Ax^j + Ay^k$. Consider in $A[[T]]$ the element $v = xy - z^{i-1}T$. Show that there exists an integer $t > 0$ and a series

in $B[[T]]$ such that $vv' \in A[[T]]$ (determine the $b_n$ inductively, proceeding in $\mathbf{N}$ by intervals of length $ij$; in the interior of each interval, take at the end of each interval, use the inequality $ijk + ij - jk - ki \geq 0$. (b) Suppose that $z^{i-1} \notin Ax + Ay$. Show that there exists in the ring $A[[T]]$ no formal power series of constant term $y^k$ ($k$ an integer $> 0$) and which is an element associated with $v$ in $B[[T]]$ (calculate the coefficient of $T$ in the product of $v$ and an invertible element of $B[[T]]$).

(c) Deduce from (a), (b) and Exercise 7 an example of a factorial domain $A$ such that the domain of formal power series $A[[T]]$ is not factorial. (In the above notation and assuming that the conditions of (a) and (b) on $x, y, z, i, j, k$ are fulfilled, show that $vv'$ cannot be a product of extremal factors $u_h$ ($1 \leq h \leq r$) in $A[[T]]$; observe that the $u_h$ are formal power series whose constant terms are powers of $y$. Then consider the ring $C = {S'}^{-1}A[[T]]$, where $S'$ consists of the series whose constant term is in $S$; $B[[T]]$ is the completion of the Zariski ring $C$; show that $v' \in C$ and that $v$ and the $u_h$ are extremal in $C$; obtain finally a contradiction with (b).)

(d) Deduce from (c) an example of a Noetherian factorial local integral domain $A$ whose completion $\hat{A}$ is a non-factorial Krull domain.