Exercise 8, § 3

Let $A$ be an integral ring, $x, y, z$ 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 $B = S^{-1}A$; we consider the rings of formal series $A[[T]]$ and $B[[T]]$.

(a) Let $i, j, k$ be three integers $\geqslant 0$ such that $ijk - ij - jk - ki \geqslant 0$ and $z^i \in \mathbf{A} x^j + \mathbf{A} y^k$. We consider in $\mathbf{A}[[T]]$ the element $v = xy - z^{i-1} T$. Now prove that there exists an integer $t > 0$ and a series

in $\mathbf{B}[[T]]$ such that $vv' \in \mathbf{A}[[T]]$ (the $b_n$ will be determined by recurrence, proceeding in $\mathbf{N}$ by intervals of length $ij$; inside each interval, take $b_{n+1} = b_n z^{i-1} x^{-1}$; at the end of each interval, use the inequality $ijk - ij - jk - ki \geqslant 0$.

(b) Suppose that $z^{i-1} \notin \mathbf{A} x + \mathbf{A} y$. Now prove that there exists no formal series in the ring $\mathbf{A}[[T]]$ having constant term $y^k$ ($k$ integer $> 0$) and which is an associated element to $v$ in $\mathbf{B}[[T]]$ (calculate the coefficient of $T$ in the product of $v$ and an invertible element of $\mathbf{B}[[T]]$).

(c) Deduce from a), b) and exerc. 7 an example of a factorial ring $\mathbf{A}$ such that the ring of formal series $\mathbf{A}[[T]]$ is not factorial. (With the preceding notations, and supposing the conditions of a) and b) on $x, y, z, i, j, k$ fulfilled, now prove that $vv'$ cannot be a product of extremal factors $u_h$ ($1 \leqslant h \leqslant r$) in $\mathbf{A}[[T]]$; one will observe that the $u_h$ are formal series whose constant terms are powers of $y$. Consider next the ring $\mathbf{C} = {S'}^{-1} \mathbf{A}[[T]]$, where $S'$ is formed of the series whose constant term is in $S$; $\mathbf{B}[[T]]$ is the completion of the Zariski ring $\mathbf{C}$; now prove that $v' \in \mathbf{C}$, and that $v$ and the $u_h$ are extremal in $\mathbf{C}$; obtain finally a contradiction with b)).

(d) Deduce from c) an example of a local factorial noetherian ring $\mathbf{A}$ whose completion $\hat{\mathbf{A}}$ is a Krull ring which is not factorial.