Exercise 7, § 3

Let $K$ be a Noetherian factorial domain and $F$ an extremal element of the polynomial ring $K[X_1, \ldots, X_n]$; suppose that, when a weight $q(i) > 0$ ($1 \leq i \leq n$) is attributed to each $X_i$, $F$ is isobaric and of weight $q > 0$. Let $A$ be the domain generated, in an algebraic closure $\Omega$ of the field of fractions of $K(X_1, \ldots, X_n)$, by $K$, the $X_i$ ($1 \leq i \leq n$) and a root $z$ of the polynomial $Z^c - F$, where $c$ is an integer prime to $q$. Show that $A$ is factorial in the two following cases: (1) $c \equiv 1 \pmod{q}$; (2) every finitely generated projective $K$-module is free (which holds for example when $K$ is a field or a principal ideal domain or a local ring). (In the first case, consider the ring of fractions $A[1/z]$; show that it is a factorial domain and apply Proposition 3 of no. 4. In the second case, consider an integer $d$ such that $cd \equiv 1 \pmod{q}$ and let $z' \in \Omega$ such that $z = {z'}^d$; the domain $B = A[z']$ is factorial by virtue of the first case and is a free $A$-module. Consider $B$ as a graded ring taking $z'$ to be of weight $q$, each of the $X_i$ of weight $cdq(i)$ and reduce it to proving that for two homogeneous elements $u, v$ of $A$ the ideal $Au \cap Av$ is principal, using Exercise 16 of § 1; consider finally the ideal $Bu \cap Bv$ in $B$ and use Chapter I, § 3, no. 6, Proposition 12.) In particular, if $K$ is a field and $a, b, c$ are three integers $> 0$ which are relatively prime in pairs, the domain

is factorial.