Exercise 7, § 3

Soient K a noetherian factorial ring, 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 assigned to each $X_i$, F is isobaric and of weight $q > 0$. Let A be the ring 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 $q$. Now prove that A is factorial in the following two cases: 1) one has $c \equiv 1 \pmod{q}$; 2) every finitely generated projective K-module is free (which is the case 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 factorial and apply prop. 3 of No. 4. In the second case, consider an integer $d$ such that $cd \equiv 1 \pmod{q}$, and let $z' \in \Omega$ be such that $z = {z'}^d$; the ring $B = A[z']$ is factorial by virtue of the first case, and is a free a-module. Consider B as a graded ring by taking $z'$ of weight $q$, each of the $X_i$ of weight $cdq(i)$, and reduce to proving that for two homogeneous elements $u, v$ of A, the ideal $Au \cap Av$ is principal, using Exercise 16 of § 1; finally consider the ideal $Bu \cap Bv$ in B and use Chapter I, § 3, No. 6, prop. 12). In particular, if K is a field, $a, b, c$ three integers $> 0$ prime to each other pairwise, the ring

is factorial.