Exercise 27, § 3

(a) Let $K$ be a field, $B$ the ring generated, in an algebraic closure of the field of rational fractions $K(U, V, X, Y)$ in 4 indeterminates, by the polynomial ring $K[U, V, X, Y]$ and by a root $z$ of the polynomial $F = Z^7 - U^5X^2 - V^4Y^3$. Now prove that $B$ is a factorial ring (cf. exerc. 7).

(b) Let $p$ be the (prime) ideal generated by $X, Y, U, V$ and $z$ in $A$, and set $C = A_p[[T]]$; show that $C$ is a local noetherian ring, which is not factorial, but whose associated graded ring $gr(C)$ is factorial (use exerc. 8).