Exercise 27, § 3

(a) Let $K$ be a field and $B$ the ring generated, in an algebraic closure of the field of rational functions $K(U, V, X, Y)$ in 4 indeterminates, by the polynomial ring $K[U, V, X, Y]$ and a root $z$ of the polynomial

Show that $B$ is a factorial domain (cf. Exercise 7).

(b) Let $\mathfrak{p}$ be the (prime) ideal generated by $X, Y, U, V$ and $z$ in $A$ and write $C = A_{\mathfrak{p}}[[T]]$; show that $C$ is a Noetherian local integral domain, which is not factorial, but whose associated graded domain $\mathrm{gr}(C)$ is factorial (use Exercise 8).