Exercise 9, § 3

(a) Let $A$ be a Noetherian domain such that, for every maximal ideal $m$ of $A$, $A_m$, is a discrete valuation ring. If $B$ is the ring of formal power series $A[[X_1, \ldots, X_n]]$, show that, for every maximal ideal $n$ of $B$, the domain $B_n$ is factorial (consider its completion, using Proposition 8 of no. 9). Deduce that every divisorial ideal of $B$ is a projective $B$-module.

(b) Let $C$ be a Noetherian ring such that every finitely generated projective $C$-module is free; show that the ring of formal power series $C[[X]]$ has the same property (cf. Chapter II, § 3, no. 2, Proposition 5).

(c) Deduce from (a) and (b) that, if $A$ is a principal ideal domain, the domain of formal power series $A[[X_1, \ldots, X_n]]$ is factorial.