Exercise 9, § 3

a) Let $\mathbf{A}$ be a noetherian integral ring such that for every maximal ideal $m$ of $\mathbf{A}$, $\mathbf{A}_m$ is a discrete valuation ring. If $\mathbf{B}$ is the formal power series ring $\mathbf{A}[[X_1, \ldots, X_n]]$, now prove that for every maximal ideal $n$ of $\mathbf{B}$, the ring $\mathbf{B}_n$ is factorial (consider its completion, using prop. 8 of No. 9). Deduce that every divisorial ideal of $\mathbf{B}$ is a $\mathbf{B}$-module projective.

b) Let $\mathbf{C}$ be a noetherian ring such that every finitely generated projective $\mathbf{C}$-module is free; now prove that the formal power series ring $\mathbf{C}[[X]]$ has the same property (cf. Chapter II, § 3, No. 2, prop. 5).

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