Proposition 8

Let C be a ring which is either a field or a discrete valuation ring. Then the domain of formal power series $C[[X_1, \ldots, X_n]]$ is factorial.

Let $p$ be the maximal ideal of C and $x$ a generator of $p$ (if C is a field, then $\pi = 0$). Let C be given the $p$-adic topology, which is Hausdorff. As C is a Noetherian local ring, $B = C[[X_1, \ldots, X_n]]$ is a Noetherian local ring and its completion is $\hat{C}[[X_1, \ldots, X_n]]$ (Chapter III, §2, no. 6, Proposition 6). By the Corollary to Proposition 4 (no. 6), it suffices to prove that $\hat{C}[[X_1, \ldots, X_n]]$ is factorial. Now, if C is a field, then $\hat{C} = C$; if C is a discrete valuation ring, the same is true of $\hat{C}$ (Chapter VI, §5, no. 3, Proposition 5). We shall therefore assume in the remainder of the proof that C is complete.

Arguing by induction starting with the trivial case $n = 0$, we shall assume that it has been proved that $A = C[[X_1, \ldots, X_{n-1}]]$ is factorial. We shall identify B with $A[[X_n]]$ and denote by m the maximal ideal of A (generated by $\pi, X_1, \ldots, X_{n-1}$). We shall prove that every non-zero element g of B is, in an essentially unique way, a product of extremal elements.

Let K be the field $C/C\pi$; as $B/B\pi$ is identified with $K[[X_1, \ldots, X_n]]$, the ideal $Bx$ is prime and $x$ is extremal. If $x \neq 0$, $B_{B\pi}$ is therefore the ring of a normed discrete valuation w (Chapter VI, §3, no. 6, Proposition 9); every non-zero element g of B may therefore be written as $g = \pi^{w(g)} f$, where $f \in B$ and f is not a multiple of $\pi$. It will therefore suffice to show that $f$ is an essentially unique product of extremal elements. Now the canonical image of $f$ in $K[[X_1, \ldots, X_n]]$ is not zero; Lemma 3 (no. 7) therefore shows that there exists an automorphism of B which maps $f$ to an element $f'$ such that the coefficients of $f'(0, \ldots, 0, X_n)$ are not all in $Cx$; this means that the coefficients of the series $f'$, considered as a formal power series in $X_n$, are not all in m. It will suffice to prove our assertion for $f'$.

In what follows, all the elements of B will be considered as formal power series in $X_n$ with coefficients in A. By Proposition 6 of no. 8 (applicable since C and therefore A are separable and complete and the reduced series off' is $\neq 0$), $f'$ is associated, in B, with a unique distinguished polynomial F. By Proposition 7 of no. 8, every series which divides $f'$ (or, what amounts to the same, which divides F) is associated with a distinguished polynomial which divides F and every decomposition off' is, to within invertible factors, of the form $f' = u F_1 \cdots F_q$, where $u$ is invertible and the $F_i$ are extremal distinguished polynomials (in B) such that $F = F_1 \cdots F_q$. By the Corollary to Proposition 7 of no. 8, the $F_i$ are also extremal in $\mathbf{A}[X_n]$. Now, as $\mathbf{A}$ is factorial by the induction hypothesis, so is $\mathbf{A}[X_n]$ (Theorem 2, no. 5); hence, since they are monic, the $F_i$ are uniquely determined by $F$ (up to a permutation). This shows the uniqueness of the decomposition $f' = u F_1 \ldots F_r$; its existence follows from the fact that $B$ is Noetherian, which completes the proof.

Remarks (1) There exist factorial rings $\mathbf{A}$ such that the ring $\mathbf{A}[[X]]$ is not factorial (Exercise 8). However, if $\mathbf{A}$ is a principal ideal domain, $\mathbf{A}[[X_1, \ldots, X_n]]$ is factorial (Exercise 9). (2) \* We shall see later, by homological methods, that every regular local ring is factorial (cf. § 4, no. 7, Corollary 3 to Proposition 16). This will give another proof, conceptually simpler, of Proposition 8. \*