Exercise A0.V.1.07

Proposition

If $R$ is a Noetherian ring then the ring $R[[x]]$ is also Noetherian.

Proof

Let the order of a power series in $x$ denote the smallest power of $x$ whose coefficient is non-zero. This coefficient is then called the dominant coefficient of a series.

Let $I$ be an ideal of $R[[x]].$

The proof of 1.6 can be adapted almost verbatim to show that $A_i$, the union of $\{0\}$ with the set of dominant coefficients of series of order $i$ in $I$, forms an ideal of $R.$

Given a non-zero series $f,$ we can always form a series with the same dominant coefficient but with degree one higher than that of $f$ by taking $x \cdot f.$ This implies that $A_n \subseteq A_{n + 1}.$

Since $R$ is Noetherian this sequence of inclusions stabilizes, say at $n = d.$ Each $A_n$ is finitely generated for the same reason. So let $(a_1, \ldots, a_m)$ be the generating set of all $A_q.$ For each $a_k$ there exists a power series $f_k \in I,$ whose dominant coefficient is $a_k$ and order is $d$ (via scaling by an appropriate power of $x$).

For each $p < d,$ let $G_p$ be a finite generating set of $A_p.$ For each element $v \in G_p$ there exists some power series $g_{v,p} \in I$ such that the dominant coefficient of $g_{v,p}$ is $v$ and its order is $p.$

Let $u \in I$ be an arbitrary power series.

Let $c_0, \ldots, c_d$ be a sequence of approximations of $u,$ such that $c_0 = 0$ and $c_h - u$ contains no terms of degree below $d,$ and let $\widehat{c_k} = c_0 + \ldots + c_k.$ The sequence is constructed as follows.

If $u - \widehat{c_i}$ has no terms of degree below $d$, we’re done. Otherwise, find the dominant term $w_r x^{r}$ of $u - \widehat{c_i}.$ Being a degree-$r$ dominant coefficient, $w_r$ is an element of $A_r$ and hence can be written down as a $R$-linear combination $\rho_1v_1 + \ldots + \rho_{|G_r|}v_{|G_r|}$ with $\rho \in R$ and $v \in G_r.$ Hence the series $c_{i+1} = \rho_1 g_{v_1, r} + \ldots + \rho_{|G_r|} g_{v_{|G_r|}, r}$ has the dominant term $w_r x^r$ and $u - \widehat{c_{i + 1}}$ has order strictly higher than $u - \widehat{c_{i}}.$ We repeat this procedure a finite number of steps until no terms of degree below $d$ remain.

Let $c_{d + 1}, \ldots$ be a potentially infinite sequence of further approximations of $u$ constructed as follows.

If $u - \widehat{c_j} = 0$ the sequence terminates at $j.$ Otherwise, take the dominant term $w_r x^r$ of $u - \widehat{c_j}.$ Since $r \geq d,$ we can choose a natural number $\gamma$ such that each series among $x^{\gamma}f_1, \ldots, x^{\gamma}f_m$ has the order $r.$ Since their dominant coefficients generate $A_q,$ we can find coefficients $\sigma_1, \ldots, \sigma_m$ such that the dominant term of $c_{j + 1} = \sigma_1x^{\gamma}f_1, \ldots, \sigma_mx^{\gamma}f_m$ is $w_rx^r.$ Just like before, $u - \widehat{c_{j + 1}}$ has order strictly higher than $u - \widehat{c_{j}}.$ This sequence may not terminate, however, for any $\iota \in \mathbb{N}$ the value of $c_\iota$ is well defined and $u - \widehat{c_\iota}$ has no terms with degree below $\iota.$

Finally, let $G^* = \{f_1, \ldots, f_m\} \cup \{ g_{v_p} \mid v \in G_p, p < d\}.$ For each $\tau \in G^*$ define the series $\xi_\tau \in R[[x]]$ by setting the coefficient of $x^\iota$ in $\xi_\tau$ to the coefficient of $x^{\iota}$ in the polynomial acting as the coefficient of $\tau$ in $\widehat{c_\iota}.$

We know that each step of the sequence $c$ adds only higher order corrections, and does not change the lower orders. After $\iota$ corrections, each increasing the order at least by one, the $\iota$-th coefficient of any generator is fixed. By construction of $c,$ we know that the $\iota$-th coefficient of $u - \sum_{\tau \in G^*} \xi_\tau \tau$ is zero. Hence $u = \sum_{\tau \in G^*} \xi_\tau \tau.$

Since $u$ was arbitrary, this demonstrates that $G^*$ is a finite generating set of $I.$ Since $I$ was also arbitrary, this demonstrates that $R[[x]]$ is Noetherian.

$\blacksquare$