Example 2

Let $A$ be a filtered commutative ring whose filtration we denote by $(a_n)_{n \in \mathbf{Z}}$; let $A'$ be the ring of formal power series $A[[X_1, \ldots, X_s]]$. For all $e = (e_1, \ldots, e_s) \in \mathbf{N}^s$, we write $|e| = \sum_{i=1}^s e_i$, $X^e = \prod_{i=1}^s X_i^{e_i}$ so that every element $P \in A'$ can be written uniquely $P = \sum_{e \in \mathbf{N}^s} \alpha_{e,P} X^e$ where $\alpha_{e,P} \in A$. For all $n \in \mathbf{Z}$, let $a'_n$ denote the set of $P \in A'$ such that $\alpha_{e,P} \in a_{n-|e|}$ for all $e \in \mathbf{N}^s$; we show that $a'_n$ is an ideal of $A'$. Clearly $a'_n$ is an additive subgroup of $A'$; on the other hand, if $P \in a'_n$ and $Q \in A'$, then, for all $e \in \mathbf{N}^s$, $\alpha_{e,PQ} = \sum_{e'+e''=e} \alpha_{e',Q}\alpha_{e'',P}$; as the relation $e' + e'' = e$ implies $|e''| \leq |e|$, $PQ \in a'_n$. Moreover, if $Q \in a'_m$, then, for $e' + e'' = e$, $\alpha_{e',Q}\alpha_{e'',P} \in a_{m-|e'|}a_{n-|e''|} \subset a_{m+n-|e|}$, which proves that $(a'_n)_{n \in \mathbf{Z}}$ is a filtration compatible with the ring structure on $A'$ (for obviously $1 \in a'_0$). When in future we speak of $A'$ as a filtered ring, we shall mean, unless otherwise stated, with the filtration (ah). Clearly $\bigcap_{n \in \mathbf{Z}} a'_n$ is the set of formal power series all of whose coefficients belong to $\bigcap_{n \in \mathbf{Z}} a_n$; then, if $A$ is Hausdorff, so is $A'$. If $a_0 = A$, then $a'_0 = A'$.