Remark 87.28.1 (Universal property restricted power series). Let $A \to C$ be a continuous map of complete linearly topologized rings. Then any $A$-algebra map $A[x_1, \ldots x_ r] \to C$ extends uniquely to a continuous map $A\{ x_1, \ldots , x_ r\} \to C$ on restricted power series.
87.28 Restricted power series
Let $A$ be a topological ring complete with respect to a linear topology (More on Algebra, Definition 15.36.1). Let $I_\lambda $ be a fundamental system of open ideals. Let $r \geq 0$ be an integer. In this setting one often denotes
\[ A\{ x_1, \ldots , x_ r\} = \mathop{\mathrm{lim}}\nolimits _\lambda A/I_\lambda [x_1, \ldots , x_ r] = \mathop{\mathrm{lim}}\nolimits _\lambda (A[x_1, \ldots , x_ r]/I_\lambda A[x_1, \ldots , x_ r]) \]
endowed with the limit topology. In other words, this is the completion of the polynomial ring with respect to the ideals $I_\lambda $. We can think of elements of $A\{ x_1, \ldots , x_ r\} $ as power series
\[ f = \sum \nolimits _{E = (e_1, \ldots , e_ r)} a_ E x_1^{e_1} \ldots x_ r^{e_ r} \]
in $x_1, \ldots , x_ r$ with coefficients $a_ E \in A$ which tend to zero in the topology of $A$. In other words, for any $\lambda $ all but a finite number of $a_ E$ are in $I_\lambda $. For this reason elements of $A\{ x_1, \ldots , x_ r\} $ are sometimes called restricted power series. Sometimes this ring is denoted $A\langle x_1, \ldots , x_ r\rangle $; we will refrain from using this notation.
Remark 87.28.2. Let $A$ be a ring and let $I \subset A$ be an ideal. If $A$ is $I$-adically complete, then the $I$-adic completion $A[x_1, \ldots , x_ r]^\wedge $ of $A[x_1, \ldots , x_ r]$ is the restricted power series ring over $A$ as a ring. However, it is not clear that $A[x_1, \ldots , x_ r]^\wedge $ is $I$-adically complete. We think of the topology on $A\{ x_1, \ldots , x_ r\} $ as the limit topology (which is always complete) whereas we often think of the topology on $A[x_1, \ldots , x_ r]^\wedge $ as the $I$-adic topology (not always complete). If $I$ is finitely generated, then $A\{ x_1, \ldots , x_ r\} = A[x_1, \ldots , x_ r]^\wedge $ as topological rings, see Algebra, Lemma 10.96.3.
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