Remark 87.28.1 (Universal property restricted power series).reference 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.
Comments (0)