Remark 23.5.2. Let (A, I, \gamma ) be a divided power ring. There is a variant of Lemma 23.5.1 for infinitely many variables. First note that if s < t then there is a canonical map
A\langle x_1, \ldots , x_ s \rangle \to A\langle x_1, \ldots , x_ t\rangle
Hence if W is any set, then we set
A\langle x_ w: w \in W\rangle = \mathop{\mathrm{colim}}\nolimits _{E \subset W} A\langle x_ e:e \in E\rangle
(colimit over E finite subset of W) with transition maps as above. By the definition of a colimit we see that the universal mapping property of A\langle x_ w: w \in W\rangle is completely analogous to the mapping property stated in Lemma 23.5.1.
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