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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