We may check this locally on
\(A\text{,}\) so by
Lemma 5.2.5 we may assume that
\(I\) is principal generated by a distinguished element
\(d\text{.}\) Let
\((A, I) \to (B, J)\) be a morphism of
\(\delta\)-pairs. Let
\(B'\) be the free
\(\delta\)-ring over
\(A\) in the generators
\(x/d\) for
\(x \in J\text{.}\) Let
\(B_1\) be the derived
\((p,d)\)-completion of
\(B\) (viewed as a
\(\delta\)-ring using
Exercise 6.7.12). If
\(B_1\) is
\(d\)-torsion-free, then
\((B_1, dB_1)\) has the desired universal property. Otherwise, we transfinitely iterate the operations of taking the maximal
\(d\)-torsion-free quotient and taking the derived
\((p,d)\)-completion; this terminates because a countably filtered colimit of derived
\((p,d)\)-complete rings is again derived
\((p,d)\)-complete (
Remark 6.3.4), so we can stop taking the completions once we get to an uncountable ordinal.