Lemma 60.2.1. Let (A, I, \gamma ) be a divided power ring. Let A \to B be a ring map. Let J \subset B be an ideal with IB \subset J. There exists a homomorphism of divided power rings
(A, I, \gamma ) \longrightarrow (D, \bar J, \bar\gamma )
such that
\mathop{\mathrm{Hom}}\nolimits _{(A, I, \gamma )}((D, \bar J, \bar\gamma ), (C, K, \delta )) = \mathop{\mathrm{Hom}}\nolimits _{(A, I)}((B, J), (C, K))
functorially in the divided power algebra (C, K, \delta ) over (A, I, \gamma ). Here the LHS is morphisms of divided power rings over (A, I, \gamma ) and the RHS is morphisms of (ring, ideal) pairs over (A, I).
Proof.
Denote \mathcal{C} the category of divided power rings (C, K, \delta ). Consider the functor F : \mathcal{C} \longrightarrow \textit{Sets} defined by
F(C, K, \delta ) = \left\{ (\varphi , \psi ) \middle | \begin{matrix} \varphi : (A, I, \gamma ) \to (C, K, \delta ) \text{ homomorphism of divided power rings}
\\ \psi : (B, J) \to (C, K)\text{ an } A\text{-algebra homomorphism with }\psi (J) \subset K
\end{matrix} \right\}
We will show that Divided Power Algebra, Lemma 23.3.3 applies to this functor which will prove the lemma. Suppose that (\varphi , \psi ) \in F(C, K, \delta ). Let C' \subset C be the subring generated by \varphi (A), \psi (B), and \delta _ n(\psi (f)) for all f \in J. Let K' \subset K \cap C' be the ideal of C' generated by \varphi (I) and \delta _ n(\psi (f)) for f \in J. Then (C', K', \delta |_{K'}) is a divided power ring and C' has cardinality bounded by the cardinal \kappa = |A| \otimes |B|^{\aleph _0}. Moreover, \varphi factors as A \to C' \to C and \psi factors as B \to C' \to C. This proves assumption (1) of Divided Power Algebra, Lemma 23.3.3 holds. Assumption (2) is clear as limits in the category of divided power rings commute with the forgetful functor (C, K, \delta ) \mapsto (C, K), see Divided Power Algebra, Lemma 23.3.2 and its proof.
\square
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