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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