The Stacks project

Proof. Denote $\mathcal{C}$ the category of divided power rings $(C, K, \delta )$. Consider the functor $F : \mathcal{C} \longrightarrow \textit{Sets}$ defined by

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. $\square$