Lemma 23.3.4. The category of divided power rings has all colimits.

Proof. The empty colimit is $\mathbf{Z}$ with divided power ideal $(0)$. Let's discuss general colimits. Let $\mathcal{C}$ be a category and let $c \mapsto (A_ c, I_ c, \gamma _ c)$ be a diagram. Consider the functor

$F(B, J, \delta ) = \mathop{\mathrm{lim}}\nolimits _{c \in \mathcal{C}} Hom((A_ c, I_ c, \gamma _ c), (B, J, \delta ))$

Note that any $f = (f_ c)_{c \in C} \in F(B, J, \delta )$ has the property that all the images $f_ c(A_ c)$ generate a subring $B'$ of $B$ of bounded cardinality $\kappa$ and that all the images $f_ c(I_ c)$ generate a divided power sub ideal $J'$ of $B'$. And we get a factorization of $f$ as a $f'$ in $F(B')$ followed by the inclusion $B' \to B$. Also, $F$ commutes with limits. Hence we may apply Lemma 23.3.3 to see that $F$ is representable and we win. $\square$

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