Lemma 60.7.4. Let $f : (T, \mathcal{J}, \delta ) \to (S, \mathcal{I}, \gamma )$ and $f' : (T', \mathcal{J}', \delta ') \to (S, \mathcal{I}, \gamma )$ be morphisms of divided power schemes. There exists a divided power scheme $(T'', \mathcal{J}'', \delta '')$ and a cartesian diagram
\[ \xymatrix{ T \ar[d]_ f & T'' \ar[d] \ar[l] \\ S & T' \ar[l]_{f'} } \]
in the category of divided power schemes. The morphsm $T'' \to T \times _ S T'$ is a closed immersion and the morphism $T''_0 \to T_0 \times _{S_0} T'_0$ is an isomorphism.
Proof.
Sketch. Note that the two final statements are compatible, via $\mathop{\mathrm{Spec}}(-)$, with what we have seen for pushouts of divided power algebras in Divided Power Algebra, Remark 23.3.5. Thus we construct $T''$ as a closed subscheme of $T \times _ S T'$ as follows: for any affine opens $U \subset S$, $V \subset T$, $V' \subset T'$ with $f(V), f'(V') \subset U$ we consider the closed subscheme of $V \times _ U V'$ determined by the construction in Divided Power Algebra, Remark 23.3.5. Since the schemes $V \times _ U V'$ are the members of an open covering of $T \times _ S T'$ we can proceed as follows: (1) we show that these closed subschemes glue, (2) the resulting divided power structures glue, and (3) the result of glueing is the fibre product in the category of divided power schemes. To see (1) is true amounts to showing that the construction of the pushout in the category of divided power rings commutes with localization (suitably formulated); this follows from the result of Divided Power Algebra, Lemma 23.4.2 which implies that divided powers extend to localizations (by flatness).
$\square$
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