The Stacks project

Lemma 60.7.4. Let $(U', T', \delta ') \to (S'_0, S', \gamma ')$ and $(S_0, S, \gamma ) \to (S'_0, S', \gamma ')$ be morphisms of divided power schemes. If $(U', T', \delta ')$ is a divided power thickening, then there exists a divided power scheme $(T_0, T, \delta )$ and

\[ \xymatrix{ T \ar[r] \ar[d] & T' \ar[d] \\ S \ar[r] & S' } \]

which is a cartesian diagram in the category of divided power schemes.

Proof. Omitted. Hints: If $T$ exists, then $T_0 = S_0 \times _{S'_0} U'$ (argue as in Divided Power Algebra, Remark 23.3.5). Since $T'$ is a divided power thickening, we see that $T$ (if it exists) will be a divided power thickening too. Hence we can define $T$ as the scheme with underlying topological space the underlying topological space of $T_0 = S_0 \times _{S'_0} U'$ and as structure sheaf on affine pieces the ring given by Lemma 60.5.3. $\square$


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