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Tag 07ME

Chapter 54: Crystalline Cohomology > Section 54.7: Divided power schemes

Lemma 54.7.4. 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 54.5.3. $\square$

    The code snippet corresponding to this tag is a part of the file crystalline.tex and is located in lines 1543–1557 (see updates for more information).

    \begin{lemma}
    \label{lemma-fibre-product}
    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.
    \end{lemma}
    
    \begin{proof}
    Omitted. Hints: If $T$ exists, then $T_0 = S_0 \times_{S'_0} U'$
    (argue as in Divided Power Algebra, Remark \ref{dpa-remark-forgetful}).
    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 \ref{lemma-affine-thickenings-colimits}.
    \end{proof}

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