Some remarks on how to globalize the previous notions.
Definition 60.7.1. Let $\mathcal{C}$ be a site. Let $\mathcal{O}$ be a sheaf of rings on $\mathcal{C}$. Let $\mathcal{I} \subset \mathcal{O}$ be a sheaf of ideals. A divided power structure $\gamma $ on $\mathcal{I}$ is a sequence of maps $\gamma _ n : \mathcal{I} \to \mathcal{I}$, $n \geq 1$ such that for any object $U$ of $\mathcal{C}$ the triple
is a divided power ring.
To be sure this applies in particular to sheaves of rings on topological spaces. But it's good to be a little bit more general as the structure sheaf of the crystalline site lives on a... site! A triple $(\mathcal{C}, \mathcal{I}, \gamma )$ as in the definition above is sometimes called a divided power topos in this chapter. Given a second $(\mathcal{C}', \mathcal{I}', \gamma ')$ and given a morphism of ringed topoi $(f, f^\sharp ) : (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}), \mathcal{O}) \to (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}'), \mathcal{O}')$ we say that $(f, f^\sharp )$ induces a morphism of divided power topoi if $f^\sharp (f^{-1}\mathcal{I}') \subset \mathcal{I}$ and the diagrams
are commutative for all $n \geq 1$. If $f$ comes from a morphism of sites induced by a functor $u : \mathcal{C}' \to \mathcal{C}$ then this just means that
is a homomorphism of divided power rings for all $U' \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C}')$.
In the case of schemes we require the divided power ideal to be quasi-coherent. But apart from this the definition is exactly the same as in the case of topoi. Here it is.
Definition 60.7.2. A divided power scheme is a triple $(S, \mathcal{I}, \gamma )$ where $S$ is a scheme, $\mathcal{I}$ is a quasi-coherent sheaf of ideals, and $\gamma $ is a divided power structure on $\mathcal{I}$. A morphism of divided power schemes $(S, \mathcal{I}, \gamma ) \to (S', \mathcal{I}', \gamma ')$ is a morphism of schemes $f : S \to S'$ such that $f^{-1}\mathcal{I}'\mathcal{O}_ S \subset \mathcal{I}$ and such that
is a homomorphism of divided power rings for all $U' \subset S'$ open.
Recall that there is a 1-to-1 correspondence between quasi-coherent sheaves of ideals and closed immersions, see Morphisms, Section 29.2. Thus given a divided power scheme $(T, \mathcal{J}, \gamma )$ we get a canonical closed immersion $U \to T$ defined by $\mathcal{J}$. Conversely, given a closed immersion $U \to T$ and a divided power structure $\gamma $ on the sheaf of ideals $\mathcal{J}$ associated to $U \to T$ we obtain a divided power scheme $(T, \mathcal{J}, \gamma )$. In many situations we only want to consider such triples $(U, T, \gamma )$ when the morphism $U \to T$ is a thickening, see More on Morphisms, Definition 37.2.1.
Definition 60.7.3. A triple $(U, T, \gamma )$ as above is called a divided power thickening if $U \to T$ is a thickening.
Fibre products of divided power schemes exist when one of the three is a divided power thickening. Here is a formal statement.
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
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). $\square$
We make the following observation. Suppose that $(U, T, \gamma )$ is a divided power scheme. Assume that $T$ is a scheme over $\mathbf{Z}_{(p)}$ and that $p$ is locally nilpotent on $U$. Then
$p$ locally nilpotent on $T \Leftrightarrow U \to T$ is a thickening (see Divided Power Algebra, Lemma 23.2.6), and
$p^ e\mathcal{O}_ T$ is locally on $T$ preserved by $\gamma $ for $e \gg 0$ (see Divided Power Algebra, Lemma 23.4.5).
This suggest that good results on divided power thickenings will be available under the following hypotheses.
Situation 60.7.5. Here $p$ is a prime number and $(S, \mathcal{I}, \gamma )$ is a divided power scheme over $\mathbf{Z}_{(p)}$. We set $S_0 = V(\mathcal{I}) \subset S$. Finally, $X \to S_0$ is a morphism of schemes such that $p$ is locally nilpotent on $X$.
It is in this situation that we will define the big and small crystalline sites.
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