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.
\[ (\mathcal{O}(U), \mathcal{I}(U), \gamma ) \]
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.
\[ (\mathcal{O}_{S'}(U'), \mathcal{I}'(U'), \gamma ') \longrightarrow (\mathcal{O}_ S(f^{-1}U'), \mathcal{I}(f^{-1}U'), \gamma ) \]
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 $(U', T', \delta ') \to (S'_0, S', \gamma ')$ 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 which is a cartesian diagram in the category of divided power schemes.
\[ \xymatrix{ T \ar[r] \ar[d] & T' \ar[d] \\ S \ar[r] & S' } \]
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. $\square$
We make the following observation. Suppose that $(U, T, \gamma )$ is triple as above. 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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