In this section we will stick with the (small) crystalline site as it seems more natural. We globalize Definition 60.6.1 as follows.
Definition 60.12.1. In Situation 60.7.5 let $\mathcal{F}$ be a sheaf of $\mathcal{O}_{X/S}$-modules on $\text{Cris}(X/S)$. An $S$-derivation $D : \mathcal{O}_{X/S} \to \mathcal{F}$ is a map of sheaves such that for every object $(U, T, \delta )$ of $\text{Cris}(X/S)$ the map is a divided power $\Gamma (V, \mathcal{O}_ V)$-derivation where $V \subset S$ is any open such that $T \to S$ factors through $V$.
\[ D : \Gamma (T, \mathcal{O}_ T) \longrightarrow \Gamma (T, \mathcal{F}) \]
This means that $D$ is additive, satisfies the Leibniz rule, annihilates functions coming from $S$, and satisfies $D(f^{[n]}) = f^{[n - 1]}D(f)$ for a local section $f$ of the divided power ideal $\mathcal{J}_{X/S}$. This is a special case of a very general notion which we now describe.
Please compare the following discussion with Modules on Sites, Section 18.33. Let $\mathcal{C}$ be a site, let $\mathcal{A} \to \mathcal{B}$ be a map of sheaves of rings on $\mathcal{C}$, let $\mathcal{J} \subset \mathcal{B}$ be a sheaf of ideals, let $\delta $ be a divided power structure on $\mathcal{J}$, and let $\mathcal{F}$ be a sheaf of $\mathcal{B}$-modules. Then there is a notion of a divided power $\mathcal{A}$-derivation $D : \mathcal{B} \to \mathcal{F}$. This means that $D$ is $\mathcal{A}$-linear, satisfies the Leibniz rule, and satisfies $D(\delta _ n(x)) = \delta _{n - 1}(x)D(x)$ for local sections $x$ of $\mathcal{J}$. In this situation there exists a universal divided power $\mathcal{A}$-derivation
Moreover, $\text{d}_{\mathcal{B}/\mathcal{A}, \delta }$ is the composition
where the first map is the universal derivation constructed in the proof of Modules on Sites, Lemma 18.33.2 and the second arrow is the quotient by the submodule generated by the local sections $\text{d}_{\mathcal{B}/\mathcal{A}}(\delta _ n(x)) - \delta _{n - 1}(x)\text{d}_{\mathcal{B}/\mathcal{A}}(x)$.
We translate this into a relative notion as follows. Suppose $(f, f^\sharp ) : (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}), \mathcal{O}) \to (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}'), \mathcal{O}')$ is a morphism of ringed topoi, $\mathcal{J} \subset \mathcal{O}$ a sheaf of ideals, $\delta $ a divided power structure on $\mathcal{J}$, and $\mathcal{F}$ a sheaf of $\mathcal{O}$-modules. In this situation we say $D : \mathcal{O} \to \mathcal{F}$ is a divided power $\mathcal{O}'$-derivation if $D$ is a divided power $f^{-1}\mathcal{O}'$-derivation as defined above. Moreover, we write
which is the receptacle of the universal divided power $\mathcal{O}'$-derivation.
Applying this to the structure morphism
(see Remark 60.9.6) we recover the notion of Definition 60.12.1 above. In particular, there is a universal divided power derivation
Note that we omit from the notation the decoration indicating the module of differentials is compatible with divided powers (it seems unlikely anybody would ever consider the usual module of differentials of the structure sheaf on the crystalline site).
Lemma 60.12.2. Let $(T, \mathcal{J}, \delta )$ be a divided power scheme. Let $T \to S$ be a morphism of schemes. The quotient $\Omega _{T/S} \to \Omega _{T/S, \delta }$ described above is a quasi-coherent $\mathcal{O}_ T$-module. For $W \subset T$ affine open mapping into $V \subset S$ affine open we have where the right hand side is as constructed in Section 60.6.
\[ \Gamma (W, \Omega _{T/S, \delta }) = \Omega _{\Gamma (W, \mathcal{O}_ W)/\Gamma (V, \mathcal{O}_ V), \delta } \]
Proof. Omitted. $\square$
Lemma 60.12.3. In Situation 60.7.5. For $(U, T, \delta )$ in $\text{Cris}(X/S)$ the restriction $(\Omega _{X/S})_ T$ to $T$ is $\Omega _{T/S, \delta }$ and the restriction $\text{d}_{X/S}|_ T$ is equal to $\text{d}_{T/S, \delta }$.
Proof. Omitted. $\square$
Lemma 60.12.4. In Situation 60.7.5. For any affine object $(U, T, \delta )$ of $\text{Cris}(X/S)$ mapping into an affine open $V \subset S$ we have where the right hand side is as constructed in Section 60.6.
\[ \Gamma ((U, T, \delta ), \Omega _{X/S}) = \Omega _{\Gamma (T, \mathcal{O}_ T)/\Gamma (V, \mathcal{O}_ V), \delta } \]
Lemma 60.12.5. In Situation 60.7.5. Let $(U, T, \delta )$ be an object of $\text{Cris}(X/S)$. Let in $\text{Cris}(X/S)$. Let $\mathcal{K} \subset \mathcal{O}_{T(1)}$ be the quasi-coherent sheaf of ideals corresponding to the closed immersion $\Delta : T \to T(1)$. Then $\mathcal{K} \subset \mathcal{J}_{T(1)}$ is preserved by the divided structure on $\mathcal{J}_{T(1)}$ and we have
\[ (U(1), T(1), \delta (1)) = (U, T, \delta ) \times (U, T, \delta ) \]
\[ (\Omega _{X/S})_ T = \mathcal{K}/\mathcal{K}^{[2]} \]
Proof. Note that $U = U(1)$ as $U \to X$ is an open immersion and as (60.9.1.1) commutes with products. Hence we see that $\mathcal{K} \subset \mathcal{J}_{T(1)}$. Given this fact the lemma follows by working affine locally on $T$ and using Lemmas 60.12.4 and 60.6.5. $\square$
It turns out that $\Omega _{X/S}$ is not a crystal in quasi-coherent $\mathcal{O}_{X/S}$-modules. But it does satisfy two closely related properties (compare with Lemma 60.11.2).
Lemma 60.12.6. In Situation 60.7.5. The sheaf of differentials $\Omega _{X/S}$ has the following two properties:
$\Omega _{X/S}$ is locally quasi-coherent, and
for any morphism $(U, T, \delta ) \to (U', T', \delta ')$ of $\text{Cris}(X/S)$ where $f : T \to T'$ is a closed immersion the map $c_ f : f^*(\Omega _{X/S})_{T'} \to (\Omega _{X/S})_ T$ is surjective.
Proof. Part (1) follows from a combination of Lemmas 60.12.2 and 60.12.3. Part (2) follows from the fact that $(\Omega _{X/S})_ T = \Omega _{T/S, \delta }$ is a quotient of $\Omega _{T/S}$ and that $f^*\Omega _{T'/S} \to \Omega _{T/S}$ is surjective. $\square$
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