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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