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

$D : \Gamma (T, \mathcal{O}_ T) \longrightarrow \Gamma (T, \mathcal{F})$

is a divided power $\Gamma (V, \mathcal{O}_ V)$-derivation where $V \subset S$ is any open such that $T \to S$ factors through $V$.

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