The Stacks project

Definition 60.11.1. In Situation 60.7.5. Let $\mathcal{C} = \text{CRIS}(X/S)$ or $\mathcal{C} = \text{Cris}(X/S)$. Let $\mathcal{F}$ be a sheaf of $\mathcal{O}_{X/S}$-modules on $\mathcal{C}$.

  1. We say $\mathcal{F}$ is locally quasi-coherent if for every object $(U, T, \delta )$ of $\mathcal{C}$ the restriction $\mathcal{F}_ T$ is a quasi-coherent $\mathcal{O}_ T$-module.

  2. We say $\mathcal{F}$ is quasi-coherent if it is quasi-coherent in the sense of Modules on Sites, Definition 18.23.1.

  3. We say $\mathcal{F}$ is a crystal in $\mathcal{O}_{X/S}$-modules if all the comparison maps ( are isomorphisms.

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