De Rham prismatic crystals over $\mathcal{O}_K$

Abstract

We study de Rham prismatic crystals on $(\mathcal{O}_K)_{\bbold{\Delta}}$. We show that a de Rham crystal is controlled by a sequence of matrices $\{A_{m,1}\}_{m \geq 0}$ with $A_{0,1}$ "nilpotent". Using this, we prove that the natural functor from de Rham crystals over $(\mathcal{O}_K)_{\bbold{\Delta}}$ to the category of nearly de Rham representations is fully faithful. The key ingredient is a Sen style decompletion theorem for de Rham representations of $G_K$.