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

We study de Rham prismatic crystals on $(\mathcal{O}_K)_{\bboldΔ}$. 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 $(\mathcal{O}_K)_{\bboldΔ}$ 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 $G_K$.

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