De Rham prismatic crystals over OK

We study de Rham prismatic crystals on . We show that a de Rham crystal is controlled by a sequence of matrices { A m , 1 } m ≥ 0 with A 0 , 1 “nilpotent”. Using this, we prove that the natural functor from the category of de Rham crystals over to the category of nearly de Rham representations is fully faithful. The key ingredient is a Sen style decompletion theorem for B dR + -representations of G K .