De Rham prismatic crystals over mathcal {O}_K

Abstract

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 ge 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_{{textrm{dR}}}^+-representations of G_K.