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

Abstract

We study de Rham prismatic crystals on (O K ) ∆ . 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 de Rham crystals over (O K ) ∆ 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 .

de Rham crystals on the prismatic site of $\mathcal{O}_K$

“…In the final stage of our redaction (after we obtained all the main results), Zeyu Liu informed us that he independently obtained some partial results (under the prismatic setting) discussed in this paper, cf. [Liu22]. For the reader's convenience, we make some rough comparisons:…”