Prismatic crystals over the de Rham period sheaf

Let ${\mathcal{O}}_K$ be a mixed characteristic complete discrete valuation ring with perfect residue field. We study ${\mathbb{B}}_{\mathrm{d}\mathrm{R}}^{+}$-crystals on the (log-) prismatic site of ${\mathcal{O}}_K$, which are crystals defined over the de Rham period sheaf. We first classify these crystals using certain log connections. By constructing a Sen–Fontaine theory for ${\mathbf{B}}_{\mathrm{d}\mathrm{R}}^{+}$-representations over a Kummer tower, we further classify these crystals by (log-) nearly de Rham representations. In addition, we compare (log-) prismatic cohomology of these crystals with the corresponding Sen–Fontaine cohomology and Galois cohomology.