Prismatic crystals over the de Rham period sheaf

TL;DR

This paper classifies prismatic crystals over the de Rham period sheaf using log connections and nearly de Rham representations, establishing links with Sen--Fontaine and Galois cohomology.

Contribution

It introduces a classification of $dr^+$-crystals via log connections and nearly de Rham representations, extending prismatic theory with a Sen--Fontaine framework.

Findings

Classified $dr^+$-crystals using log connections.

Developed a Sen--Fontaine theory for $dr^+$-representations.

Connected prismatic cohomology with Galois cohomology.

Abstract

Let $\mathcal{O}_K$ be a mixed characteristic complete discrete valuation ring with perfect residue field. We study $\mathbb{B}_\mathrm{dR}^+$-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 $\mathbf{B}_{\mathrm{dR}}^+$-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.