Prismatic Crystals for schemes in characteristic $p$

TL;DR

This paper develops a new classification of crystalline prismatic crystals on schemes in characteristic p, generalizing previous equivalences and establishing a cohomological comparison theorem.

Contribution

It introduces a classification of prismatic crystals via p-connections and proves a more general equivalence than Ogus' by gluing local correspondences.

Findings

Established a cohomological comparison theorem.

Proved the equivalence between crystalline and prismatic crystals.

Generalized Ogus' equivalence to a broader setting.

Abstract

In this paper, we consider the (crystalline) prismatic crystals on a scheme $\mathfrak{X}$. We classify the crystals by $p$-connections on a certain ring and prove a cohomological comparison theorem. This equivalence is more general than the Ogus' equivalence. Moreover, we will prove the equivalence between the crystalline crystals and prismatic crystals by gluing local Ogus-Vologodsky correspondence.