Prismatic Crystals for schemes in characteristic $p$

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.