Prismatic Crystals for schemes in characteristic $p$

Abstract

Let $(A,\delta_A)$ be a crystalline prism and let $\mathfrak X$ be a finite type $A/p$-scheme admitting a Koszul-regular closed immersion into a smooth formal $A$-scheme $Y$. We construct a sheaf of prismatic envelopes $\Delta_Y(\mathfrak X)$ attached to a Frobenius lift modulo $p^2$ on $Y$, prove that prismatic crystals on $(\mathfrak X/A)_{\Delta}$ are equivalent to integrable topologically quasi-nilpotent $p$-connections on $\Prism_Y(\mathfrak X)$, and identify their prismatic cohomology with the corresponding de Rham complex. When a global Frobenius lift is available, a lifted Ogus--Vologodsky functor gives an equivalence between $p$-connections on the prismatic envelope of the Frobenius twist and connections on the $p$-complete PD-envelope. Gluing this local correspondence yields an equivalence between prismatic crystals on $\mathfrak X^{(1)}$ and crystalline crystals on $\mathfrak X$ for l.c.i. $\mathfrak X$ over $A/p$.