An integral analogue of Fontaine's crystalline functor

TL;DR

This work constructs an integral crystalline realization functor $ ext{D}_{ ext{crys}}$ from prismatic $F$-crystals to filtered $F$-crystals on a smooth formal scheme over $W$, and proves it yields strong divisibility in the locally filtered free (lff) case. It establishes a precise equivalence between prismatic $F$-gauges with $H$odge–Tate weights in $[0,p-2]$ and Fontaine–Laffaille modules, thereby linking the prismatic and Fontaine–Laffaille frameworks and clarifying cohomological consequences for $p$-divisible groups. The paper also develops a robust interrelation between prismatic, crystalline, and Dieudonné theories (prismatic Dieudonné, Grothendieck–Messing crystalline Dieudonné, and BK–Kim variants), showing how integral $D_{ ext{crys}}$--type lattices recover and unify classical Dieudonné theories and their prismatic avatars. Technically, the authors deploy crystalline–de Rham comparisons, Nygaard filtrations, and Tsuji’s framework to obtain exactness and stacky interpretations, enabling cohomological identifications that underpin Shimura variety applications and integral $p$-adic Hodge theory. The results provide a comprehensive lattice-theoretic bridge across several contemporary approaches to integral $p$-adic Hodge structures and Dieudonné theory.

Abstract

For a smooth formal scheme $\mathfrak{X}$ over the Witt vectors $W$ of a perfect field $k$, we construct a functor $\mathbb{D}_\mathrm{crys}$ from the category of prismatic $F$-crystals $(\mathcal{E},\varphi_\mathcal{E})$ (or prismatic $F$-gauges) on $\mathfrak{X}$ to the category of filtered $F$-crystals on $\mathfrak{X}$. We show that $\mathbb{D}_\mathrm{crys}(\mathcal{E},\varphi_\mathcal{E})$ enjoys strong properties (e.g., strong divisibility in the sense of Faltings) when $(\mathcal{E},\varphi_\mathcal{E})$ is what we call locally filtered free (lff). Most significantly, we show that $\mathbb{D}_\mathrm{crys}$ actually induces an equivalence between the category of prismatic $F$-gauges on $\mathfrak{X}$ with Hodge--Tate weights in $[0,p-2]$ and the category of Fontaine--Laffaille modules on $\mathfrak{X}$. Finally, we use our functor $\mathbb{D}_\mathrm{crys}$ to enhance the study of prismatic Dieduonné theory of $p$-divisible groups (as initiated by Anschütz--Le Bras) allowing one to recover the filtered crystalline Dieudonné crystal from the prismatic Dieudonné crystal. This in turn allows us to clarify the relationship between prismatic Dieudonné theory and the work of Kim on classifying $p$-divisible groups using Breuil--Kisin modules.

Theorems & Definitions (111)

• Theorem A: see Theorem \ref{['thm:big-equiv-diagram']}
• Remark 1
• Theorem B: see Theorem \ref{['thm:crys-dR-comparison']}
• Remark 2
• Remark 3
• Theorem C: see Theorem \ref{['thm:integral-dcrys-strongly-divisible-and-rational-agreeance']}
• Proposition A: see Proposition \ref{['prop:filtered-equiv']}
• Proposition B: see Proposition \ref{['prop:G_X-bi-exact']}
• Proposition C: see Proposition \ref{['prop: Gauge coh']}
• Theorem D: see Theorem \ref{['thm:ALB-dJ-comparison']}