An integral analogue of Fontaine's crystalline functor

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 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.