Divided prismatic Frobenius crystals of small height and the category $\mathcal{M}\mathcal{F}$

Abstract: Let $\mathcal{X}$ be a smooth $p$-adic formal scheme over a mixed characteristic complete discrete valuation ring $\mathcal{O}{K}$ with perfect residue field. We introduce a general category $\mathcal{M}\mathcal{F}{[0, p-2]}^{tor-free}(\mathcal{X})$ of $p$-torsion free crystalline coefficient objects and show that this category is equivalent to the category of completed prismatic Frobenius crystals of height $p-2$, recently introduced by Du-Liu-Moon-Shimizu. In particular this shows that the category $\mathcal{M}\mathcal{F}^{tor-free}_{[0, p-2]}(\mathcal{X})$ is equivalent to the category of crystalline $\mathbb{Z}_p$-local systems on $\mathcal{X}$ with Hodge-Tate weights in ${0,\ldots , p-2}$, which generalizes the crystalline part of a theorem of Breuil-Liu to higher dimensions.