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 $\mathcal{M}{\mathcal{F}}_{[0,p-2]}^{tor-free}(\mathcal{X})$ is equivalent to the category of crystalline ${\mathbb{Z}}_p$-local systems on $\mathcal{X}$ with Hodge-Tate weights in $\{0,\dots ,p-2\}$, which generalizes the crystalline part of a theorem of Breuil-Liu to higher dimensions.

41 pages, comments welcome!