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.
Categories: math.AG math.NT