Prismatic $F$-gauges and a result of T. Liu

We give a new proof of a recent result of Tong Liu, which gives a general control on the torsion in the graded pieces of the so-called integral Hodge filtration associated to a crystalline Galois lattice. Our approach is stack-theoretic, and is inspired on the one hand by a result of Gee–Kisin on the shape of mod $p$ crystalline Breuil–Kisin modules, and on the other hand by the structures seen on the diffracted Hodge complex studied by Bhatt–Lurie. Along the way, we also obtain an explicit description of the Hodge–Tate locus in the Nygaard stack ${\mathcal{O}}_K^{\mathcal{N}}$ for a general extension $K\mathrm{/}{\mathbf{Q}}_p$.