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

Abstract

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 $K/{\mathbf{Q}}_p$.

v2: improved exposition; include description of the Hodge--Tate locus for a general extension; details added on various identifications with the stacky picture