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

TL;DR

This paper offers a new stack-theoretic proof of a recent result by Tong Liu concerning torsion control in the integral Hodge filtration of crystalline Galois lattices, connecting advanced algebraic structures.

Contribution

It introduces a novel stack-theoretic approach to analyze the integral Hodge filtration, inspired by recent developments in crystalline Breuil--Kisin modules and the diffracted Hodge complex.

Findings

New proof of Liu's torsion control result

Explicit description of the Hodge--Tate locus in the Nygaard stack

Connections between crystalline Galois lattices and advanced algebraic structures

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 $K/\mathbf{Q}_p$.