Prismatic $F$-gauges and a result of T. Liu
Dat Pham
TL;DR
The paper provides a new stack-theoretic proof of Tong Liu's result on torsion control in the integral Hodge filtration attached to crystalline Galois lattices. It introduces a Sen operator $ heta$ on the HT quotient $ rak M_{HT}$ constructed via a differential operator $D$ and a prismatized framework, embedding the problem in the Nygaard/Hodge stacky picture and connecting with the diffracted Hodge complex. A key advancement is the explicit description of the Hodge--Tate locus for general $K/f Q_p$ and the identification of Nygaard and Hodge filtrations within this stacky setup, yielding torsion-vanishing conclusions away from a specific arithmetic set $I=igl brace r+mpigr brace$. The approach unifies ideas from Gee--Kisin, Gao--Liu, and Bhatt--Lurie, showing how prismatic $F$-gauges and the stacky prismatization framework illuminate classical $p$-adic Hodge structures. In particular, it recovers torsion-free behavior for effective Breuil--Kisin lattices when the index $i<p$, and strengthens the conceptual bridge between the classical theory and the modern diffracted/Hodge-theoretic perspective.
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 $K/\mathbf{Q}_p$.