Arithmetic {B}reuil-{K}isin-{F}argues modules and comparison of integral {p}-adic Hodge theories

TL;DR

This paper explores the relationship between various $p$-adic Hodge theories by analyzing arithmetic Breuil-Kisin-Fargues modules and their connection to filtered $(, N, G_K)$-modules over a discretely valued field with perfect residue field.

Contribution

It introduces the concept of arithmetic BKF modules, studies their properties, and compares different $p$-adic Hodge theories at the ${A}_{ ext{inf}}$ level using a functorial approach.

Findings

Established a functor from filtered $(, N, G_K)$-modules to Breuil-Kisin-Fargues modules.

Proved a rigidity result for arithmetic BKF modules.

Compared existing $p$-adic Hodge theories at the ${A}_{ ext{inf}}$ level.

Abstract

Let $K$ be a discrete valuation field with perfect residue field, we study the functor from weakly admissible filtered $(\varphi,N,G_K)$-modules over $K$ to the isogeny category of Breuil-Kisin-Fargues $G_K$-modules. This functor is the composition of a functor defined by Fargues-Fontaine from weakly admissible filtered $(\varphi,N,G_K)$-modules to $G_K$-equivariant modifications of vector bundles over the Fargues-Fontaine curve $X_{FF}$, with the functor of Fargues-Scholze that between the category of admissible modifications of vector bundles over $X_{FF}$ and the isogeny category of Breuil-Kisin-Fargues modules. We study those objects appear in the essential image of the above functor and call them arithmetic BKF modules. We show certain rigidity result of arithmetic BKF modules and use it to compare existing $p$-adic Hodge theories at ${A}_{\mathrm{inf}}$ level.