[Paper Review] Breuil-Kisin modules and integral $p$-adic Hodge theory

Hui Gao|arXiv (Cornell University)|May 21, 2019

Algebraic Geometry and Number Theory19 references4 citations

TL;DR

This paper constructs a category of Breuil-Kisin $G_K$-modules to classify integral semi-stable Galois representations, using Breuil-Kisin and Breuil-Kisin-Fargues modules with Galois actions. It resolves a foundational gap in prior work by replacing incomplete $p$-adic topologies with $p$-adically complete rings, ensuring $G_K$-stability and establishing a complete algebraic framework for integral $p$-adic Hodge theory.

ABSTRACT

We construct a category of Breuil-Kisin $G_K$-modules to classify integral semi-stable Galois representations. Our theory uses Breuil-Kisin modules and Breuil-Kisin-Fargues modules with Galois actions, and can be regarded as the algebraic avatar of the integral $p$-adic cohomology theories of Bhatt-Morrow-Scholze and Bhatt-Scholze. As a key ingredient, we classify Galois representations that are of finite $E(u)$-height.

Motivation & Objective

• To construct a category of Breuil-Kisin $G_K$-modules that algebraically classify integral semi-stable Galois representations.
• To provide an algebraic avatar of the integral $p$-adic cohomology theories of Bhatt-Morrow-Scholze and Bhatt-Scholze.
• To resolve a foundational gap in earlier works (e.g., [Liu07], [Gao18]) concerning the $p$-adic completeness of key rings like $\widehat{\mathcal{R}}$.
• To classify Galois representations of finite $E(u)$-height using the new framework.
• To establish a fully integral, $p$-adically complete version of $(\varphi,\tau)$-module theory for semi-stable representations.

Proposed method

• Constructs a category of finite free Breuil-Kisin $G_K$-modules using $\mathfrak{S} = W(k)[[u]]$ and Galois actions on $W(R)$, where $R = \mathcal{O}_{C_p}^\flat$.
• Uses the Teichmüller lift $[\underline{\pi}] \in W(R)$ to embed $\mathfrak{S}$ into $W(R)$, ensuring compatibility with Frobenius and Galois actions.
• Applies a Weierstrass preparation-style argument to show that inverse limits of torsion models $\mathfrak{M}_n'$ are finite free $\mathfrak{S}$-modules of finite $E(u)$-height.
• Replaces the use of $\widehat{\mathcal{R}}$ (which lacks known $p$-adic completeness) with $W(R)$, which is $p$-adically complete, to ensure $G_K$-stability of the module.
• Establishes that $\widetilde{\mathfrak{M}} \otimes_{\varphi,\mathfrak{S}} W(R)$ is $G_K$-stable by leveraging the $p$-adic completeness of $W(R)$ and $W(k)$.
• Uses the fact that $\mathfrak{M}_n' / u\mathfrak{M}_n'$ is $G_K$-fixed to show $\widetilde{\mathfrak{M}} / u\widetilde{\mathfrak{M}}$ is also $G_K$-fixed, via $p$-adic completeness.

Experimental results

Research questions

• RQ1 Can a complete algebraic classification of integral semi-stable Galois representations be achieved via a $p$-adically complete module category?
• RQ2 What is the correct replacement for the incomplete $\widehat{\mathcal{R}}$ in the construction of $(\varphi,\hat{G})$-modules to ensure $G_K$-stability?
• RQ3 How can the gap in [Liu07] and [Gao18]—concerning the $p$-adic completeness of $\mathcal{R}_{K_0} \cap \mathbf{A}_{\mathrm{cris}}$—be resolved?
• RQ4 Can the theory of finite $E(u)$-height representations be fully classified using Breuil-Kisin $G_K$-modules?
• RQ5 To what extent can the integral $p$-adic cohomology theories of Bhatt-Morrow-Scholze and Bhatt-Scholze be realized algebraically through such modules?

Key findings

• The category of finite free Breuil-Kisin $G_K$-modules provides a complete classification of integral semi-stable Galois representations.
• The use of $W(R)$ instead of $\widehat{\mathcal{R}}$ resolves the $p$-adic completeness issue that invalidated earlier proofs involving $\hat{G}$-stability.
• The inverse limit $\widetilde{\mathfrak{M}} = \varprojlim \mathfrak{M}_n'$ is a finite free $\mathfrak{S}$-module of finite $E(u)$-height, ensuring integrality and finiteness.
• The module $\widetilde{\mathfrak{M}} \otimes_{\varphi,\mathfrak{S}} W(R)$ is $G_K$-stable, proving that the associated Galois representation is semi-stable.
• The $G_K$-action on $\widetilde{\mathfrak{M}} / u\widetilde{\mathfrak{M}}$ is trivial, confirming that the representation is unramified modulo $u$.
• The framework resolves the gap in [Liu07, Prop. 6.1.1] and [Gao18, Thm. 3.1] by ensuring all rings and ideals involved are $p$-adically complete.