Breuil-Kisin modules and integral $p$-adic Hodge theory

TL;DR

This paper develops a new algebraic framework using Breuil-Kisin modules to classify integral semi-stable Galois representations, connecting to modern $p$-adic cohomology theories.

Contribution

It introduces a category of Breuil-Kisin Galois modules that captures integral semi-stable representations, extending the algebraic understanding of $p$-adic Hodge theory.

Findings

Classification of Galois representations of finite $E(u)$-height.

Establishment of an algebraic model for integral $p$-adic cohomology.

Connection to recent advances in $p$-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.