Breuil–Kisin modules and integral $p$-adic Hodge theory (with Appendix A by Yoshiyasu Ozeki, and Appendix B by Hui Gao and Tong Liu)

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.