We construct a category of Breuil–Kisin -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 -adic cohomology theories of Bhatt–Morrow–Scholze and Bhatt–Scholze. As a key ingredient, we classify Galois representations that are of finite -height.
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Hui Gao, Breuil–Kisin modules and integral -adic Hodge theory (with Appendix A by Yoshiyasu Ozeki, and Appendix B by Hui Gao and Tong Liu). J. Eur. Math. Soc. 25 (2023), no. 10, pp. 3979–4032DOI 10.4171/JEMS/1278