Breuil-Kisin modules and integral $p$-adic Hodge theory
Hui Gao
TL;DR
This work constructs Breuil-Kisin G_K-modules as an algebraic avatar for integral p-adic cohomology theories and proves they classify Z_p-lattices in semi-stable G_K-representations, preserving integral data through finite E(u)-height constraints. Central to the approach are overconvergent φ,τ-modules, the monodromy operator N_∇ defined via locally analytic vectors, and a Frobenius-regularization mechanism that descends monodromy to the O-ring, yielding potential semi-stability. The main contributions include an exact equivalence between Breuil-Kisin G_K-modules and semistable lattices, a crystallinity criterion, and a robust bridge to Kisin’s O-modules and MF categories, strengthening integral p-adic Hodge theory and its connections to prismatic cohomology. The results provide an algebraic framework compatible with Bhatt–Morrow–Scholze’s integral cohomology theories, with applications to deformation theory and automorphy lifting, and they clarify the relationship with Liu’s φ,Ĝ- and Gee–Liu’s semi-stable formalisms while avoiding gaps in previous approaches.
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.