A prismatic approach to $(\phi ,\hat{G})$-modules and $F$-crystals

Abstract

We give a new construction of $(\phi ,\hat{G})$-modules using the theory of prisms developed by Bhatt and Scholze. As an application, we give a new proof about the equivalence between the category of prismatic $F$-crystals in finite locally free ${\mathcal{O}}_{\Delta }$-modules over $({\mathcal{O}}_K{)}_{\Delta }$ and the category of lattices in crystalline representations of $G_K$, where $K$ is a complete discretely valued field of mixed characteristic with perfect residue field. We also generalize this result to semi-stable representations using the absolute logarithmic prismatic site defined by Koshikawa.

arXiv commentSubmitted version, 46 pages

References in corpus (8)

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• Relative p-adic Hodge theory, II: Imperfect period rings
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• Logarithmic Prismatic Cohomology I
• Comparison of prismatic cohomology and derived de Rham cohomology
• Breuil-Kisin modules and integral $p$-adic Hodge theory
• Galois representations, $(\phi ,\Gamma )$-modules and prismatic F-crystals
• Lattices in crystalline representations and Kisin modules associated with iterate extensions