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

TL;DR

This paper introduces a new prismatic framework for $(, g)$-modules and $F$-crystals, providing simplified proofs of their equivalences with crystalline and semi-stable Galois representations, extending existing theories with new constructions.

Contribution

It offers a novel prismatic construction of $(, g)$-modules and $F$-crystals, and generalizes the equivalence to semi-stable representations using logarithmic prisms.

Findings

New prismatic construction of $(, g)$-modules

Simplified proof of equivalence with crystalline representations

Extension to semi-stable representations using logarithmic prisms

Abstract

We give a new construction of $(\varphi, \hat G)$-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.