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. We give two applications of our results. Firstly, we provide a new proof for the equivalence between the category of prismatic $F$-crystals in finite locally free ${\mathcal{O}}_{△△}$-modules over $({\mathcal{O}}_K{)}_{△△}$ 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. Moreover, we generalize this result to semistable representations using the absolute logarithmic prismatic site defined by Koshikawa.