Prismatic F-Crystals and Wach Modules

by Abhinandan (IMJ-PRG, Sorbonne Université)

Thursday Oct 24, 2024, 11:00 AM → 12:30 PM Europe/Paris

Amphithéâtre Léon Motchane (IHES)

Séminaire de géométrie arithmétique

For an absolutely unramified extension $K/\mathbb{Q}_p$ with perfect residue field, by the works of Fontaine, Colmez, Wach and Berger, it is well known that the category of Wach modules over a certain integral period ring is equivalent to the category of lattices inside crystalline representations of $G_K$ (the absolute Galois group of $K$). Moreover, by the recent works of Bhatt and Scholze, we also know that lattices inside crystalline representations of $G_K$ are equivalent to the category of prismatic $F$-crystals on the absolute prismatic site of $O_K$, the ring of integers of $K$. The goal of this talk is to present a direct construction of the categorical equivalence between Wach modules and prismatic $F$-crystals over the absolute prismatic site of $O_K$. If time permits, we will also mention a natural generalisation of these results to the case of a "small" base ring and intended application (work in progress).

 

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