Crystalline representations and Wach modules in the imperfect residue field case

TL;DR

This paper extends the theory of Wach modules to the setting of imperfect residue fields in unramified extensions of ululp, establishing an equivalence with crystalline Galois representations and linking Wach modules to filtered i-modules.

Contribution

It introduces and studies Wach modules for imperfect residue fields, establishing an equivalence with crystalline Galois representations and relating Wach modules to filtered i-modules.

Findings

Equivalence between lattices in crystalline representations and integral Wach modules.

Relation between rational Wach modules with Nygaard filtration and filtered i-modules.

Extension of Wach module theory to imperfect residue fields.

Abstract

For an absolutely unramified field extension $L/\mathbb{Q}_p$ with imperfect residue field, we define and study Wach modules in the setting of $(\varphi,\Gamma)$-modules for $L$. Our main result establishes a direct equivalence between the category of lattices inside crystalline representations of the absolute Galois group of $L$ and the category of integral Wach modules for $L$. Moreover, we provide a direct relation between a rational Wach module equipped with the Nygaard filtration and the filtered $\varphi$-module of its associated crystalline representation.