A note on Sen's theory in the imperfect residue field case

TL;DR

This paper extends Sen's theory to imperfect residue fields by comparing Galois cohomology with Lie algebra cohomology, utilizing Hyodo's calculations and properties of solvable Lie algebras.

Contribution

It provides a comparison theorem linking Galois and Lie algebra cohomology in the imperfect residue field setting, enhancing understanding of Sen's theory.

Findings

Established a comparison theorem between Galois and Lie algebra cohomology.

Utilized Hyodo's Galois cohomology calculations.

Proved effaceability of Lie algebra cohomology for solvable Lie algebras.

Abstract

In Sen's theory in the imperfect residue field case, Brinon defined a functor from the category of C_p-representations to the category of linear representations of certain Lie algebra. We give a comparison theorem between the continuous Galois cohomology of C_p-representations and the Lie algebra cohomology of the associated representations. The key ingredients of the proof are Hyodo's calculation of Galois cohomology and the effaceability of Lie algebra cohomology for solvable Lie algebras.