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

Abstract

In Sen’s theory in the imperfect residue field case, Brinon defined a functor from the category of \({{\mathbb C}_p}\)-representations to the category of linear representations of a certain Lie algebra. We give a comparison theorem between the continuous Galois cohomology of \({{\mathbb C}_p}\)-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.