Comparisons of Lie algebra cohomologies of $(\phi ,\Gamma )$-modules

Abstract

We generalise a result of Fourquaux and Xie thereby completely determining the relationship between ${\mathbb{Q}}_p$ and $L$-analytic Lie algebra cohomology of analytic $({\phi }_L,{\Gamma }_L)$-modules. We use the results to conclude that for $L\ne {\mathbb{Q}}_p,$ there exist examples of étale $({\phi }_L,{\Gamma }_L)$-modules over Robba rings whose ${\mathbb{Q}}_p$-analytic cohomology does not arise as a base change of Galois cohomology.

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