(ϕ,Γ)-modules over noncommutative overconvergent and Robba rings

Abstract

We construct noncommutative multidimensional versions of overconvergent power series rings and Robba rings. We show that the category of étale $(\phi ,\Gamma )$-modules over certain completions of these rings are equivalent to the category of étale $(\phi ,\Gamma )$-modules over the corresponding classical overconvergent, resp. Robba rings (hence also to the category of $p$-adic Galois representations of ${\mathbb{Q}}_p$). Moreover, in the case of Robba rings, the assumption of étaleness is not necessary, so there exists a notion of trianguline objects in this sense.

41 pages, revised