$(\varphi,\Gamma)$-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 $\left(\phi ,\Gamma \right)$-modules over certain completions of these rings is equivalent to the category of étale $\left(\phi ,\Gamma \right)$-modules over classical overconvergent or Robba rings as the case may be (hence also to the category of $p$-adic Galois representations of ${\mathbb{Q}}_p$). In the case of Robba rings, the assumption of étaleness is not necessary, so there exists a notion of trianguline objects in this sense.