(\phi,\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 \'etale $(\varphi,\Gamma)$-modules over certain completions of these rings are equivalent to the category of \'etale $(\varphi,\Gamma)$-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 \'etaleness is not necessary, so there exists a notion of trianguline objects in this sense.