(\phi,\Gamma)-modules over noncommutative overconvergent and Robba rings

TL;DR

This paper develops noncommutative multidimensional overconvergent and Robba rings, establishing equivalences with classical categories of $(\

Contribution

It introduces noncommutative multidimensional overconvergent and Robba rings and proves categorical equivalences with classical $(\phi,\Gamma)$-modules, extending to non-étale cases.

Findings

Categories of $(\phi,\Gamma)$-modules over these rings are equivalent to classical categories.

The equivalence holds for both overconvergent and Robba rings, including non-étale Robba rings.

Existence of a notion of trianguline objects in the noncommutative Robba ring setting.

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.