$(\varphi,\Gamma)$-modules over relatively discrete algebras

TL;DR

This paper advances the theory of $(\varphi,\Gamma)$-modules over complex rings combining discrete and affinoid algebras, establishing foundational results crucial for $p$-adic Langlands correspondence and representation classification.

Contribution

It introduces new results on the structure, functoriality, and classification of $(\varphi,\Gamma)$-modules over these rings, extending the theory in a novel algebraic setting.

Findings

Existence of a fully faithful functor from Galois representations to $(\varphi,\Gamma)$-modules.

Deperfection of $(\varphi,\Gamma)$-modules over perfect period rings.

Classification of rank 1 $(\varphi,\Gamma)$-modules.

Abstract

In this paper, we study $(\varphi,\Gamma)$-modules over rings which are "combinations of discrete algebras and affinoid $\mathbb{Q}_p$-algebras", and prove basic results such as the existence of a fully faithful functor from the category of Galois representations, the deperfection of $(\varphi,\Gamma)$-modules over perfect period rings, and the dualizability of the cohomology of $(\varphi,\Gamma)$-modules, and the classification of $(\varphi,\Gamma)$-modules of rank $1$. This work is motivated by the categorical $p$-adic Langlands correspondence for locally analytic representations, as proposed by Emerton-Gee-Hellmann, and the $GL_1$ case, as formulated and proved by Rodrigues Jacinto-Rodr\'iguez Camargo.