Study of various categories gravitating around $(\varphi,\Gamma)$-modules

TL;DR

This paper develops a general framework for categories related to $(, )$-modules, exploring their properties under base change, invariants, and coinduction, with applications to Fontaine equivalences and $p$-adic representations.

Contribution

It introduces a unified categorical framework for $(, )$-modules and studies their behavior under various functors, extending the understanding of Fontaine equivalences and $p$-adic representations.

Findings

Categories of étale projective modules are preserved under base change.

Invariants under a normal submonoid are well-behaved within these categories.

Finite type continuous representations are characterized via $(r, u)$-dévissage and topological étale modules.

Abstract

Functors involved in Fontaine equivalences decompose as extension of scalars and taking of invariants between full subcategories of modules over a topological ring equipped with semi-linear continuous action of a topological monoid. We give a general framework for these categories and the functors between them. We define the categories of \'etale projective $\mathcal{S}$-modules over $R$ to englobe categories that will correspond by Fontaine-type equivalences to finite free representations of a group. We study their preservation by base change, taking of invariants by a normal submonoid of $\mathcal{S}$ and coinduction to a bigger monoid. We define and study categories corresponding to finite type continuous representations over $\mathbb{Z}_p$ through the notions of finite projective $(r,\mu)$-d\'evissage and of topological \'etale $\mathcal{S}$-modules over $R$.