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

Nataniel Marquis

TL;DR

This work develops a unifying categorical framework for Fontaine-type equivalences by organizing semilinear actions of topological monoids on rings into the category $\mathrm{Mod}(\mathcal{S},R)$ and its étale and étale-projective subcategories. It introduces systematic operations—extension of scalars, invariants under a normal submonoid, and coinduction—and provides an automated dévissage theory for modules carrying $r$-adic structures, including a detailed structure theorem describing decompositions into free and torsion pieces. The paper further incorporates topology, defining initial topologies and topological variants of the module categories and proving stability and monoidal properties under Fontaine-type functors in this setting. These foundations are then specialized to recover Fontaine’s equivalence for $\mathbb{Q}_p$, via explicit rings $\mathcal{O}_{\mathcal{E}}$, $\mathcal{O}_{\widehat{\mathcal{E}^{\mathrm{ur}}}}$, and associated Galois actions, and to extend the framework toward Lubin–Tate and multivariable generalizations. Overall, it provides a flexible, automatable toolkit for translating Galois representations into $\varphi$-$\Gamma$-module data across broad coefficient rings and topologies.

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 étale 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 $(r,μ)$-dévissage and of topological étale $\mathcal{S}$-modules over $R$.

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

TL;DR

This work develops a unifying categorical framework for Fontaine-type equivalences by organizing semilinear actions of topological monoids on rings into the category and its étale and étale-projective subcategories. It introduces systematic operations—extension of scalars, invariants under a normal submonoid, and coinduction—and provides an automated dévissage theory for modules carrying -adic structures, including a detailed structure theorem describing decompositions into free and torsion pieces. The paper further incorporates topology, defining initial topologies and topological variants of the module categories and proving stability and monoidal properties under Fontaine-type functors in this setting. These foundations are then specialized to recover Fontaine’s equivalence for , via explicit rings , , and associated Galois actions, and to extend the framework toward Lubin–Tate and multivariable generalizations. Overall, it provides a flexible, automatable toolkit for translating Galois representations into --module data across broad coefficient rings and topologies.

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 étale projective -modules over 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 and coinduction to a bigger monoid. We define and study categories corresponding to finite type continuous representations over through the notions of finite projective -dévissage and of topological étale -modules over .

Paper Structure

This paper contains 15 sections, 52 theorems, 143 equations.

Table of Contents

  1. Introduction
  2. The category of $\mathcal{S}$-modules over $R$ and some subcategories
  3. The $\mathcal{S}$-modules over $R$
  4. Étale and étale projective modules
  5. Closed symmetric monoidal structure
  6. Operations on $\mathcal{S}$-modules over $R$
  7. Extension of scalars
  8. Invariants by a normal submonoid
  9. Coinduction to a bigger monoid
  10. The category of $\mathcal{S}$-modules over $R$ with projective $r$-dévissage
  11. Adding topology
  12. Initial topology and associated subcategories
  13. Preservation by operations
  14. Fontaine's equivalence for $\mathbb{Q}_p$ revisited
  15. Study of modules with finite projective dévissage

Key Result

Proposition 1.5

[See Proposition ex_cdetaledvproj] Let $\mathcal{S}$ be a topological monoid, let $A$ and $R$ be topological rings equipped with continuous $\mathcal{S}$-actions and let $f\, : \, A\rightarrow R$ be an $\mathcal{S}$-equivariant continuous ring morphism. Let $a\in A$ be such that: Then, the functor sends $\mathscr{M}\mathrm{od}_{a \mathrm{\text{-}prjdv}}^{\mathrm{\acute{e}t}}(\mathcal{S},A)$ to $

Theorems & Definitions (146)

  • Definition 1.1
  • Definition 1.2
  • Definition 1.3
  • Definition 1.4
  • Proposition 1.5
  • Theorem 1.6
  • Theorem 1.7
  • Definition 2.1
  • Example 2.2
  • Definition 2.3
  • ...and 136 more