Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

The overconvergence of multivariable $(\varphi_q,\mathcal{O}_K^{\times})$-modules at the perfectoid level

Changjiang Du

Abstract

Let $K$ be a finite unramified extension of $\mathbb{Q}_p$, and $E$ a finite extension of $K$ with ring of integers $\mathcal{O}_E$. We define the overconvergence of multivariable $(\varphi_q,\mathcal{O}_K^{\times})$-modules over $A_{\mathrm{mv},E}$ and explore some basic properties. We prove the overconvergence at the perfectoid level using the geometry of relative Fargues-Fontaine curve.

The overconvergence of multivariable $(\varphi_q,\mathcal{O}_K^{\times})$-modules at the perfectoid level

Abstract

Let be a finite unramified extension of , and a finite extension of with ring of integers . We define the overconvergence of multivariable -modules over and explore some basic properties. We prove the overconvergence at the perfectoid level using the geometry of relative Fargues-Fontaine curve.
Paper Structure (16 sections, 41 theorems, 173 equations)

This paper contains 16 sections, 41 theorems, 173 equations.

Table of Contents

  1. Introduction
  2. Overconvergent $(\varphi_q,\mathcal{O}_K^{\times})$-modules over $A_{\mathrm{mv},E}$
  3. The coefficient ring $A_{\mathrm{mv},E}^{\dagger}$
  4. The endomorphism $\varphi_q$ and the $\mathcal{O}_K^{\times}$-action
  5. The structure of étale $\varphi_q$-modules over $A_{\mathrm{mv},E}^{\dagger}$
  6. Overconvergent multivariable $(\varphi_q,\mathcal{O}_K^{\times})$-modules
  7. Overconvergence at the perfectoid level
  8. A reminder on the relative Fargues-Fontaine curve
  9. Overconvergent perfectoid $(\varphi_q,\mathcal{O}_K^{\times})$-modules
  10. Lubin-Tate case
  11. Multivariable case
  12. The relation between $\widetilde{\mathcal{R}}^{\mathrm{int}}_{A_{\infty}}$ and $A_{\mathrm{mv},E}^{\dagger}$
  13. Some technical lemmas
  14. Finiteness of $A^{\circ}$-submodules of étale $(\varphi_q,\mathcal{O}_K^{\times})$-modules over $A$
  15. Étaleness of submodules of étale $(\varphi_q,\mathcal{O}_K^{\times})$-modules over $A$
  16. ...and 1 more sections

Key Result

Theorem 1.2

The base change functor $j^{\dagger,*}$ is exact and admits a right adjoint functor $j_{*}^{\dagger}$ such that the natural transformation $\mathrm{id}\to j_{*}^{\dagger}\circ j^{\dagger,*}$ is an isomorphism. In particular, $j^{\dagger,*}$ is fully faithful.

Theorems & Definitions (87)

  • Definition 1.1
  • Theorem 1.2: Theorem \ref{['general facts']}
  • Proposition 1.3: Corollary \ref{['fin proj over int Robba is free']}
  • Proposition 1.4: fargues2024geometrizationlocallanglandscorrespondence, WS2020berkeley
  • Theorem 1.5: Theorem \ref{['overconvergence of mv module at the perfectoid level']}, Proposition \ref{['for r large is free']}
  • Definition 2.1
  • Lemma 2.2
  • Definition 2.3
  • Lemma 2.4
  • Lemma 2.5
  • ...and 77 more