Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Multivariable $(\varphi_q,\mathcal{O}_K^{\times})$-modules associated to $p$-adic representations of $\mathrm{Gal}(\overline{K}/K)$

Changjiang Du

Abstract

Let $K$ be an unramified extension of $\mathbb{Q}_p$, and $E$ a finite extension of $K$ with ring of integers $\mathcal{O}_E$. We associate to every finite type continuous $\mathcal{O}_E$-representation $ρ$ of $\mathrm{Gal}(\overline{K}/K)$ an étale $(\varphi_q,\mathcal{O}_K^{\times})$-module $D_{A_{\mathrm{mv},E}}^{(0)}(ρ)$ over $A_{\mathrm{mv},E}$, where $A_{\mathrm{mv},E}$ is the $p$-adic completion of a completed localization of the Iwasawa algebra $\mathcal{O}_E[\negthinspace[\mathcal{O}_K]\negthinspace]$. Furthermore, we prove that the functor $D_{A_{\mathrm{mv},E}}^{(0)}$ is fully faithful and exact. This functor is a $p$-adic analogue of $D_A^{(0)}$ in the recent work of Breuil, Herzig, Hu, Morra and Schraen.

Multivariable $(\varphi_q,\mathcal{O}_K^{\times})$-modules associated to $p$-adic representations of $\mathrm{Gal}(\overline{K}/K)$

Abstract

Let be an unramified extension of , and a finite extension of with ring of integers . We associate to every finite type continuous -representation of an étale -module over , where is the -adic completion of a completed localization of the Iwasawa algebra . Furthermore, we prove that the functor is fully faithful and exact. This functor is a -adic analogue of in the recent work of Breuil, Herzig, Hu, Morra and Schraen.
Paper Structure (19 sections, 56 theorems, 245 equations)

This paper contains 19 sections, 56 theorems, 245 equations.

Table of Contents

  1. Introduction
  2. The multivariable topological ring $A_{\mathrm{mv},E}$
  3. Two $(\varphi_q,\mathcal{O}_K^{\times})$-equivariant injections from non-perfect coefficient rings to the ring of Witt vectors
  4. A $(\varphi_q,\mathcal{O}_K^{\times})$-equivariant injection from $A_{\mathrm{mv},E_0}$ to $W_E(A_{\infty})$
  5. A $(\varphi_q,\mathcal{O}_K^{\times})$ injection from $A_{\mathrm{LT},E,\sigma_{0}}$ to $W_E\left(\mathbb{F}((T_{\mathrm{LT}}^{1/p^{\infty}}))\right)$
  6. Frobenius Descent for finitely presented étale $(\varphi_q,\mathcal{O}_K^{\times})$-modules over $W_E(A_{\infty})$
  7. Finite projective étale $\varphi_q$-modules
  8. Finitely presented étale $(\varphi_q,\mathcal{O}_K^{\times})$-modules
  9. Descent from $W_E(A_{\infty}')$ to $W_E(A_{\infty})$
  10. A vanishing result on certain continuous cohomology groups
  11. Descent for finite projective modules
  12. Descent for finitely presented modules
  13. Review of classical $(\varphi,\Gamma)$-modules
  14. Lubin-Tate and cyclotomic $(\varphi,\Gamma)$-modules
  15. Comparison between $D_{\mathrm{LT},\sigma_{0}}(\rho)$ and $D_{\mathrm{cycl},\sigma_{0}}(\rho)$
  16. ...and 4 more sections

Key Result

Theorem 1.1

Theorems & Definitions (108)

  • Theorem 1.1
  • Theorem 1.2
  • Lemma 2.1
  • Proof 1
  • Definition 2.2
  • Lemma 2.3
  • Proof 2
  • Corollary 2.4
  • Proof 3
  • Lemma 2.5
  • ...and 98 more