Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Galois representations, $(\varphi, Γ)$-modules and prismatic F-crystals

Zhiyou Wu

Abstract

We prove that both local Galois representations and $(\varphi,Γ)$-modules can be recovered from prismatic F-crystals, from which we obtain a new proof of the equivalence of Galois representations and $(\varphi,Γ)$-modules.

Galois representations, $(\varphi, Γ)$-modules and prismatic F-crystals

Abstract

We prove that both local Galois representations and -modules can be recovered from prismatic F-crystals, from which we obtain a new proof of the equivalence of Galois representations and -modules.

Paper Structure

This paper contains 7 sections, 31 theorems, 154 equations.

Table of Contents

  1. Introduction
  2. Prisms
  3. Definitions and Examples
  4. Perfect prisms and perfectoid rings
  5. Prismatic F-crystals
  6. étale $\varphi$-modules
  7. $(\varphi,\Gamma)$-modules and prismatic $F$-crystals

Key Result

Theorem 1.1

Let $k$ be a perfect field of characteristic $p$, and $K$ be a finite extension of $W(k)[\frac{1}{p}]$. Then $(\varphi,\Gamma)$-modules over $\bold{A}_K$ are equivalent to prismatic F-crystals in $\mathcal{O}_{{\mathlarger{\mathbbl{\Delta}}}}[\frac{1}{I_{{\mathlarger{\mathbbl{\Delta}}}}}]^{\wedge}_p

Theorems & Definitions (74)

  • Theorem 1.1
  • Corollary 1.2
  • Definition 2.1
  • Definition 2.2
  • Example 2.3
  • Example 2.4
  • Lemma 2.5
  • proof
  • Corollary 2.6
  • proof
  • ...and 64 more