(φ,Γ)-modules over relatively discrete algebras

2024/09/21 by Yutaro Mikami, Mikami, Yutaro
Computer Science · Mathematics · #Advanced Algebra and Logic #Advanced Topics in Algebra #Algebraic Geometry (math.AG) #FOS: Mathematics #Number Theory (math.NT) #Rings, Modules, and Algebras

paper · pdf · doi:10.48550/arxiv.2409.14145

openalex publication_date 2024/09/21 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/01

Abstract

In this paper, we study (φ,Γ)-modules over rings which are "combinations of discrete algebras and affinoid ℚ_p-algebras", and prove basic results such as the existence of a fully faithful functor from the category of Galois representations, the deperfection of (φ,Γ)-modules over perfect period rings, and the dualizability of the cohomology of (φ,Γ)-modules, and the classification of (φ,Γ)-modules of rank 1. This work is motivated by the categorical p-adic Langlands correspondence for locally analytic representations, as proposed by Emerton-Gee-Hellmann, and the GL₁ case, as formulated and proved by Rodrigues Jacinto-Rodríguez Camargo.

Related

• Finiteness and duality of cohomology of (φ,Γ)-modules and the 6-functor formalism of locally analytic representations
• Arithmetic families of (φ, Γ)-modules and locally analytic representations of GL₂(Q_p)
• Geometric translations of (φ,Γ)-modules for GL₂(ℚ_p)
• Families of Galois representations and (φ, τ)-modules
• Cohomology of (φ, τ)-modules