On categories of (phi, Gamma)-modules

TL;DR

This paper explores various natural descriptions of the category of (phi, Gamma)-modules over p-adic fields, connecting classical and modern frameworks, and clarifies how to translate key constructions across these contexts.

Contribution

It provides multiple descriptions of (phi, Gamma)-modules and demonstrates how to adapt core constructions from power series rings to perfectoid algebras.

Findings

Multiple natural descriptions of (phi, Gamma)-modules are presented.

Connections between classical and modern frameworks are established.

Guidance on translating constructions to perfectoid contexts is provided.

Abstract

Let K be a complete discretely valued field of mixed characteristics (0, p) with perfect residue field. One of the central objects of study in p-adic Hodge theory is the category of continuous representations of the absolute Galois group of K on finite-dimensional Qp-vector spaces. In recent years, it has become clear that this category can be studied more effectively by embedding it into the larger category of (phi, Gamma)-modules; this larger category plays a role analogous to that played by the category of vector bundles on a compact Riemann surface in the Narasimhan-Seshadri theorem on unitary representations of the fundamental group of said surface. This category turns out to have a number of distinct natural descriptions, which on one hand suggests the naturality of the construction, but on the other hand forces one to use different descriptions for different applications. We provide several of these descriptions and indicate how to translate certain key constructions, which were originally given in the context of modules over power series rings, to the more modern context of perfectoid algebras and spaces.