Descent of $(\varphi,\tau)$-modules in characteristic $p$

TL;DR

This paper investigates the descent properties of $(, au)$-modules over perfectoid rings in characteristic $p$, extending existing theories and clarifying their relationship with $(, abla)$-modules without Galois representations.

Contribution

It generalizes Berger and Rozensztajn's work on $(, abla)$-modules to $(, au)$-modules and answers a key question about their connection to $(, abla)$-modules.

Findings

Established descent criteria for $(, au)$-modules

Extended super-Hölder vector theory to new module classes

Clarified the link between $(, au)$- and $(, abla)$-modules

Abstract

In this article, we study the descent of $(\varphi,\tau)$-modules over perfectoid period rings in characteristic $p$ via Berger and Rozensztajn's theory of super-H\"{o}lder vectors. This is a generalization of their work on $(\varphi,\Gamma)$-modules. As an application, we answer a question of Caruso regarding the connection between $(\varphi,\tau)$-modules and $(\varphi,\Gamma)$-modules without involving Galois representations as intermediaries.