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

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.