Perfectoid Spaces in Multivariate $p$-adic Hodge Theory

TL;DR

This paper advances the understanding of perfectoid spaces in multivariate $p$-adic Hodge theory, highlighting their role in connecting characteristic $0$ and $p$ geometries.

Contribution

It introduces a systematic framework for analyzing perfectoid spaces within multivariate $p$-adic Hodge theory over specialized rings.

Findings

Develops a new approach to study perfectoid spaces in multivariate settings

Establishes connections between characteristic $0$ and $p$ geometries

Provides tools for future research in $p$-adic Hodge theory

Abstract

Perfectoid spaces have become a crucial tool in $p$-adic geometry, serving as a bridge between adic spaces in characteristic $0$ and those in characteristic $p$. In this article, we develop a systematic way to study the structure of perfectoid spaces within the setting of multivariate $p$-adic Hodge theory over a variant of the rings introduced in \cite{Bri}.