Some ring-theoretic properties of rings via Frobenius and monoidal maps

TL;DR

This paper explores how tilting operations in perfectoid geometry relate ring-theoretic properties of rings in mixed and positive characteristics, using the structure of monoidal maps on p-adically complete rings.

Contribution

It introduces a novel approach to connect ring properties across characteristics via monoidal maps in perfectoid geometry.

Findings

Establishes relationships between ring properties through tilting.

Utilizes monoidal maps to analyze p-adically complete rings.

Provides a framework for understanding mixed characteristic rings.

Abstract

The purpose of this note is to relate certain ring-theoretic properties of rings in mixed and positive characteristics that are related to each other by a tilting operation used in perfectoid geometry. To this aim, we exploit the multiplicative structure of the ``monoidal map", which is constructed on arbitrary $p$-adically complete rings.