An application of Fontaine's monoidal maps to perfectoid towers

TL;DR

This paper introduces monoidal maps for perfectoid towers to relate their arithmetic properties across characteristics, demonstrating their applications in almost integrality, normality, and connecting to ramification theory.

Contribution

It develops monoidal maps for perfectoid towers and applies them to establish properties like almost integrality and normality, linking ramification theory to perfectoid structures.

Findings

Perfectoid towers are almost integral using monoidal maps.

F. Andreatta's ramification towers are shown to be perfectoid.

Monoidal maps help prove the normality of small tilts.

Abstract

To connect arithmetic and ring-theoretic properties of rings of mixed characteristic with those of positive characteristic, we introduce monoidal maps for perfectoid towers. Using these maps, we discuss the almost integrality of perfectoid towers and of their tilts. We also show that the towers constructed by F. Andreatta via ramification theory become perfectoid towers, and we apply the monoidal maps to deduce the normality of their small tilts.