Reified valuations and adic spectra

TL;DR

This paper introduces reified valuations with real-valued groups to develop a new framework for adic spectra, bridging Huber's and Berkovich's theories and extending perfectoid spaces.

Contribution

It proposes reified adic spectra that integrate real-valued valuations, creating a unified approach compatible with nonarchimedean analytic geometry.

Findings

Reified adic spectra align with Berkovich spaces.

Extension of perfectoid space theory to reified setting.

Framework for a unified nonarchimedean analytic theory.

Abstract

We revisit Huber's theory of continuous valuations, which give rise to the adic spectra used in his theory of adic spaces. We instead consider valuations which have been reified, i.e., whose value groups have been forced to contain the real numbers. This yields reified adic spectra which provide a framework for an analogue of Huber's theory compatible with Berkovich's construction of nonarchimedean analytic spaces. As an example, we extend the theory of perfectoid spaces to this setting.