Open Access

Reified valuations and adic spectra

Research in Number Theory20151:20

https://doi.org/10.1007/s40993-015-0021-7

Received: 20 March 2015

Accepted: 19 August 2015

Published: 9 December 2015

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.