Global analytic geometry

TL;DR

This paper introduces a new valuation concept that unifies archimedean and non-archimedean valuations, leading to a global analytic space framework that bridges Berkovich and Huber's theories.

Contribution

It proposes a novel valuation combining Krull valuations and seminorms, enabling a unified approach to global analytic geometry.

Findings

Defined a new valuation type blending Krull and seminorm concepts

Constructed the spectrum of Z and Z[X] within this framework

Developed analytic spectra and sheaves of functions on them

Abstract

We define a new type of valuation of a ring that combines the notion of Krull valuation with that of multiplicative seminorm. This definition partially restores the broken symmetry between archimedean and non-archimedean valuations. This also allows us to define a notion of global analytic space that reconciles Berkovich's notion of analytic space of a (Banach) ring with Huber's notion of non-archimedean analytic space. After defining natural generalized valuation spectra and computing the spectrum of Z and Z[X], we define analytic spectra and sheaves of analytic functions on them.