Analytic geometry over F_1 and the Fargues-Fontaine curve

TL;DR

This paper develops a new framework for analytic geometry over the field with one element, connecting it to the Fargues-Fontaine curve and Witt vectors, and extending Toen-Vaquie scheme theory.

Contribution

It introduces an analytic approach over F_1 using norms, and studies base change functors to Banach rings, linking to classical analytic spaces.

Findings

Recovered basic spaces like polydisks via base change from F_1

Established a foundation for analytic geometry over F_1

Connected the theory to the Fargues-Fontaine curve and Witt vectors

Abstract

This paper develops a theory of analytic geometry over the field with one element. The approach used is the analytic counter-part of the Toen-Vaquie theory of schemes over F_1, i.e. the base category relative to which we work out our theory is the category of sets endowed with norms (or families of norms). Base change functors to analytic spaces over Banach rings are studied and the basic spaces of analytic geometry (like polydisks) are recovered as a base change of analytic spaces over F_1. We end by discussing some applications of our theory to the theory of the Fargues-Fontaine curve and to the ring Witt vectors.