Analytic geometry over [formula omitted] and the Fargues-Fontaine curve

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 Toën-Vaquié theory of schemes over F1, 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 (e.g. polydisks) are recovered as a base change of analytic spaces over F1. We conclude by discussing some applications of our theory to the theory of the Fargues-Fontaine curve and to the ring of Witt vectors.