Analytic geometry over F_1 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 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.

Small corrections have been made in the last section of the paper and some typos have been corrected

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Cited by in corpus (7)

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• Topologization and Functional Analytification I: Intrinsic Morphisms of Commutative Algebras
• Topics on Geometric and Representation Theoretic Aspects of Period Rings I