## 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 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 (e.g. polydisks) are recovered as a base change of analytic spaces over F_{1}. We conclude by discussing some applications of our theory to the theory of the Fargues-Fontaine curve and to the ring of Witt vectors.

Original language | English |
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Article number | 106815 |

Journal | Advances in Mathematics |

Volume | 356 |

DOIs | |

State | Published - 7 Nov 2019 |

### Bibliographical note

Funding Information:The first author acknowledges the University of Regensburg with the support of the DFG funded CRC 1085 ?Higher Invariants. Interactions between Arithmetic Geometry and Global Analysis?.

Publisher Copyright:

© 2019 Elsevier Inc.

## Keywords

- Bornological spaces
- Fargue-Fontaine curve
- Geometry over the field with one element
- Witt vectors
- p-adic Hodge Theory

## ASJC Scopus subject areas

- Mathematics (all)

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