## Abstract

In this article, we look at analytic geometry from the perspective of relative algebraic geometry with respect to the categories of bornological and Ind-Banach spaces over valued fields (both Archimedean and non-Archimedean). We are able to recast the theory of Grosse-Klönne dagger affinoid domains with their weak G-topology in this new language. We prove an abstract recognition principle for the generators of their standard topology (the morphisms appearing in the covers) and for the condition of a family of morphisms to be a cover. We end with a sketch of an emerging theory of dagger affinoid spaces over the integers, or any Banach ring, where we can see the Archimedean and non-Archimedean worlds coming together.

Original language | English |
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Pages (from-to) | 391-462 |

Number of pages | 72 |

Journal | Journal of Number Theory |

Volume | 162 |

DOIs | |

State | Published - 1 May 2016 |

Externally published | Yes |

### Bibliographical note

Funding Information:The first author acknowledges the support of the University of Padova by MIUR PRIN2010-11 “Arithmetic Algebraic Geometry and Number Theory”, and the University of Regensburg with the support of the DFG funded CRC 1085 “Higher Invariants. Interactions between Arithmetic Geometry and Global Analysis” that permitted him to work on this project. The second author acknowledges the University of Oxford and the support of the European Commission under the Marie Curie Programme for the IEF grant which enabled this research to take place. The contents of this article reflect the views of the two authors and not the views of the European Commission. We would like to thank Francesco Baldassarri, Elmar Grosse-Klönne, Kobi Kremnizer, Frédéric Paugam, and Jérôme Poineau for interesting conversations.

Publisher Copyright:

© 2015 Elsevier Inc.

## Keywords

- Global analytic geometry
- Over-convergent structure sheaf
- Rigid geometry

## ASJC Scopus subject areas

- Algebra and Number Theory