On a generalization of affinoid varieties

TL;DR

This thesis develops a unified theory of analytic geometry over valued fields, extending classical concepts to both non-archimedean and archimedean cases, and generalizes key theorems like Gerritzen-Grauert.

Contribution

It introduces dagger affinoid algebras as a foundational tool and constructs a category of dagger analytic spaces, bridging various existing frameworks.

Findings

Generalization of Gerritzen-Grauert theorem

Construction of dagger analytic spaces

Comparison with Berkovich and complex analytic spaces

Abstract

In this thesis we develop the foundations for a theory of analytic geometry over a valued field, uniformly encompassing the case when the base field is equipped with a non-archimedean valuation and the case when it has an archimedean one. Our building blocks are dagger affinoid algebras, i.e. algebras of germs of analytic functions, equipped with their canonical bornology. We obtain results akin to the ones of affinoid algebras and affinoid spaces theory in our context. In particular, we give a generalization of the celebrated Gerritzen-Grauert theorem. Finally, we construct the category of dagger analytic spaces and compare its objects with classical objects from Berkovich geometry, dagger spaces of Grosse-Klonne and complex analytic spaces.