On a generalization of affinoid varieties

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.

The whole text has been improved, many details added and a gap in the proof of the main theorem has been fixed. The numeration has been maintained as close as possible to the former version

References in corpus (1)

• Overconvergent global analytic geometry

Cited by in corpus (11)

• Non-Archimedean analytic geometry as relative algebraic geometry
• Dagger Geometry As Banach Algebraic Geometry
• Stein Domains in Banach Algebraic Geometry
• Analytic geometry over F_1 and the Fargues-Fontaine curve
• Closed graph theorems for bornological spaces
• Dagger completions and bornological torsion-freeness
• On the Sheafyness Property of Spectra of Banach Rings
• Fréchet Modules and Descent
• Nonarchimedean bornologies, cyclic homology and rigid cohomology
• Analytification, localization and homotopy epimorphisms
• The Monsky--Washnitzer topos