Dagger Geometry As Banach Algebraic Geometry

TL;DR

This paper develops a new geometric framework for dagger affinoid spaces using Banach algebraic geometry, unifying Archimedean and non-Archimedean analytic geometries through a categorical approach.

Contribution

It introduces a novel application of relative algebraic geometry to bornological and Ind-Banach spaces, recasting dagger affinoid domains within this framework.

Findings

Recast of dagger affinoid domains in Banach algebraic geometry

Recognition principle for generators of the standard topology

Sketch of a unified theory over integers and Banach rings

Abstract

In this article, we apply the approach of relative algebraic geometry towards analytic geometry to the category of bornological and Ind-Banach spaces (non-Archimedean or not). We are able to recast the theory of Grosse-Kl\"onne 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). 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.