Logical Berkovich Geometry: A Point-free Perspective

TL;DR

This paper applies point-free topology techniques to Berkovich geometry, extending foundational results without requiring the field to be non-trivially valued, and explores the relationship between topology and logic in non-Archimedean settings.

Contribution

It introduces a point-free perspective to Berkovich geometry, removing the need for non-trivially valued fields in key results, and connects topology with logical frameworks.

Findings

Points of the Berkovich Spectrum correspond to R-good filters.

The field K need not be non-trivially valued for the main results.

Point-free techniques sharpen the understanding of Berkovich geometry.

Abstract

Extending our insights from \cite{NVOstrowski}, we apply point-free techniques to sharpen a foundational result in Berkovich geometry. In our language, given the ring $\mathcal{A}:=K\{R^{-1}T\}$ of convergent power series over a suitable non-Archimedean field $K$, the points of its Berkovich Spectrum $\mathcal{M}(\mathcal{A})$ correspond to $R$-good filters. The surprise is that, unlike the original result by Berkovich, we do not require the field $K$ to be non-trivially valued. Our investigations into non-Archimedean geometry can be understood as being framed by the question: what is the relationship between topology and logic?