Continuous functions on Berkovich spaces

TL;DR

This paper investigates the structure of continuous functions on Berkovich spaces over non-archimedean fields, establishing an inverse limit description of certain function rings for reduced, normal analytic spaces.

Contribution

It provides a new inverse limit characterization of the ring of functions on Berkovich spaces, extending understanding of their algebraic and topological properties.

Findings

The ring of functions is isomorphic to an inverse limit of quotient rings.

The results apply to reduced and normal $k$-analytic spaces.

Provides foundational insights into non-archimedean analytic geometry.

Abstract

Let $k$ be a perfect complete valued field with a nontrivial non-archimedean norm $|\cdot|$ and $\omega\in k$ with $0<|\omega|<1.$ Let $X$ be a reduced and normal $k$-analytic space. Then $O^{\circ}\simeq \lim\limits_{\leftarrow}O^{\circ}/\omega^n.$