# The topology on Berkovich affine lines over complete valuation rings

## TL;DR

This paper characterizes the topology of the Berkovich affine line over complete valuation rings with algebraically closed fraction fields, showing it is connected, locally path connected, and a completion of a specific metric space.

## Contribution

It provides a comprehensive description of the topology of the Berkovich affine line over such rings, including its connectedness and local path connectedness, and relates it to a canonical uniform structure.

## Key findings

- $ ext{A}_R^1$ is connected and locally path connected.
- $ ext{A}_R^1$ is the completion of $K imes (1, ext{infinity})$ under a canonical uniform structure.
- Description of the Berkovich spectrum of $ ext{Z}_p[G]$ for cyclic $p$-groups.

## Abstract

In this article, we give a full description of the topology of the one dimensional affine analytic space $\mathbb{A}_R^1$ over a complete valuation ring $R$ (i.e. a valuation ring with "real valued valuation" which is complete under the induced metric), when its field of fractions $K$ is algebraically closed. In particular, we show that $\mathbb{A}_R^1$ is both connected and locally path connected. Furthermore, $\mathbb{A}_R^1$ is the completion of $K\times (1,\infty)$ under a canonical uniform structure. As an application, we describe the Berkovich spectrum $\mathfrak{M}(\mathbb{Z}_p[G])$ of the Banach group ring $\mathbb{Z}_p[G]$ of a cyclic $p$-group $G$ over the ring $\mathbb{Z}_p$ of $p$-adic integers.