The topology on Berkovich affine lines over complete valuation rings

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.

14 pages; to appear in Quarterly Journal of Mathematics