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arXiv · 1703.05460

The topology on Berkovich affine lines over complete valuation rings

Chi-Wai Leung·Chi-Keung Ng
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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.

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Chi-Wai Leung, Chi-Keung Ng. 2017-03-16. The topology on Berkovich affine lines over complete valuation rings. https://arxiv.org/abs/1703.05460

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