Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: The topology on Berkovich affine lines over complete valuation rings   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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The categories of this paper are: astro-ph
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The categories of this paper are: quant-ph
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The categories of this paper are: cmp-lg
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The categories of this paper are: cs
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The categories of this paper are: hep-ex
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The categories of this paper are: hep-ph
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The categories of this paper are: nlin
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The categories of this paper are: physics
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The categories of this paper are: econ
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The categories of this paper are: chem-ph
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The categories of this paper are: supr-con
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The categories of this paper are: hep-th
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The categories of this paper are: stat
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The categories of this paper are: math-ph
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The categories of this paper are: ao-sci
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The categories of this paper are: q-bio