Characterisation of the Berkovich Spectrum of the Banach Algebra of Bounded Continuous Functions

Berkovich spectra of elements in Banach Rings

Adapting the notion of the spectrum $Σ_a$ for an element $a$ in an ultrametric Banach algebra (as defined by Berkovich), we introduce and briefly study the Berkovich spectrum $σ^{Ber}_R(u)$ of an element $u$ in a Banach ring $R$. This spectrum is a compact subset of the affine analytic space $A_Z^1$ over $Z$, and the later can be identified with the "equivalence classes" of all elements in all complete valuation fields. If $R$ is generated by $u$ as a unital Banach ring, then $σ^{Ber}_R(u)$ coincides with the spectrum of $R$ (as defined by Berkovich). If $R$ is a unital complex Banach algebra,…

完美oid空间的Berkovich方法

A Berkovich Approach to Perfectoid Spaces

本文从Berkovich空间视角重建完美oid空间基础，利用arc_pi下降技术使仿射完美oid空间的底层拓扑空间成为紧Hausdorff空间，并建立了arc_pi层的基础结果。

Overconvergent subanalytic subsets in the framework of Berkovich spaces

We study the class of overconvergent subanalytic subsets of a $k$-affinoid space $X$ when $k$ is a non-archimedean field. These are the images along the projection $X \times B^n \to X$ of subsets defined with inequalities between functions of $X\times B^n$ which are overconvergent in the variables of $B^n$. In particular, we study the local nature, with respect to $X$, of overconvergent subanalytic subsets. We show that they behave well with respect to the Berkovich topology, but not to the $G$-topology. This gives counter-examples to previous results on the subject, and a way to correct them.…

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 …

Stein Domains in Banach Algebraic Geometry

In this article we give a homological characterization of the topology of Stein spaces over any valued base field. In particular, when working over the field of complex numbers, we obtain a characterization of the usual Euclidean (transcendental) topology of complex analytic spaces. For non-Archimedean base fields the topology we characterize coincides with the topology of the Berkovich analytic space associated to a non-Archimedean Stein algebra. Because the characterization we used is borrowed from a definition in derived geometry, this work should be read as a contribution towards the found…

整体Berkovich空间：范畴、拓扑、上同调

Espaces de Berkovich globaux : catégorie, topologie, cohomologie

本文研究整体Berkovich空间的基础理论，定义了性质良好的态射并建立相关构造，证明其局部道路连通性，还证明任意维开/闭圆盘无高阶凝聚上同调，将算术幂级数环的Noether性定理推广到多变量情形。