Derived Analytic Geometry for Z-Valued Functions. Part I -- Topological Properties

Abstract

We study the Banach algebras $\mathrm{C}(X,R)$ of continuous functions from a compact Hausdorff topological space $X$ to a Banach ring $R$ whose topology is discrete. We prove that the Berkovich spectrum of $\mathrm{C}(X,R)$ is homeomorphic to $\zeta (X)\times \mathcal{M}(R)$, where $\zeta (X)$ is the Banaschewski compactification of $X$ and $\mathcal{M}(R)$ is the Berkovich spectrum of $R$. We study how the topology of the spectrum of $\mathrm{C}(X,R)$ is related to the notion of homotopy Zariski open embedding used in derived geometry. We find that the topology of $\zeta (X)$ can be easily reconstructed from the homotopy Zariski topology associated to $\mathrm{C}(X,R)$. We also prove some results about the existence of Schauder bases on $\mathrm{C}(X,R)$ and a generalisation of the Stone--Weierstrass Theorem, under suitable hypotheses on $X$ and $R$.

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