On Shilov boundaries, Rees valuations and integral extensions

Abstract

We explore an analogy between, on one hand, the notions of integral closure of ideals and Rees valuations in commutative algebra and, on the other hand, the notions of spectral seminorm and Shilov boundary in nonarchimedean geometry. For any Tate ring $\mathcal{A}$ with a Noetherian ring of definition $\mathcal{A}_{0}$ and pseudo-uniformizer $\varpi\in\mathcal{A}_{0}$, we prove that the Shilov boundary for $\mathcal{A}$ naturally coincides with the set of Rees valuation rings of the principal ideal $(\varpi)_{\mathcal{A}_{0}}$ of $\mathcal{A}_{0}$. Furthermore, we characterize the Shilov boundary for a wide class of Tate rings by means of minimal open prime ideals in the subring of power-bounded elements. For affinoid algebras, in the sense of Tate, whose underlying rings are integral domains, this recovers a well-known result of Berkovich. Moreover, under some mild assumptions, we prove stability of our characterization of the Shilov boundary under (completed) integral extensions. In particular, for every mixed-characteristic Noetherian domain $R$, we obtain a description of the Shilov boundary for the Tate ring $\widehat{R^{+}}[p^{-1}]$, where $\widehat{R^{+}}$ is the $p$-adic completion of the absolute integral closure of the domain $R$.