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On Shilov boundaries, Rees valuations and integral extensions

Dimitri Dine

TL;DR

This work connects integral closure and Rees valuations from commutative algebra with spectral seminorms and Shilov boundaries from nonarchimedean geometry in the context of Tate rings. It proves that the Shilov boundary of a Tate ring $\mathcal{A}=A[\varpi^{-1}]$ is precisely the finite set of $\varpi$-normalized Rees valuations $\mathcal{RV}(\varpi)$ of $(\varpi)_{A}$ in Noetherian settings, and provides broad algebraic criteria (strong/weak $\varpi$-Shilov, $\varpi$-valuative) ensuring Berkovich's description of the Shilov boundary for a wide class of Tate rings. The arc$_{\varpi}$-closure is introduced as a key $\varpi$-local closure operator tied to the spectral seminorm, and $\varpi$-$\ast$-operations generalize multiplicative ideal theory to the $\varpi$-local setting, enabling finiteness results and coherence criteria. Stability results show Berkovich's description is preserved under completed integral extensions, yielding explicit boundary descriptions for mixed-characteristic Noetherian domains via constructions like $\widehat{R^{+}}[p^{-1}]$. Overall, the paper provides a unified algebraic framework linking valuation theory, integral closures, and nonarchimedean geometry, with practical identifications of Shilov boundaries in broad contexts.

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 $R$.

On Shilov boundaries, Rees valuations and integral extensions

TL;DR

This work connects integral closure and Rees valuations from commutative algebra with spectral seminorms and Shilov boundaries from nonarchimedean geometry in the context of Tate rings. It proves that the Shilov boundary of a Tate ring is precisely the finite set of -normalized Rees valuations of in Noetherian settings, and provides broad algebraic criteria (strong/weak -Shilov, -valuative) ensuring Berkovich's description of the Shilov boundary for a wide class of Tate rings. The arc-closure is introduced as a key -local closure operator tied to the spectral seminorm, and --operations generalize multiplicative ideal theory to the -local setting, enabling finiteness results and coherence criteria. Stability results show Berkovich's description is preserved under completed integral extensions, yielding explicit boundary descriptions for mixed-characteristic Noetherian domains via constructions like . Overall, the paper provides a unified algebraic framework linking valuation theory, integral closures, and nonarchimedean geometry, with practical identifications of Shilov boundaries in broad contexts.

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 with a Noetherian ring of definition and pseudo-uniformizer , we prove that the Shilov boundary for naturally coincides with the set of Rees valuation rings of the principal ideal of . 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 , we obtain a description of the Shilov boundary for the Tate ring , where is the -adic completion of the absolute integral closure of the domain .

Paper Structure

This paper contains 16 sections, 129 theorems, 224 equations.

Table of Contents

  1. Introduction
  2. Background
  3. First main theorem: Rees valuations
  4. Second main theorem: Berkovich's description of the Shilov boundary
  5. Third main theorem: Completed integral extensions
  6. Main ideas behind the proofs
  7. Outline of the paper
  8. Notation and terminology
  9. Acknowledgements
  10. Power-multiplicative seminorms
  11. The ${\rm arc}_{\varpi}$-operation, Shilov boundary and Rees valuations
  12. Fractional ideals and $\varpi$-valuative rings
  13. Generalities on $\varpi$-$\ast$-operations
  14. Characterizing the Shilov boundary
  15. The $\varpi$-Shilov property and coherence
  16. ...and 1 more sections

Key Result

Proposition 1.1

For any ideal $I$ in a Noetherian ring $A$ and any $f\in A$ the limit where exists. Moreover, for any integer $n>0$, an element $f\in A$ belongs to the integral closure $\overline{I^{n}}$ of $I^{n}$ if and only if $\overline{\nu_{I}}(f)\geq n$.

Theorems & Definitions (306)

  • Proposition 1.1: Swanson-Huneke, Cor. 6.9.1 and Lemma 6.9.2
  • Proposition 1.2: Reformulation of Proposition \ref{['Starting point']}
  • Definition 1.3: Boundary, Shilov boundary
  • Definition 1.4: Shilov boundary for a Tate ring
  • Theorem 1.5: Theorem \ref{['Rees valuations']}
  • Theorem 1.5: Theorem \ref{['Rees valuations']}
  • Corollary 1.6: Corollary \ref{['Finite Shilov boundary']}
  • Proposition 1.7: Berkovich Berkovich, Prop. 2.4.4, see also Bhatt-Hansen Bhatt-Hansen, Prop. 2.2
  • Definition 1.8: Berkovich's description of the Shilov boundary, Definition \ref{["Berkovich's description of the Shilov boundary"]}
  • Theorem 1.9: Theorem \ref{['Strongly Shilov']}, Proposition \ref{['Large value groups and strong Shilov rings']}
  • ...and 296 more