2025/07/09 by Dimitri Dine, Dine, Dimitri · 1 citation
Mathematics · #13A15 #13A18 #13B21 #13B22 #13F05 #14G22 #14G45 #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Commutative Algebra (math.AC) #Commutative Algebra and Its Applications #FOS: Mathematics #Number Theory (math.NT) #Rings, Modules, and Algebras
paper · pdf · doi:10.48550/arxiv.2507.07091
openalex publication_date 2025/07/09 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 A with a Noetherian ring of definition A0 and pseudo-uniformizer \varpi\inA0, we prove that the Shilov boundary for A naturally coincides with the set of Rees valuation rings of the principal ideal (\varpi)_A0 of A0. 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 \widehatR+[p-1], where \widehatR+ is the p-adic completion of the absolute integral closure of the domain R.