arXiv subjects

Dimitri Dine

Publications and source records attributed to Dimitri Dine.

6 recordsLinked to original sources

arXiv2025

Abhyankar valuations, Pr\"ufer-Manis valuations, and perfectoid Tate algebras

Let $K$ be a perfectoid field. We describe all quotient fields of the perfectoid Tate algebra\begin{equation*}T_{n,K}^{\text{perfd}}=K\langle X_{1}^{1/p^{\infty}},\dots, X_{n}^{1/p^{\infty}}\rangle\end{equation*}in any number $n\geq1$ of variables in terms of (completed perfections of) the nonarchimedean fields $K_{r_1,\dots,r_l}$ occuring in Berkovich geometry. We prove that every quotient field\begin{equation*}L=T_{n,K}^{\text{perfd}}/\mathfrak{m}\end{equation*}is a so-called \textit{semi-immediate} extension of $K_{r_1,\dots,r_l}^{\text{perfd}}$ for some\begin{equation*}l\leq\min(n-\text{ht}(\mathfrak{m}^{\flat}\cap (T_{n,K^{\flat}})^{\text{coperf}}),n-1), \end{equation*}which pins down the value groups and the residue fields of the possible quotient fields $L$. Moreover, we show that if\begin{equation*}\mathfrak{m}^{\flat}\cap(T_{n,K^{\flat}})^{\text{coperf}}\neq 0,\end{equation*} at least one of the radii $r_{i}$ has to be irrational, i.e.,\begin{equation*}r_{i}\not\in\sqrt{|K^{\times}|}.\end{equation*} The main ingredient in our proof is the notion of \textit{topologically simple} valuations, which generalize type (IV) points in the classification of points on $\text{Spa}(K\langle T\rangle)$ to the case of higher-dimensional polydisks. We also consider \textit{rational Abhyankar} valuations and \textit{irrational Abhyankar} valuations, which generalize type (II) and (III) points, respectively. We deduce our main result from a description of topologically simple absolute values and of Abhyankar absolute values on usual Tate algebra. Along the way, we also show that our topologically simple valuations are the same as Pr\"ufer-Manis valuations in the sense of Knebusch-Zhang. Finally, we also show that all allowed possibilities for the quotient fields $L$ do indeed occur (i.e., the above bound $l\leq n-1$ is optimal) by generalizing an example of Gleason.

math.NT↗

arXiv2025

Banach modules, almost mathematics and condensed mathematics

We study the relationship between almost mathematics, condensed mathematics and the categories of seminormed and Banach modules over a Banach ring $A$, with submetric (norm-decreasing) $A$-module homomorphisms for morphisms. If $A$ is a Banach ring with a norm-multiplicative topologically nilpotent unit $\varpi$ contained in the closed unit ball $A_{\leq1}$ such that $\varpi$ admits a compatible system of $p$-power roots $\varpi^{1/p^{n}}$ with \begin{equation*}\lVert\varpi^{1/p^{n}}\rVert=\lVert\varpi\rVert^{1/p^{n}}\end{equation*}for all $n$, we prove that the "almost closed unit ball" functor \begin{equation*}M\mapsto M_{\leq1}^{a}\end{equation*}is an equivalence between the category of Banach $A$-modules and submetric $A$-module maps and the category of $\varpi$-adically complete, $\varpi$-torsion-free almost $(A_{\leq1}, (\varpi^{1/p^{\infty}}))$-modules. We also obtain an analogous result for Banach algebras and almost algebras. The main novelty in our approach is that we show that the norm on the Banach module $M$ is completely determined by the corresponding almost $A_{\leq1}$-module $M_{\leq1}^{a}$, rather than being determined only up to equivalence. We deduce from our results the existence of a natural fully faithful embedding of the category of Banach $A$-modules and submetric $A$-module maps into the category of (static) condensed almost $(A_{\leq1}, (\varpi^{1/p^{\infty}}))$-modules in the sense of Mann, which factors through the full subcategory of solid condensed $(A_{\leq1}, (\varpi^{1/p^{\infty}}))$-almost modules. If $A$ is perfectoid and the adic spectrum of $(A, A^{\circ})$ is totally disconnected, we show that this embedding transforms the complete tensor product of Banach $A$-modules into (an almost analog of) the solid tensor product of solid condensed almost modules.

math.NT↗

arXiv2025

Formal models for relative adic spaces

We extend Raynaud's theory of formal models from rigid-analytic spaces over a nonarchimedean field to uniform qcqs adic spaces $X$, with no finite-type assumptions, over an arbitrary Tate affinoid base $S$. The key new ingredient is the notion of a normalized formal blow-up which takes on the role played by admissible formal blow-ups in the classical theory.

math.AG↗

arXiv2025

On Shilov boundaries, Rees valuations and integral extensions

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 $R$.

math.AC↗

arXiv2023

Tilting and untilting for ideals in perfectoid rings

For an (integral) perfectoid ring $R$ of characteristic $0$ with tilt $R^{\flat}$, we introduce and study a tilting map $(-)^{\flat}$ from the set of $p$-adically closed ideals of $R$ to the set of ideals of $R^{\flat}$ and an untilting map $(-)^{\sharp}$ from the set of radical ideals of $R^{\flat}$ to the set of ideals of $R$. The untilting map $(-)^{\sharp}$ is defined purely algebraically and generalizes the analytically defined untilting map on closed radical ideals of a perfectoid Tate ring of characteristic $p$ introduced by the first author. We prove that these two maps, $(-)^{\flat}$ and $(-)^{\sharp}$, define an inclusion-preserving bijection between the set of ideals $J$ of $R$ such that the quotient $R/J$ is perfectoid and the set of $p^{\flat}$-adically closed radical ideals of $R^{\flat}$, where $p^{\flat}\in R^{\flat}$ corresponds to a compatible system of $p$-power roots of a unit multiple of $p$ in $R$. Furthermore, we prove that the maps send (closed) prime ideals to prime ideals and thus define a homeomorphism between the subspace of the spectrum of $R$ consisting of prime ideals $\mathfrak{p}$ of $R$ such that $R/\mathfrak{p}$ is perfectoid and the subspace of the spectrum of $R^{\flat}$ consisting of $p^{\flat}$-adically closed prime ideals of $R^{\flat}$. In particular, we obtain a generalization and a new proof of the main result of the first author's previous research which concerned prime ideals in perfectoid Tate rings.

math.AG↗

arXiv2020

Topological spectrum and perfectoid Tate rings

We study the topological spectrum of a seminormed ring $R$ which we define as the space of prime ideals $\mathfrak{p}$ such that $\mathfrak{p}$ equals the kernel of some bounded power-multiplicative seminorm. For any seminormed ring $R$ we show that the topological spectrum is a quasi-compact sober topological space. When $R$ is a perfectoid Tate ring we construct a natural homeomorphism between the topological spectrum of $R$ and the topological spectrum of its tilt $R^{\flat}$. As an application, we prove that a perfectoid Tate ring $R$ is an integral domain if and only if its tilt is an integral domain.

math.AG↗