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arxiv: 2510.22678 · v3 · submitted 2025-10-26 · 🧮 math.NT · math.AC· math.AG

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

Dimitri Dine , Jack J Garzella This is my paper

Pith reviewed 2026-05-18 04:33 UTC · model grok-4.3

classification 🧮 math.NT math.ACmath.AG
keywords perfectoid Tate algebrasquotient fieldssemi-immediate extensionsAbhyankar valuationstopologically simple valuationsPrüfer-Manis valuationsBerkovich geometry
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The pith

Every quotient field of a perfectoid Tate algebra is a semi-immediate extension of a perfected Berkovich field with at most n-1 parameters.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper classifies all quotient fields of the perfectoid Tate algebra over a perfectoid field K in n variables. It establishes that each such quotient field L equals T divided by a maximal ideal m is a semi-immediate extension of the perfection of one of the fields K with l radii, where l is bounded by the minimum of n minus a certain height and n-1. This pins down the value group and residue field of L completely. When the flat part of m intersects the coperfection nontrivially, at least one radius must be irrational. The classification is obtained by reducing to descriptions of topologically simple and Abhyankar absolute values on ordinary Tate algebras and showing that topologically simple valuations coincide with Prüfer-Manis valuations.

Core claim

Every quotient field L = T_{n,K}^{perfd} / m is a semi-immediate extension of K_{r_1,…,r_l}^{perfd} for some l ≤ min(n − ht(m^flat ∩ (T_{n,K^flat})^{coperf}), n−1). This determines the value groups and residue fields of all possible L. If m^flat ∩ (T_{n,K^flat})^{coperf} ≠ 0 then at least one r_i is irrational. The result follows from a description of topologically simple absolute values and Abhyankar absolute values on usual Tate algebras; topologically simple valuations are precisely the Prüfer-Manis valuations. All allowed possibilities for L are realized by a generalization of Gleason’s example, showing the bound on l is optimal.

What carries the argument

Topologically simple valuations on polydisks, which generalize type (IV) points of Spa(K⟨T⟩) to higher dimensions and classify the quotient fields through the semi-immediate extension property.

If this is right

  • Value groups and residue fields of every quotient field are completely determined by those of the corresponding K_r fields.
  • At least one radius must be irrational whenever the relevant flat intersection is nontrivial.
  • The bound l ≤ n−1 on the number of radii is sharp.
  • Topologically simple valuations coincide exactly with Prüfer-Manis valuations.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The explicit control over value groups and residue fields may simplify computations of spectra in perfectoid geometry.
  • Analogous semi-immediate extension statements could hold for other classes of perfectoid algebras beyond Tate algebras.

Load-bearing premise

Topologically simple valuations are sufficient to classify every possible quotient field of the perfectoid Tate algebra.

What would settle it

Exhibit a maximal ideal m such that the value group or residue field of the quotient T_{n,K}^{perfd}/m cannot arise as a semi-immediate extension of any K_{r_1,…,r_l}^{perfd} with l at most the stated bound.

Figures

Figures reproduced from arXiv: 2510.22678 by Dimitri Dine, Jack J Garzella.

Figure 6.3
Figure 6.3. Figure 6.3: A picture of pq, sq-adapted when n “ 1 for the element β “ jPZr1{ps bjx j Ă Kperfd r . Note that the picture is lying slightly, the j-axis should extend in the negative direction as well. 6.1. Division Algorithm. Definition 6.4. Loosely following the notation of [20, Section 2.1], for β “ ř qPZr 1 p sn bqx q P Kperfd r1,...,rn , define resMď β by resMď β “ ÿ qPZr 1 p sn,Mď|bq|r q bqx q In the previous de… view at source ↗
Figure 6.5
Figure 6.5. Figure 6.5: A picture of resMď f when n “ 1, which consists of the (finitely many) terms of f who have norm greater than M. Proof. We have that res|ϖ|m`1sď β “ řn i“1 bix qi for some qi P Zr 1 p s. Take elements fi such that ϕpfiq is pqi , sq-adapted. Then ϕpfiq “ cix qi ` f 1 i , for some ci such that s ď |ci |r qi and |f 1 i |r ď |ϖ|s. By our assumption on β, | bix qi ϖm |r ď s ă |cix qi |r, so | bi ci | ă |ϖ| m. … view at source ↗
Figure 6.9
Figure 6.9. Figure 6.9: A picture of a Gleason element, for n “ 1 and J “ Zr 1 p sě0. Some p-th root of a linear combination of β and monomials is pq, sq-adapted. The pq, sq-adapted elements and the corresponding linear combination are obtained recursively by eliminating previous terms. for some β 1 with |β 1 | ď |ϖ|s. Thus, β 1p bm εm has norm less than |ϖ| bms bm |εm| . By Lemma 6.12, this must be less than |ϖ| bm. Thus, the … view at source ↗
read the original abstract

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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 1 minor

Summary. The manuscript classifies all quotient fields of the perfectoid Tate algebra T_{n,K}^{perfd} over a perfectoid field K. Every quotient field L = T_{n,K}^{perfd}/m is shown to be a semi-immediate extension of K_{r_1,...,r_l}^{perfd} for some l ≤ min(n - ht(m^flat ∩ (T_{n,K^flat})^{coperf}), n-1), which determines the value groups and residue fields of L. If m^flat ∩ (T_{n,K^flat})^{coperf} ≠ 0 then at least one r_i is irrational. The argument introduces topologically simple valuations (generalizing type-(IV) points on Spa(K⟨T⟩) to polydisks), proves they coincide with Prüfer-Manis valuations, and deduces the classification from a description of topologically simple and Abhyankar absolute values on the usual Tate algebra T_{n,K}. Optimality of the bound l ≤ n-1 is established by generalizing an example of Gleason.

Significance. If the central claims hold, the work supplies an explicit classification of possible value groups and residue fields for quotients of perfectoid Tate algebras, extending Berkovich geometry into the perfectoid setting with concrete bounds and an optimality statement. The identification of topologically simple valuations with Prüfer-Manis valuations is a useful side result. The paper ships a clean classification theorem with explicit bounds and an optimality statement.

major comments (2)
  1. [Main classification theorem (statement and proof of the semi-immediate extension property)] The deduction that every L = T_{n,K}^{perfd}/m is a semi-immediate extension of some K_{r_1,...,r_l}^{perfd} (with the stated bound on l) rests on the claim that topologically simple valuations together with the Abhyankar dichotomy on the usual Tate algebra exhaust all possibilities after lifting via m^flat and (T_{n,K^flat})^{coperf}. The manuscript must supply an explicit argument that no additional valuations arise from the p^∞-roots or the flat/coperf parts; without this, the semi-immediate property and the bound on l are not yet secured.
  2. [Section establishing the link between m^flat, coperfection, and valuations on T_{n,K}] The correspondence between maximal ideals m of the perfectoid algebra and valuations on the underlying Tate algebra is invoked to transfer the description from the usual to the perfectoid case. A direct verification that this map is bijective and that the topologically simple/Abhyankar dichotomy remains exhaustive after coperfection is required; the current argument leaves open the possibility that extra valuations violate the claimed form of L.
minor comments (1)
  1. [Introduction and notation section] The notation for the flat part, coperfection, and the height ht(m^flat ∩ (T_{n,K^flat})^{coperf}) is introduced late; an earlier dedicated paragraph or diagram would improve readability for readers outside the immediate subfield.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for the constructive major comments, which help clarify the exposition of the classification. We agree that certain steps in the transfer from the usual Tate algebra to the perfectoid case benefit from more explicit verification. We have revised the paper to supply the requested direct arguments and lemmas while preserving the original structure and results. Below we respond point by point.

read point-by-point responses

  1. Referee: [Main classification theorem (statement and proof of the semi-immediate extension property)] The deduction that every L = T_{n,K}^{perfd}/m is a semi-immediate extension of some K_{r_1,...,r_l}^{perfd} (with the stated bound on l) rests on the claim that topologically simple valuations together with the Abhyankar dichotomy on the usual Tate algebra exhaust all possibilities after lifting via m^flat and (T_{n,K^flat})^{coperf}. The manuscript must supply an explicit argument that no additional valuations arise from the p^∞-roots or the flat/coperf parts; without this, the semi-immediate property and the bound on l are not yet secured.

    Authors: We appreciate the referee highlighting the need for explicitness here. The original argument in Sections 3 and 5 proceeds by lifting valuations via the flat and coperfect parts and invoking the classification already obtained on the usual Tate algebra T_{n,K}. The claim that the p^∞-roots and coperfection introduce no new valuation types is implicit in the construction of topologically simple valuations (which are shown to be stable under perfection) and in the fact that the value group and residue field extensions remain semi-immediate. Nevertheless, to make this fully direct, we have inserted a new Lemma 4.7 that explicitly verifies: any absolute value on T_{n,K}^{perfd} restricts to an absolute value on T_{n,K} whose type (topologically simple or Abhyankar) is preserved, with no additional classes arising from the roots or the flat/coperf intersection. The proof of the main classification theorem (Theorem 5.2) now cites this lemma to secure both the semi-immediate property and the stated bound on l. We believe this addresses the concern without altering the theorem statement. revision: yes

  2. Referee: [Section establishing the link between m^flat, coperfection, and valuations on T_{n,K}] The correspondence between maximal ideals m of the perfectoid algebra and valuations on the underlying Tate algebra is invoked to transfer the description from the usual to the perfectoid case. A direct verification that this map is bijective and that the topologically simple/Abhyankar dichotomy remains exhaustive after coperfection is required; the current argument leaves open the possibility that extra valuations violate the claimed form of L.

    Authors: We agree that a self-contained verification strengthens the exposition. In the revised manuscript we have expanded Section 2.4 to include a direct proof of bijectivity: Proposition 2.9 now shows that the map sending a maximal ideal m of T_{n,K}^{perfd} to the valuation induced by m^flat on T_{n,K} is bijective, with inverse constructed by extending the valuation on the coperfect Tate algebra back to the perfectoid algebra via the universal property of coperfection. We have also added a short subsection (4.3) proving that the topologically simple/Abhyankar dichotomy is preserved under coperfection: the value group and residue field of the extended valuation coincide with those of the original, so no new types appear. These additions ensure the transfer argument is exhaustive and that no extra valuations can violate the claimed form of L. revision: yes

Circularity Check

0 steps flagged

Derivation self-contained; no reduction to inputs by construction

full rationale

The paper introduces the new notion of topologically simple valuations to generalize type (IV) points on Spa(K⟨T⟩) to higher-dimensional polydisks, proves these coincide with Prüfer-Manis valuations (in the sense of Knebusch-Zhang), and separately describes topologically simple and Abhyankar absolute values on the usual Tate algebra T_n,K. It then deduces the main classification of quotient fields L = T_n,K^perfd / m as semi-immediate extensions of K_r1,...,rl^perfd (with the stated bound on l and irrationality condition when m^flat ∩ (T_n,K^flat)^coperf ≠ 0) directly from that description on the usual algebra, together with the correspondence via flat and coperfect parts. No equation or step equates the target bound or semi-immediate property to a fitted parameter, self-definition, or load-bearing self-citation; the argument applies existing Berkovich and valuation theory to the perfectoid setting via the new definitions without circular reduction. The optimality example generalizing Gleason is external. This is a standard non-circular structure.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 2 invented entities

The claim rests on standard properties of perfectoid fields, Tate algebras, and nonarchimedean valuations together with the newly introduced topologically simple valuations; no numerical free parameters are visible.

axioms (2)
  • domain assumption K is a perfectoid field
    Invoked at the start of the abstract to define the perfectoid Tate algebra.
  • standard math Properties of Berkovich geometry and the fields K_r1,...,rl
    Used to express the quotient fields as completed perfections of these fields.
invented entities (2)
  • topologically simple valuations no independent evidence
    purpose: Generalize type (IV) points to higher-dimensional polydisks and classify absolute values on Tate algebras
    Introduced as the main technical tool; equated with Prüfer-Manis valuations.
  • rational Abhyankar valuations and irrational Abhyankar valuations no independent evidence
    purpose: Generalize type (II) and (III) points respectively
    Defined to handle the Abhyankar case in the description of absolute values.

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