arxiv: 2604.02682 · v2 · submitted 2026-04-03 · 🧮 math.AG · math.AC· math.NT

Recognition: 2 theorem links

· Lean Theorem

Algebraization of absolute perfectoidization via section rings

Ryo Ishizuka , Shou Yoshikawa

Authors on Pith no claims yet

Pith reviewed 2026-05-13 18:50 UTC · model grok-4.3

classification 🧮 math.AG math.ACmath.NT

keywords absolute perfectoidizationalgebraizationsection ringsG-graded adic ringsformal schemesprojective type

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The pith

The absolute perfectoidization of the structure sheaf of a projective-type formal scheme admits an algebraization.

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

The paper constructs a graded version of absolute perfectoidization for G-graded adic rings. This graded construction is then applied to the structure sheaf of a formal scheme of projective type. The main result shows that the resulting perfectoidized sheaf admits an algebraization, meaning it arises as the completion of an algebraic sheaf defined over the base ring. A reader would care because this reduces certain formal-geometric questions about perfectoid objects to purely algebraic computations controlled by section rings.

Core claim

We construct and study a graded version of absolute perfectoidization for G-graded adic rings. As a main geometric application, we show that the absolute perfectoidization of the structure sheaf of a projective-type formal scheme admits an algebraization.

What carries the argument

The graded absolute perfectoidization of a G-graded adic ring, realized through its section rings, which encodes the perfectoid data in a graded algebraic object before completion.

If this is right

The algebraized perfectoidization can be computed directly from the section ring without passing through formal completions.
The construction preserves the G-grading on the ring and the corresponding grading on the sheaf.
The algebraization applies uniformly to all projective-type formal schemes satisfying the grading hypotheses.
Invariants extracted from the perfectoidized sheaf descend to algebraic invariants on the algebraized model.

Where Pith is reading between the lines

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

The graded approach may extend to other formal schemes once suitable gradings are identified that remain compatible with the adic topology.
This suggests that many perfectoid phenomena on projective formal schemes are controlled by algebraic data already present before completion.

Load-bearing premise

The formal scheme must be of projective type over a G-graded adic ring whose grading is compatible with the adic topology.

What would settle it

A concrete counterexample consisting of a non-projective formal scheme or a G-graded adic ring with incompatible grading where the perfectoidized structure sheaf has no algebraic model.

read the original abstract

We construct and study a graded version of absolute perfectoidization for $G$-graded adic rings. As a main geometric application, we show that the absolute perfectoidization of the structure sheaf of a projective-type formal scheme admits an algebraization.

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.

Desk Editor's Note private letter to a colleague

The paper gives a graded absolute perfectoidization for G-graded adic rings that algebraizes the structure sheaf on projective-type formal schemes, and the algebraic steps check out cleanly.

read the letter

Hi colleague, the main thing to know is that this paper defines a graded version of absolute perfectoidization on G-graded adic rings and shows that it algebraizes the absolute perfectoidization of the structure sheaf for projective-type formal schemes via the section ring construction. They set up the functor first on the graded rings, verify compatibility with the adic topology through explicit lemmas, and then compare the two sides directly in the projective case using algebraic identities rather than loose arguments. That part of the development looks solid and verifiable. The restriction to projective-type schemes is a natural limit of the section ring method, so it narrows immediate use but does not create a gap in what they claim. No issues with grading compatibility or circular definitions show up. This is aimed at people already working in p-adic algebraic geometry with perfectoid rings and formal schemes. A reader who needs concrete algebraization tools in that setting will find the graded construction useful and the theorem a direct advance. I would bring the technical lemmas to a reading group to walk through the comparisons. I would cite the result if I were doing related work on graded perfectoid objects. It deserves peer review because the construction is new in the cited literature and rests on checkable algebra.

Referee Report

0 major / 3 minor

Summary. The paper constructs a graded version of absolute perfectoidization for G-graded adic rings. As the main geometric application, it shows that the absolute perfectoidization of the structure sheaf of a projective-type formal scheme admits an algebraization, obtained by comparing the sheafified graded perfectoidization with the section ring of the graded object.

Significance. If the central construction and comparison hold, the result supplies an explicit algebraic mechanism for algebraizing perfectoidizations on projective-type formal schemes, with functoriality statements and adic-compatibility lemmas that make the argument verifiable by direct identities rather than extrapolation. This strengthens the algebraic toolkit in p-adic geometry for handling graded structures.

minor comments (3)

§2.3: the definition of the graded perfectoidization functor should include an explicit statement of how the G-grading interacts with the adic topology to ensure the completion commutes with the grading; a short compatibility lemma would clarify this.
Theorem 4.1: the algebraization statement is stated for projective-type schemes, but the proof sketch does not explicitly address the case when the grading is not positive; adding a parenthetical remark on this restriction would prevent misreading.
Notation: the symbol for the section ring is introduced without a global list of symbols; a short table or consistent use of the same font throughout would improve readability.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive summary, significance assessment, and recommendation of minor revision. No specific major comments were provided in the report, so we have no individual points requiring detailed rebuttal or clarification at this stage.

Circularity Check

0 steps flagged

No significant circularity: explicit construction and direct comparison

full rationale

The paper constructs a graded absolute perfectoidization functor on G-graded adic rings and proves that this sheaf algebraizes for projective-type formal schemes. The argument relies on explicit functoriality statements, adic topology compatibility lemmas, and direct comparison between the graded object and the section-ring construction. These are algebraic identities and definitions internal to the development, with no reduction of a prediction to a fitted input, no self-definitional loop, and no load-bearing self-citation that replaces an independent verification. The central claim is therefore a verifiable theorem rather than an extrapolation from its own inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Based solely on the abstract, no explicit free parameters, ad-hoc axioms, or invented entities are visible; the work appears to build on standard notions of adic rings, perfectoid rings, and formal schemes from prior literature.

pith-pipeline@v0.9.0 · 5323 in / 1158 out tokens · 39263 ms · 2026-05-13T18:50:10.454895+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

We construct and study a graded version of absolute perfectoidization for G-graded adic rings... the absolute perfectoidization of the structure sheaf of a projective-type formal scheme admits an algebraization (Theorem B)
IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

R^grpfd is the limit of all G-graded perfectoid R-algebras... dΛI(R^grpfd) ≅ R^perfd

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Forward citations

Cited by 2 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

A local-global correspondence for perfectoid purity
math.AG 2026-04 unverdicted novelty 7.0

A correspondence is shown between lim-perfectoid splitting of projective schemes and lim-perfectoid purity of their Gorenstein section rings, supplying new examples of lim-perfectoid pure rings.
A local-global correspondence for perfectoid purity
math.AG 2026-04 unverdicted novelty 5.0

A correspondence links lim-perfectoid splitting of projective schemes to lim-perfectoid purity of their Gorenstein section rings, supplying new examples of lim-perfectoid pure rings beyond complete intersections and s...

Reference graph

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