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arxiv: 2604.25265 · v2 · submitted 2026-04-28 · 🧮 math.AG · math.AC· math.NT

Recognition: no theorem link

A local-global correspondence for perfectoid purity

Ryo Ishizuka , Shou Yoshikawa
Authors on Pith no claims yet

Pith reviewed 2026-05-12 03:08 UTC · model grok-4.3

classification 🧮 math.AG math.ACmath.NT
keywords perfectoid puritylim-perfectoid splittingprojective schemesGorenstein ringssection ringsalgebraic geometryring purity
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0 comments X

The pith

Lim-perfectoid splitting of projective schemes corresponds exactly to lim-perfectoid purity of their Gorenstein section rings.

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

The paper introduces lim-perfectoid splitting as a global variant of lim-perfectoid purity. It proves that for a projective scheme whose section ring is Gorenstein, the scheme admits lim-perfectoid splitting if and only if the ring is lim-perfectoid pure. This equivalence is used to produce new families of lim-perfectoid pure rings that fall outside the previously studied complete-intersection and splinter cases. A reader would care because the result supplies a concrete bridge between global geometric properties and local ring-theoretic ones in the perfectoid setting.

Core claim

Our main result establishes a correspondence between the lim-perfectoid splitting of projective schemes and the lim-perfectoid purity of their Gorenstein section rings. As an application, we construct a new supply of examples of lim-perfectoid pure rings that go beyond the previously known complete intersection or splinter-type cases.

What carries the argument

Lim-perfectoid splitting, introduced as the global counterpart to lim-perfectoid purity and shown to be equivalent to purity of the Gorenstein section ring.

If this is right

  • New lim-perfectoid pure rings arise directly from projective schemes that satisfy the splitting condition.
  • Purity of a Gorenstein section ring can be checked by verifying the global splitting property of the corresponding projective scheme.
  • The construction supplies examples outside the complete-intersection and splinter classes previously studied.

Where Pith is reading between the lines

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

  • The same local-global pattern could be tested for non-Gorenstein rings or non-projective schemes if the splitting and purity notions are suitably extended.
  • Results about singularities or F-singularities in characteristic p might transfer across this correspondence to produce further examples.
  • Geometric constructions on projective schemes become a systematic source of pure rings in arithmetic and mixed-characteristic settings.

Load-bearing premise

The schemes are projective and the section rings are Gorenstein so that the notions of lim-perfectoid splitting and purity are defined and the stated correspondence applies.

What would settle it

A single projective scheme with Gorenstein section ring that possesses lim-perfectoid splitting while its section ring fails to be lim-perfectoid pure would disprove the claimed equivalence.

read the original abstract

We introduce (lim-)perfectoid splitting, which is a global variant of (lim-)perfectoid purity. Our main result establishes a correspondence between the lim-perfectoid splitting of projective schemes and the lim-perfectoid purity of their Gorenstein section rings. As an application, we construct a new supply of examples of lim-perfectoid pure rings that go beyond the previously known complete intersection or splinter-type cases.

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

0 major / 3 minor

Summary. The paper introduces (lim-)perfectoid splitting as a global variant of (lim-)perfectoid purity for schemes. Its main result is a correspondence theorem asserting that a projective scheme admits lim-perfectoid splitting if and only if its Gorenstein section ring is lim-perfectoid pure. The correspondence is applied to produce new families of lim-perfectoid pure rings outside the previously known complete-intersection and splinter cases.

Significance. If the stated correspondence holds, the work supplies a concrete local-global bridge between geometric splitting conditions on projective schemes and algebraic purity conditions on their graded rings. This enlarges the supply of examples of lim-perfectoid pure rings and may facilitate future applications in mixed-characteristic commutative algebra and algebraic geometry. The introduction of the new global notion and the explicit construction of examples are clear strengths.

minor comments (3)
  1. The abstract and introduction should explicitly state the precise hypotheses on the base ring (e.g., whether it is assumed to be a perfectoid ring or a complete local ring) under which the correspondence is proved.
  2. Notation for the lim-perfectoid splitting and purity functors should be introduced once in a dedicated notation subsection and used consistently thereafter.
  3. The application section would benefit from a short table comparing the new examples with the previously known complete-intersection and splinter cases.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive summary, significance assessment, and recommendation of minor revision. No major comments were raised in the report.

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The paper introduces the new notions of (lim-)perfectoid splitting (as a global variant of purity) and (lim-)perfectoid purity, then states a correspondence theorem relating the splitting of projective schemes to the purity of their Gorenstein section rings. This is a definitional setup followed by a proof of the stated relation under the given hypotheses, with applications to new examples constructed directly from the correspondence. No equations, fitted parameters, self-citation chains, or ansatzes are visible that would reduce the central claim to its own inputs by construction; the argument remains self-contained as a theorem in algebraic geometry.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 1 invented entities

Abstract provides no explicit free parameters, axioms, or invented entities beyond the new definitions themselves; the correspondence and new splitting notion rest on background from perfectoid geometry whose details are not visible here.

invented entities (1)
  • lim-perfectoid splitting no independent evidence
    purpose: global variant of lim-perfectoid purity for projective schemes
    Introduced in the paper as the central new concept enabling the correspondence.

pith-pipeline@v0.9.0 · 5352 in / 1217 out tokens · 42984 ms · 2026-05-12T03:08:38.262930+00:00 · methodology

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Works this paper leans on

30 extracted references · 30 canonical work pages · 4 internal anchors

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