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Computation method for perfectoid purity and perfectoid BCM-regularity

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Pith's one-line read This paper introduces quasi-F-splitting for rings in mixed characteristic and proves that, for complete intersections, the quasi-F-splitting height $n$ exactly characterizes perfectoid purity, with the perfectoid pure threshold pinned…

desk verdict New invariant and comparison theorem are promising, but the proof of the key test-perfectoid flatness in Appendix A is a load-bearing gap. read the letter →

arxiv 2502.06108 v7 pith:VQG7WU52 submitted 2025-02-10 math.AG

Shou Yoshikawa This is my paper · ORCID
classification math.AG MSC 14B0513A3513D4514F30
keywords quasi-F-splittingperfectoidpuritypurethresholdBCM-regularitymixedcharacteristicsingularitiesFedder-typecriterioninversionofadjunctionWittvectors
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper introduces quasi-F-splitting, a mixed-characteristic analogue of Frobenius splitting, and proves that for complete intersections it provides a numerical certificate for perfectoid purity, meaning the ring admits a pure map into a perfectoid ring. The certificate is the quasi-F-splitting height $ht(R)$, the smallest length $n$ for which the map $R \to W_n(R)/pW_n(R)$ sending $a$ to the class of $a^p$ splits as an $R$-module. Theorem A states that $ht(R)=n$ exactly when $R$ is perfectoid pure and the perfectoid pure threshold satisfies $1-\frac{1}{p}-\cdots-\frac{1}{p^{n-1}} \ge \mathrm{ppt}(R;\mathrm{div}(p)) \ge 1-\frac{p+\cdots+p^{n-1}}{p^n-1}$; in particular, if $R/pR$ is quasi-F-split then $R$ is perfectoid pure. The paper also gives a Fedder-type criterion that makes the height computable, uses it to build rings with thresholds different from 1, and proves an inversion-of-adjunction result connecting quasi-F-splitting to perfectoid BCM-regularity for graded cones.

What carries the argument

The load-bearing object is the quasi-F-splitting height $ht(R)$. With $W_n(R)$ the length-$n$ Witt vectors, the map $\Phi_{R,n}:R\to W_n(R)/pW_n(R)$ sending $a$ to the class of $a^p$ gives $Q_{R,n}$ the structure of an $R$-module; $R$ is $n$-quasi-F-split when this map splits as $R$-modules, and $ht(R)$ is the smallest such $n$. This recovers F-splitting of $R/pR$ when $n=1$, so the height is a mixed-characteristic analogue of the Frobenius-splitting criterion. Two further tools carry the comparison with perfectoid purity: the functorial test perfectoid $T(R)$ constructed in Appendix A by perfectoidization, which is p-completely faithfully flat over $R$ and p-torsion free, and the local cohomology recursion of Theorem 5.9, which writes $\mathrm{ppt}(R;\mathrm{div}(p))=\sum_{m\ge1}a_m/p^m$ with digits $a_m$ determined by the sequence of heights appearing in an iterative splitting procedure. The Fedder-type criterion of Theorem 4.13 computes $ht(R)$ from ideals $I_n$ in a regular ambient ring, which is what makes the examples concrete.

What would settle it

A single complete intersection with finite ht(R) for which the Fedder-type criterion gives one value but direct computation of the perfectoid pure threshold gives another would falsify Theorem A. The cleanest check is to compute, for a concrete quotient of $\mathbb Z_p[[x_1,\ldots,x_N]]$, the local cohomology map $H^d_{\mathfrak m}(R)\to H^d_{\mathfrak m}(R_\infty)$ induced by the test perfectoid and see whether the p-adic expansion of Theorem 5.9 is reproduced.

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Extended reading notes

Core claim

The central discovery is a numerical characterization of perfectoid purity in complete intersections. For a p-torsion free complete local ring $R$ with $p$ in the maximal ideal and a Frobenius-finiteness condition on $R/pR$, the paper proves that $ht(R)=n$ if and only if $R$ is perfectoid pure and \[ 1-\frac{1}{p}-\cdots-\frac{1}{$p^{{n-1}}$}\ \ge\ \mathrm{ppt}(R;\mathrm{div}(p))\ \ge\ 1-\frac{p+\cdots+$p^{{n-1}}$}{p^n-1}. \] The lower bound is exactly the threshold value obtained in the graded case with $a(S)=0$, while the upper bound $1-\frac{1}{p}-\cdots-\frac{1}{p^{n-1}}$ is exactly the threshold of a quasi-$(F,F_\infty)$-split ring of height $n$. A separate quantization theorem, Theorem B, shows that any perfectoid pure complete intersection with threshold above $(p-2)/(p-1)$ must have its threshold in one of these intervals. For graded cones over strongly F-regular Fano-type pairs with negative $a$-invariant, quasi-F-splitting implies perfectoid BCM-regularity, meaning every perfectoid big Cohen-Macaulay extension is pure, and for $p=2$ the converse holds. Explicit computations include a perfectoid pure ring $R'=\mathbb Z_p[[x,y,z]]/(x^3+y^3+z^3)$ with $p\equiv 2\bmod 3$ whose self-tensor product is not perfectoid pure.

Load-bearing premise

The entire comparison rests on the functorial test perfectoid T(R) being p-torsion free and flat over R in the p-adic sense; if that fails for some complete intersection, the p-adic threshold formula and Theorem A collapse.

Editorial extensions

If this is right

  • If $R/pR$ is quasi-F-split and $R$ is a complete intersection, then $R$ is perfectoid pure, giving a new inversion-of-adjunction style implication.
  • The Fedder-type criterion turns quasi-F-splitting height and, via Theorem 5.9, the perfectoid pure threshold into explicit ideal computations in regular rings, producing concrete threshold values such as $1/8$, $5/9$, and $4/5$ for $\mathbb Z_{(p)}[[x,y,z]]/(z^2+y^3+z^5)$ when $p=2,3,5$.
  • Perfectoid pure thresholds are quantized: any perfectoid pure complete intersection with threshold greater than $(p-2)/(p-1)$ must have its threshold in one of the intervals determined by the height $n$.
  • For graded cones over strongly F-regular Fano-type pairs with negative $a$-invariant, quasi-F-splitting of the localization implies perfectoid BCM-regularity; when $p=2$ the two properties are equivalent.
  • The construction yields new perfectoid pure rings, including $\mathbb Z_p[[x,y,z]]/(x^3+y^3+z^3)$ with $p\equiv 2\bmod 3$, which is perfectoid pure even though its self-tensor product is not perfectoid pure.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • Editorial inference: if Theorem A holds, quasi-F-splitting height can serve as the mixed-characteristic analogue of the F-purity certificate in positive characteristic, giving an algorithmic route to perfectoid purity via Fedder-type ideal computations; implementing this computation in a computer algebra system is a direct testable extension.
  • Editorial inference: the quantization in Theorem B suggests that the perfectoid pure threshold takes values in a sparse, p-adically defined set for complete intersections, analogous to F-threshold jumping numbers; computing thresholds for a larger family of hypersurfaces would test how sharp the interval bounds are.
  • Editorial inference: the example where $R\otimes_{\mathbb Z_p} R$ fails perfectoid purity even though $R$ is perfectoid pure indicates perfectoid purity is not stable under products; exploring a perfectoid purity locus or a filtered closure might behave better, mirroring known F-purity pathologies.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

2 major / 4 minor

Summary. The paper introduces quasi-F-splittings in mixed characteristic, together with a quasi-F-splitting height ht(R), and compares this invariant with perfectoid purity and the perfectoid pure threshold ppt(R;div(p)). The central result, Theorem A, states that for a complete intersection local ring R one has ht(R)=n if and only if R is perfectoid pure and 1-1/p-...-1/p^{n-1} >= ppt(R;div(p)) >= 1-(p+...+p^{n-1})/(p^n-1). The paper also proves a Fedder-type criterion for quasi-F-splittings (Theorem 4.13), computes perfectoid pure thresholds in several explicit examples, and gives applications to perfectoid BCM-regularity of graded rings (Theorems C and E). Appendix A constructs a functorial test perfectoid T(R) for every Z_(p)-algebra R.

Significance. If the claims are correct, this is a substantial contribution: it provides a computable, purely module-theoretic certificate for perfectoid purity of complete intersections, it locates the perfectoid pure threshold via a p-adic expansion determined by the quasi-F-splitting height, and it supplies a Fedder-type criterion that is useful for explicit computations. The paper is also valuable for its explicit examples, including rings with perfectoid pure threshold different from 1, a perfectoid BCM-regular ring whose closed fiber is not strongly F-regular, and a perfectoid pure ring whose self-tensor product is not perfectoid pure. The proofs are detailed and the appendix supplies the promised Fedder-type and test-ideal statements. The main reservation concerns the construction of the functorial test perfectoid in Appendix A, whose p-complete faithful flatness is asserted rather than fully justified.

major comments (2)
  1. [Appendix A, proof of Theorem A.1] The proof that T(R) is a test perfectoid and is p-torsion free when R is p-torsion free rests on the assertion that the endomorphism phi:A{R}->A{R} is (p,d)-completely faithfully flat, and hence that A{R}->A{R}_infty is so. This is not proved and no precise reference is given. For a general free delta-ring this is not an immediate consequence of the Frobenius lift property, and the p-complete faithful flatness of the perfection/colimit is exactly the property needed for R->T(R) to be p-completely faithfully flat. Since Proposition 2.9 and Theorems 5.7 and 5.9 rely on this test-perfectoid property, the central Theorem A is not fully supported unless this point is justified. Please add a proof (for example, by reducing modulo (p,d) to the Frobenius on a polynomial ring over F_p and then applying a p-complete flatness criterion) or cite a precise lemma from [BS22].
  2. [Example 5.13(1)] The paper states without proof that for p >= 7 the ring R=Z_(p)[[x,y,z]]/(z^2+y^3+z^5) is F-pure, and this is used to conclude ppt(R;div(p))=1 in that range. This is a load-bearing step for the example, not for Theorem A, but it is still an unproved assertion of a nontrivial F-purity check. Please supply a proof or a precise reference; if the assertion is not correct, the displayed value of ppt for p >= 7 would need to be recomputed.
minor comments (4)
  1. [Appendix A, proof of Theorem A.1] The notation Z{R} appears in the sentence 'Therefore, we obtain that Z(p)[R] -> Z{R} is p-completely faithfully flat', but only Z{R}_infty was defined earlier. Please clarify whether Z{R} denotes Z{R}_infty or a different intermediate ring.
  2. [Throughout] The notation W_n(R) and W bar_n(R) is easy to confuse, especially in displayed formulas where the bar is not always visible. A short sentence in Notation 2.1 or in Section 4 would help.
  3. [Theorem 6.3] In the proof of Theorem 6.3, the sentence 'Since B is a BCM-algebra over R, we obtain that R is Gorenstein, thus so is R' appears garbled and should be rephrased; the intended argument seems to be that quasi-Gorensteinness of S together with the injectivity of local cohomology gives purity, but the current wording is confusing.
  4. [Proposition 2.9(3)] The notation RS_infty^perfd in the statement is not defined in the same notation block as the other symbols; please spell out its construction or refer explicitly to [BMP+24a, Lemma 4.23].

Circularity Check

0 steps flagged · score 2.0 of 10

No construction-level circularity: the quasi-F-splitting height/perfectoid pure threshold equivalence is proved, not defined; cited prior work is parameter-free and the Fedder-type criterion is reproved.

full rationale

The central equivalence (Theorem A / Theorem 5.11) is not circular. ht(R) is defined independently via splittings of Φ_{R,n}: R → Q_{R,n} (Definition 4.4), and ppt(R; div(p)) is defined independently via purity of maps to perfectoid R-algebras (Definition 2.8). The proof connecting them runs through the functorial test perfectoid T(R) of Appendix A and local-cohomology bookkeeping (Proposition 5.2, Theorem 5.9); neither side is substituted into the other by construction. Appendix A constructs T(R) from free δ-rings, perfection, p-adic completion, and perfectoidization, and its status as a test perfectoid is verified against an arbitrary perfectoid R-algebra B using André flatness and [BS22, Theorem 3.10]; the construction does not assume perfectoid purity or the height threshold. The Fedder-type criterion (Theorem 4.13) is cited as cf. [KTY22, Theorem A] but is proved in Appendix C from the Witt-vector decomposition of Section 3, so it is not load-bearing self-citation. Uses of [KTY22] and [KTT+22] in examples (e.g., Example 5.13(2) citing [KTY22, Example 7.11]) are peripheral and parameter-free. No fitted value is renamed as a prediction and no uniqueness theorem is imported to force the chosen formalism. Two non-circular concerns are flagged for correctness rather than circularity: Example 5.13(1) asserts without proof that 'if p ≥ 7, then R is F-pure', and Appendix A's claim that the Frobenius-lift map A{R}→A{R}_∞ is p-completely faithfully flat is asserted tersely. These do not make the derivation circular.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

The central claims rest on standard Witt vector algebra, prismatic and perfectoid technology (Andre flatness), and the prior Fedder-type and quasi-F-splitting framework, several papers of which are co-authored by the present author. These are treated as independent support because they are parameter-free derivations with stated assumptions, not because they are self-citations.

assumptions (5)
  • standard math Serre's Witt vector formalism and ghost component identities (Section 2.2, Proposition 2.3).
    The R-modules Q_{R,n} and all splitting criteria are built on the Witt vector constructions; the paper cites [Ser79] for existence of the ghost polynomials and proves the identities it uses.
  • standard math Andre's flatness lemma and existence of perfectoidization ([BS22, Theorem 7.14, Corollary 7.3]).
    Used in Theorem 5.7 and Appendix A to produce perfectoid algebras with p-power roots and to define the functorial test perfectoid T(R).
  • domain assumption The theory of perfectoid purity and perfectoid BCM-regularity from [BMP+24a] and [MS21], including [BMP+24a, Theorem 6.6] and [MS21, Proposition 6.10].
    These target notions and their basic properties (purity, BCM-regularity, threshold) are imported from prior work and assumed correct.
  • domain assumption The positive-characteristic Fedder-type and quasi-F-splitting framework from [Yob19], [KTY22], and [TWY24], including [Gab04, Lemma 13.1].
    Theorem 4.13 is stated as a Fedder-type criterion almost identical to [TWY24, Theorem A]; the trace map u and the ideal sequences I_n are borrowed from that framework.
  • domain assumption Assumptions of Theorem C: S is normal quasi-Gorenstein with a(S)<0 and Spec S minus the irrelevant ideal is strongly F-regular, with S0 a divisorial valuation ring.
    These hypotheses define the scope of the graded-ring theorem; the theorem's validity is conditional on them.

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Pith. "Pith review of Computation method for perfectoid purity and perfectoid BCM-regularity." pith.science (2026). https://pith.science/paper/VQG7WU52

@misc{pith2026250206108,
  author       = {Pith},
  title        = {Pith review of: Computation method for perfectoid purity and perfectoid BCM-regularity},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/VQG7WU52}},
  note         = {Machine review of arXiv:2502.06108}
}
abstract

In this paper, we introduce the notion of quasi-$F$-splitting for rings in mixed characteristic. By comparing quasi-$F$-splitting with perfectoid purity, we obtain a new inversion of adjunction-type result. Furthermore, we study the possible values of the perfectoid pure threshold of $\mathrm{div}(p)$ and construct new examples of perfectoid pure rings.

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