{"id":"522f0491-3bd8-441c-b15b-29b4c9db62e5","arxiv_id":"2502.06108","paper_version":7,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Quasi-F-splitting height in mixed characteristic characterizes perfectoid purity and computes the perfectoid pure threshold of p for complete intersection rings.","lead":"This paper introduces a new mixed-characteristic invariant called quasi-F-splitting height and proves it controls whether a ring is perfectoid pure. It also gives a Fedder-style criterion that computes the perfectoid pure threshold of p in explicit examples.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The central equivalence rests on the unverified p-complete faithful flatness of the functorial test perfectoid T(R); Appendix A's proof of this flatness is too quick to support Theorem A.","rationale":"The reader's verdict CONDITIONAL is appropriate. I searched for an internal contradiction in §§4-5 and did not find a decisive one; the Fedder-type criterion, the local cohomology recursion, and the graded applications are intricate but coherent given the test perfectoid input. The load-bearing spot is exactly the functorial test perfectoid: all reductions to T(R) depend on its p-complete faithful flatness and p-torsion freeness. Appendix A's proof of these properties is abbreviated and contains the suspicious inference from flatness of a Frobenius lift to flatness of the perfection. This is not an external consensus dispute but an internal verification gap. The paper would be acceptable as a conditional contribution pending this check. I also note the unproved assertion in Example 5.13(1) for p≥7, but it does not affect Theorem A.","tokens_in":97,"tokens_out":15881,"duration_ms":1106072,"concrete_test":"Locate the precise statement in [BS22] used for the assertion 'the endomorphism φ on A{R} is (p,d)-completely faithfully flat, so A{R}→A{R}_∞ is p-completely faithfully flat' (Appendix A, proof of Theorem A.1). Either identify a cited lemma that applies to arbitrary Frobenius lifts on free δ-rings, or compute Tor_1^{A{R}}(A{R}_∞, A{R}/(p,d)); a nonzero Tor group would disprove the needed p-complete faithful flatness. If the cited lemma exists, the proof is sound and the concern is resolved.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Proposition 2.9 and Theorem 5.9 reduce perfectoid purity and the perfectoid pure threshold to a single functorial test perfectoid T(R). The key property of T(R) is that R→T(R) is p-completely faithfully flat and T(R) is p-torsion free when R is (Theorem A.1). The proof of Theorem A.1 in Appendix A asserts that the map A{R}→A{R}_∞ is p-completely faithfully flat because the Frobenius lift φ is (p,d)-completely faithfully flat. For a general free δ-ring, a Frobenius lift is not known to be flat; the perfection/colimit along φ need not preserve p-complete faithful flatness without additional hypotheses. If this flatness fails, R→T(R) need not be p-completely faithfully flat, so T(R) may fail to be a test perfectoid, Proposition 2.9(1)-(2) can no longer compute ppt from T(R), and Theorem 5.9 collapses; Theorem A's characterization of ht(R) is then unsupported. The subsequent algebra in §§4-5 appears internally consistent conditional on this descent property. A separate, less central gap is the unproved assertion in Example 5.13(1) that for p≥7 the ring is F-pure.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":37519,"tokens_out":16598,"duration_ms":179629,"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":[{"comment":"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].","section":"Appendix A, proof of Theorem A.1"},{"comment":"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.","section":"Example 5.13(1)"}],"minor_comments":[{"comment":"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.","section":"Appendix A, proof of Theorem A.1"},{"comment":"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.","section":"Throughout"},{"comment":"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.","section":"Theorem 6.3"},{"comment":"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].","section":"Proposition 2.9(3)"}],"recommendation":"major_revision","confidential_remarks":"The central method is novel and appears internally coherent, and the paper contains many useful explicit computations. The main issue is the under-justified p-complete faithful flatness in Appendix A, which is foundational for the test-perfectoid formalism. This is fixable by adding a proof or precise citation, but it is load-bearing enough that I would not accept the paper in its current form. The unproved F-purity assertion in Example 5.13(1) should also be addressed."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The headline: this paper introduces a genuinely new invariant, quasi-F-splitting height, in mixed characteristic and connects it to perfectoid purity. If the main theorem is right, it gives the first general way to compute perfectoid pure thresholds. That is worth a serious look.\n\nWhat is new and good: The definition of ht(R) via truncated Witt vectors is natural, and the Fedder-type criterion (Theorem 4.13) is a useful computational tool, explicitly built on prior work of KTY22 and TWY24. The comparison theorems (A, B, E) are nontrivial, and the examples are concrete and checkable. I found no internal contradiction in the main body; the algebra in Sections 3–5 is careful and mostly self-contained.\n\nThe soft spot is in Appendix A, and it is not minor. Theorem A.1 constructs a functorial test perfectoid T(R) and asserts that the map from a free δ-ring A{R} to its perfection A{R}_∞ is (p,d)-completely faithfully flat because the Frobenius lift φ is assumed to be so. But for a general free δ-ring, a Frobenius lift is not known to be flat, and passing to the perfection/colimit along φ need not preserve p-complete faithful flatness. This flatness is exactly what makes T(R) a test perfectoid; without it, Proposition 2.9 cannot compute the perfectoid pure threshold from T(R), and Theorem 5.9—hence Theorem A—is unsupported. The paper gives no proof or reference for this key step. That is a genuine gap, not a technicality.\n\nA smaller issue: Example 5.13(1) asserts that for p ≥ 7 the ring is F-pure, with no argument. It may well be true, but as written it is an unsupported assertion.\n\nConditional on fixing the Appendix A flatness, the rest of the paper looks solid. The examples and the Fedder criterion will be useful even if the main comparison theorem needs adjustment. This is a paper for people working on mixed characteristic singularities and perfectoid methods.\n\nMy recommendation: send it to peer review. A good referee should focus on the flatness of the perfection map in Theorem A.1, and ask for either a proof or a modification of the test perfectoid construction. If that can be fixed, the paper is a real contribution.","headline":"New invariant and comparison theorem are promising, but the proof of the key test-perfectoid flatness in Appendix A is a load-bearing gap.","tokens_in":38111,"tokens_out":3313,"would_cite":true,"duration_ms":32417,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["14B05","13A35","13D45","14F30"],"pacs":[],"model":"deepseek-v4-flash","headline":"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…","keywords":["quasi-F-splitting","perfectoid purity","perfectoid pure threshold","perfectoid BCM-regularity","mixed characteristic singularities","Fedder-type criterion","inversion of adjunction","Witt vectors"],"falsifier":"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.","tokens_in":37054,"feed_emoji":"🧮","tokens_out":13286,"duration_ms":118074,"temperature":0.7,"pith_summary":"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.","feed_headline":"Quasi-F-splitting height computes perfectoid purity","feed_subtitle":"For complete intersections, height n pins the perfectoid pure threshold to an explicit interval.","key_machinery":"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.","core_discovery":"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\n\\[\n1-\\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}.\n\\]\nThe 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.","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"Supplies the prismatic perfectoidization and the flatness lemma used to construct the functorial test perfectoid T(R) as p-completely faithfully flat and p-torsion free.","marker":"[BS22]"},{"why":"Defines perfectoid purity and the perfectoid pure threshold, and provides the lemmas that reduce purity checks to the test perfectoid.","marker":"[BMP+24a]"},{"why":"Sets the perfectoid terminology used throughout and supplies the p-complete faithful flatness criterion used in Remark 2.5.","marker":"[BMS19]"},{"why":"Provides Lemma 3.12 on Witt vectors of perfectoids, adapted in Theorem 4.10 to express Q_{R,n} in terms of a quotient of R.","marker":"[BMS18]"},{"why":"Gives the earlier Fedder-type criterion and an example of a ring with ht(R)=infinity; Theorem 4.13 extends this criterion to quasi-(F,F_infinity)-splitting.","marker":"[KTY22]"},{"why":"Supplies the quasi-F^e splitting height framework whose Fedder-type proof is adapted in Appendix C.","marker":"[TWY24]"},{"why":"Defines perfectoid BCM-regularity and supplies Proposition 6.10 used in Corollary 5.3 and the graded-ring Theorem C.","marker":"[MS21]"},{"why":"Introduced quasi-F-splitting in positive characteristic; Definition 4.4 is the mixed-characteristic analogue.","marker":"[Yob19]"}],"fun_headline_variants":["Height n computes perfectoid purity in complete intersections","Perfectoid pure threshold pinned to explicit interval","Quasi-F-splitting height decides perfectoid purity","Inversion of adjunction via quasi-F-splitting","Perfectoid BCM-regularity from quasi-F-splitting"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Height n computes perfectoid purity in complete intersections","Perfectoid pure threshold pinned to explicit interval","Quasi-F-splitting height decides perfectoid purity","Inversion of adjunction via quasi-F-splitting","Perfectoid BCM-regularity from quasi-F-splitting"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000533,"raw_usage":{"total_tokens":2563,"prompt_tokens":939,"completion_tokens":1624,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":555,"completion_tokens_details":{"reasoning_tokens":1546}},"tokens_in":555,"tokens_out":1624,"duration_ms":12031,"temperature":1.0,"reasoning_tokens":1546,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-08T16:46:12.508794+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[],"review_version":1}