{"id":"af739c05-fd30-45a7-81b2-6c1c9cf15ad8","arxiv_id":"2506.10626","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For F-finite animated rings in characteristic p, every map factors as a free finite-type map, a relatively perfect map, and a surjective map, and relatively perfect maps coincide with formally etale maps in the Noetherian case.","lead":"This paper constructs a relative version of inverse limit perfection for maps of derived commutative rings in positive characteristic, and uses it to factor arbitrary maps of F-finite rings into a free map, a relatively perfect map, and a surjective map. It gives commutative algebraists a new tool for studying Frobenius and etale-like behavior outside the Noetherian and perfect settings.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Noetherian conclusions in Theorems 5.7 and 5.3 depend on Gabber's Remark 13.6, quoted without statement; a direct verification of its hypotheses and of Proposition 4.18's base-change step is needed.","rationale":"The reader's weakest assumption identifies exactly the load-bearing point: the Noetherian and regularity assertions in the main factorization theorem and in Theorem E are imported from Gabber's unpublished-style note rather than proved. I agree that the existence part of the factorization is likely robust, but the specific claims (Theorem 5.7(ii), Lemma 5.2(ii), Corollary 5.8, and the proof of Theorem 5.3) rest on Proposition 4.18. I also checked the most likely alternative source of internal error: the relative Frobenius direction and the tower construction. With the correct formula, identity maps are relatively perfect, so the adjunction framework is self-consistent. The remaining risk is genuinely the black-box citation together with a one-line base-change comparison in Proposition 4.18. A conditional verdict is therefore appropriate; the author should either reproduce Gabber's statement and hypotheses or give a direct proof of Proposition 4.18. My read does not change the reader's conditional verdict.","tokens_in":20378,"tokens_out":37826,"duration_ms":491242,"concrete_test":"Obtain [7, Remark 13.6] and verify verbatim hypotheses; in particular check whether it requires the map A -> S to be relatively semiperfect or only F-finite, and whether S must be reduced, finite type over A, or otherwise restricted. Then perform the base-change check in Proposition 4.18 for a concrete regular non-polynomial base, e.g. R = F_p[x,y]/(y^p - y - x) over A = F_p[x], with S a Noetherian relatively semiperfect R-algebra; compute both S^{perf/A} ⊗_{R^{perf/A}} R and S^{perf/R} and compare. If they differ, Proposition 4.18 is false; if Gabber's hypotheses are stricter than used, the Noetherian conclusions need revision.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim that survives the loss of the black box is the existence factorization; what does not survive are the Noetherian and regularity conclusions. Proposition 4.18 invokes [7, Remark 13.6] via Example 4.17 for the assertion that, for A = F_p[x_1,...,x_n], a relatively semiperfect map A -> S with S Noetherian and F-finite makes S^{perf/A} regular Noetherian. This is the only proof of Noetherianness of S^{perf/R} for a general regular Noetherian F-finite base R. Moreover, in the same proposition the identity S^{perf/A} ⊗_{R^{perf/A}} R ≃ S^{perf/R} is asserted in one sentence using preservation of limits; if this base-change comparison requires an additional flatness or finiteness hypothesis, the reduction to Gabber's statement fails even when the remark is true. Since [7] is not reproduced and the exact hypotheses are not quoted, the Noetherian part of Theorem 5.7(ii), Lemma 5.2(ii), Corollary 5.8, and the proof of Theorem 5.3 should be treated as conditional. I found no internal inconsistency in the remaining formal parts.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper introduces relative analogues of Frobenius finiteness, semiperfectness, and perfectness for maps of derived commutative F_p-algebras, constructs a relative Frobenius tower and its inverse limit perfection, and proves that this perfection is a right adjoint to the inclusion of relatively perfect algebras when the base is F-dualizable. In the animated setting it proves a factorization theorem for F-finite maps into a free finite-type map, a relatively perfect map, and a surjection on H0, with coconnectivity and Noetherian conclusions, and uses this to prove a converse statement relating vanishing of the cotangent complex to relative perfectness for Noetherian F-finite rings, yielding a formal étaleness characterization.","tokens_in":20603,"tokens_out":24875,"duration_ms":309073,"significance":"The construction is natural and the main factorization theorem is a useful structural result in positive characteristic. The paper is well organized and largely explicit, with clean adjunction statements, Tor-independence results, and applications such as Corollaries C through E and the factorization of formally smooth morphisms. Its strongest advertised consequences, however, currently depend on Gabber's unpublished Remark 13.6 and the unpublished manuscript [3], as well as on several sketched arguments; the Noetherian and regularity conclusions should be considered conditional until those inputs are stated and proved.","major_comments":[{"comment":"The proof that S^{perf/R} is regular Noetherian has two gaps. First, it does not state the hypotheses or content of Gabber's Remark 13.6 beyond the citation in Example 4.17. Second, the displayed chain of equivalences, particularly S^{perf/A} \\otimes_{R^{perf/A}} R \\simeq S^{perf/R}, is asserted in one sentence using preservation of limits; base change does not commute with inverse limit perfection merely because the base is a perfect complex, and the tower over R^{perf/A} must be shown to base-change to the tower over R. Since Lemma 5.2(ii), Theorem 5.7(ii), Corollary 5.8, and the proof of Theorem 5.3 all use this proposition, these claims are not yet established.","section":"Proposition 4.18"},{"comment":"The main computation is omitted. The proof reduces to F_p[X] -> F_p[X,Y] and says the vanishing follows from a \"straightforward calculation\", but this vanishing is the key input for Proposition 3.11 and hence for the acyclicity of cotangent complexes of relatively perfect maps. Please write out the calculation and justify the passage from the polynomial case to a general animated ring through the sifted colimit; cotangent complexes do not in general commute with arbitrary colimits, so the relevant preservation statement needs to be cited or proved.","section":"Lemma 2.9"},{"comment":"The proof of Theorem 5.3 invokes Lemma 5.2 to obtain a Noetherian T, and then uses the implication \"étale implies relatively perfect\". This implication is asserted in Example 3.6 but not proved for the relative notion; the cited references support étale implies weakly étale and related statements, but the paper does not give the direct verification that the relative Frobenius map is an equivalence. Since Theorem 5.3 is the converse direction behind Theorem E, this step should be proved directly for Noetherian F-finite rings or supported by a precise reference proving exactly this implication.","section":"Theorem 5.3"},{"comment":"Several auxiliary results depend on the unpublished manuscript [3]: Proposition 4.6, Proposition 4.8, and Proposition 4.19 quote specific numbered results from [3] without reproducing their statements. This is acceptable for a preprint, but for a journal submission the dependence should be made self-contained, or the statements should be quoted in enough detail for the reader to verify the hypotheses.","section":"Propositions 4.6, 4.8, 4.19"}],"minor_comments":[{"comment":"The word \"Frobenus\" should be \"Frobenius\".","section":"Lemma 2.9"},{"comment":"In part (i), \"an derived\" should read \"a derived\".","section":"Lemma 3.8"},{"comment":"In the displayed square, the upper horizontal map is described as \"the Frobenius on R\", but the rings displayed are polynomial rings over S; the description should be rephrased to identify the map precisely.","section":"Example 4.17"},{"comment":"The phrase \"the discrete relative inverse limit perfection\" should be reconciled with Definition 3.1, since the relative inverse limit perfection was defined for derived rings; please clarify the precise object being claimed relatively perfect.","section":"Corollary 5.4"},{"comment":"The same sifted colimit presentation is cited to two different locations in [20]; the references should be unified.","section":"Remark 2.2 and Lemma 2.9"}],"recommendation":"major_revision","confidential_remarks":"The central construction is promising and the existence part of the factorization theorem appears defensible, but the Noetherian and regularity claims are currently conditional on Gabber's Remark 13.6 and on unpublished material in [3]. I would ask the editor to require the author either to provide full statements and proofs of these inputs or to state the affected results conditionally. The omitted calculation in Lemma 2.9 and the unproved étale-implies-relatively-perfect step in Theorem 5.3 should also be addressed before further review."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Short version: this is a genuine contribution, and you should send it to referees. But the Noetherian conclusions are not as solid as the formal parts, and the referee should insist on closing a real gap.\n\nThe new material is real. The relative inverse limit perfection functor, the adjunction relative to relatively perfect algebras, the free-by-relatively-perfect-by-surjective factorization for F-finite animated rings (Theorem 5.7), and the converse to cotangent-complex vanishing for Noetherian F-finite rings (Theorem 5.3) are all new and natural. The formal framework is coherent, and the proofs I checked are mostly standard. I do not share the reader's worry about Lemma 2.9: the 'straightforward calculation' is fine, since d(r^p)=0 in characteristic p.\n\nThe soft spot is Proposition 4.18, and the stress-test note is right to put its finger there. The paper invokes Gabber's Remark 13.6 without stating the hypotheses or the content of the result. That might be acceptable in a research article if the application were straightforward, but here the remark is load-bearing for the Noetherianness of the relative inverse limit perfection. Worse, the same proposition asserts\n  S^{perf/A} ⊗_{R^{perf/A}} R ≃ S^{perf/R}\nin one sentence, justified by saying that R is a perfect complex over the regular ring R^{perf/A} and hence base change preserves limits. That step is not automatic: a quotient of a regular ring need not be perfect as a module over that regular ring. The base-change comparison needs an argument, or an extra hypothesis. If it fails, the Noetherian and regularity conclusions in Lemma 5.2(ii), Theorem 5.7(ii), Corollary 5.8, and the proof of Theorem 5.3 do not go through. The existence part of the factorization would likely survive, since it does not need the Noetherian conclusions.\n\nTo be clear, I did not find an internal contradiction in the remaining formal parts. The reliance on the unpublished manuscript [3] is a smaller issue; those references are mostly for Adams completion facts, and quoting the precise statements would make the paper more checkable.\n\nThe paper is for specialists in F-singularities and derived commutative algebra. It deserves a serious referee. The referee should ask the author to either prove the needed version of Gabber's remark, or quote it exactly, and to give a complete proof of the base-change identity in Proposition 4.18. My recommendation: accept for peer review, with the expectation of a revision.","headline":"A substantial new construction with a real black-box problem: the Noetherian conclusions depend on Gabber's Remark 13.6, quoted without statement, and on a base-change step in Proposition 4.18 that needs a real proof.","tokens_in":21137,"tokens_out":5111,"would_cite":true,"duration_ms":66947,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["13A35","13B40","13D03"],"pacs":[],"model":"deepseek-v4-flash","headline":"Relative perfectness coincides with formal étaleness for Noetherian F-finite rings in characteristic p, and every F-finite map factors through a relative inverse limit perfection.","keywords":["relative Frobenius","F-finiteness","relatively perfect algebras","inverse limit perfection","animated rings","derived commutative rings","cotangent complex","formal étaleness"],"falsifier":"Exhibit a map $R \\to S$ of Noetherian F-finite $\\mathbb{F}_p$-algebras with vanishing cotangent complex $L_{S/R}$ whose relative Frobenius $F_{S/R}: S \\otimes_{R,F} R \\to S$ is not an isomorphism; Theorem E says no such map exists. Equivalently, compute the relative inverse limit perfection $T = S^{\\mathrm{perf}/R}$ for a relatively semiperfect map $R = \\mathbb{F}_p[x_1,\\dots,x_n] \\to S$ with $S$ Noetherian and check whether $T$ is regular Noetherian, since a single failure would falsify Proposition 4.18 and the Noetherian part of Theorem 5.7.","tokens_in":20155,"feed_emoji":"","tokens_out":8765,"duration_ms":84835,"temperature":0.7,"pith_summary":"Over a field of characteristic $p$, the Frobenius map has a relative version for a map $R \\to S$, and the paper's goal is to understand how far a general map is from being relatively perfect. It constructs the relative inverse limit perfection $S^{\\mathrm{perf}/R}$ as the inverse limit of the relative Frobenius tower, and shows that under an F-dualizability condition on $R$ this is a right adjoint to the inclusion of relatively perfect $R$-algebras. With this tool, it proves that every map of F-finite animated (connective derived) $\\mathbb{F}_p$-algebras factors as a free finite-type map, then a relatively perfect map, then a surjection on $H^0$; for Noetherian input the middle term is Noetherian, and in the discrete Noetherian case it is regular. It also proves that for maps of Noetherian F-finite $\\mathbb{F}_p$-algebras, relative perfectness is exactly formal étaleness, equivalently the vanishing of the cotangent complex. A reader should care because this gives a uniform finite-generation-style decomposition and a Frobenius-theoretic characterization of étale maps in positive characteristic.","feed_headline":"Relative perfection factors every F-finite map in characteristic p","feed_subtitle":"A Frobenius tower gives the factorization and shows formally étale maps are exactly relatively perfect ones.","key_machinery":"The load-bearing construction is the relative Frobenius tower of $S$ over $R$: the inverse system $\\cdots \\to S \\otimes^{\\mathrm{L}}_{R,F^3} R \\to S \\otimes^{\\mathrm{L}}_{R,F^2} R \\to S \\otimes^{\\mathrm{L}}_{R,F} R \\to S$, where $F$ is the absolute Frobenius and the transition maps are relative Frobenius maps. Its inverse limit $S^{\\mathrm{perf}/R}$ is the relative inverse limit perfection. The key identity is that if $F_*R$ is a dualizable $R$-module, then the functor $(-) \\otimes^{\\mathrm{L}}_{R,F} R$ commutes with limits, so the limit $S^{\\mathrm{perf}/R}$ is itself relatively perfect; Lemma 2.9 supplies the companion identity $L_{F_{S/R}} \\simeq L_{S/R} \\oplus (L_{S/R} \\otimes^{\\mathrm{L}}_{S,F} S[1])$, which immediately gives vanishing of $L_{S/R}$ for relatively perfect maps. The category of relatively perfect $R$-algebras is presentable and closed under the needed limits, so the inclusion admits the relative inverse limit perfection as a right adjoint. For discrete rings, relative perfectness also requires Tor-independence of $S$ and $F_*R$ over $R$; in the derived setting the equivalence of the derived relative Frobenius map is enough.","core_discovery":"The paper's central claim is that relative perfectness is the right measure of Frobenius-invariant structure in derived positive-characteristic algebra, and that every map of F-finite animated $\\mathbb{F}_p$-algebras can be built from one free finite-type step, one relatively perfect step, and one surjection. Concretely, Theorem 5.7 factors $R \\to S$ as $R \\to R[x_1,\\dots,x_n] \\to T \\to S$ with the first map the obvious free extension, the second relatively perfect (relative Frobenius an equivalence), and the third surjective on $H^0$; if $H^0(R)$ is Noetherian, $T$ is Noetherian as well, and the construction controls coconnectivity in the derived setting. In the discrete Noetherian case Corollary 5.8 makes $T$ discrete and regular when both rings are Noetherian. The companion Theorem 5.3 states that a map $R \\to S$ of Noetherian F-finite $\\mathbb{F}_p$-algebras is relatively perfect if and only if the cotangent complex $L_{S/R}$ vanishes, which for these rings is the same as formal étaleness. This equivalence turns the factorization into a geometric statement: formally smooth morphisms of locally Noetherian F-finite $\\mathbb{F}_p$-schemes factor locally as a projection from affine $n$-space followed by a formally étale morphism.","pith_inferences":["The realization of derived completion as a relative Frobenius tower limit suggests that relative inverse limit perfection is a completion operation interpolating between $p$-adic completion and absolute perfection; the paper exhibits this for polynomial bases, but its scope over general bases is left implicit.","The right-adjoint formulation may allow a relative perfection operation for arbitrary maps of schemes, giving a Frobenius-theoretic closure whose fixed points are exactly the maps that are étale in the appropriate derived sense.","Theorem E's equivalence between relative perfectness and formal étaleness may extend to non-Noetherian F-finite rings if the Tor-independence condition in the definition is replaced by a derived version; the paper establishes the Noetherian case only.","The same machinery could be tested as a tool for constructing p-bases or proving regularity criteria in settings where the Noetherian hypothesis is dropped, since the inverse limit perfection is defined without assuming Noetherianity."],"forward_implications":["Every F-finite animated $\\mathbb{F}_p$-algebra $S$ admits a polynomial $\\mathbb{F}_p$-algebra $R'$ and a relatively perfect $R'$-algebra $T$ with $T \\to S$ surjective on $H^0$; when $H^0(S)$ is Noetherian, $T$ is regular Noetherian.","A map of Noetherian F-finite $\\mathbb{F}_p$-algebras with acyclic cotangent complex is relatively perfect, so the $I$-adic completion of a Noetherian F-finite ring is a relatively perfect algebra over it.","Formally smooth morphisms of locally Noetherian F-finite $\\mathbb{F}_p$-schemes decompose, Zariski locally, as a projection from affine $n$-space followed by a formally étale morphism.","The discrete factorization gives an L-smooth-by-surjective factorization in the sense of earlier work, reproducing and refining that statement by an explicit construction.","Relative perfectness, rather than plain perfectness, is the Frobenius-theoretic property that matches formal étaleness and vanishing cotangent complex in the Noetherian F-finite setting."],"supporting_citations":[{"why":"This is the external result asserting that the inverse limit perfection over a polynomial ring of a Noetherian F-finite ring is regular Noetherian; the Noetherian and regularity conclusions of the paper lean on it.","marker":"[7, Remark 13.6]"},{"why":"Shows the absolute Frobenius of a derived $\\mathbb{F}_p$-algebra induces zero on negative cohomology, underlying the coconnectivity and discreteness results for relatively perfect algebras.","marker":"[4, Proposition 11.6]"},{"why":"This is the prior L-smooth factorization result that the paper's discrete factorization reproduces and refines with a new proof.","marker":"[5, Theorem 4.4]"},{"why":"Relates formal smoothness and formal étaleness to relative perfectness and p-bases, and is used in Theorem 5.3 and Corollary 5.6.","marker":"[8, Théorème 21.2.7]"},{"why":"Provides the derived commutative algebra Frobenius functor on which the relative Frobenius tower is built.","marker":"[10, Construction 2.4.1]"},{"why":"Gives flatness of relatively perfect schemes over regular locally Noetherian bases, needed for the Noetherian conclusions.","marker":"[11, Proposition 5.2]"},{"why":"Kunz's characterization of regular local rings in characteristic $p$ in terms of flat Frobenius is used to prove regularity.","marker":"[12, Theorem 2.1]"},{"why":"Gives that étale maps of rings are relatively perfect, closing the converse direction of Theorem 5.3.","marker":"[23, Lemma 0EBS]"}],"fun_headline_variants":["F-finite maps factor: free, relatively perfect, surjective","Cotangent complex vanishes iff relatively perfect over F-finite","Relative perfection captures formal etaleness for F-finite"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The Noetherian and regularity conclusions rest on an external black box, the remark cited as [7, Remark 13.6], which asserts that the inverse limit perfection over a polynomial ring of a Noetherian F-finite ring is regular Noetherian; if that assertion requires extra hypotheses, the regularity and Noetherian statements in the main theorems would need revision, although the existence of the factorization itself may survive.","fun_headline_variants_meta":{"raw":{"variants":["F-finite maps factor: free, relatively perfect, surjective","Cotangent complex vanishes iff relatively perfect over F-finite","Relative perfection captures formal etaleness for F-finite"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000677,"raw_usage":{"total_tokens":3098,"prompt_tokens":983,"completion_tokens":2115,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":599,"completion_tokens_details":{"reasoning_tokens":2059}},"tokens_in":599,"tokens_out":2115,"duration_ms":19194,"temperature":1.0,"reasoning_tokens":2059,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-07T04:28:27.453511+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Exhibit a map $R \\to S$ of Noetherian F-finite $\\mathbb{F}_p$-algebras with vanishing cotangent complex $L_{S/R}$ whose relative Frobenius $F_{S/R}: S \\otimes_{R,F} R \\to S$ is not an isomorphism; Theorem E says no such map exists. Equivalently, compute the relative inverse limit perfection $T = S^{\\mathrm{perf}/R}$ for a relatively semiperfect map $R = \\mathbb{F}_p[x_1,\\dots,x_n] \\to S$ with $S$ Noetherian and check whether $T$ is regular Noetherian, since a single failure would falsify Proposition 4.18 and the Noetherian part of Theorem 5.7.","supporting_citations":[],"review_version":1}