{"id":"905e3eb6-8c08-4c10-847c-43150dadbb56","arxiv_id":"1908.06764","paper_version":2,"verdict":"REJECT","confidence":"HIGH","novelty_score":5.0,"correctness_risk":"high","formal_verification":"none","parameter_count":0,"one_line_summary":"The paper constructs Hochschild-Serre and comparison spectral sequences for commutative Lie algebra cohomology in characteristic 2, but the example computations in Section 4 are incorrect.","lead":"This paper builds spectral sequences, a standard algebraic tool, for computing commutative cohomology of Lie algebras over fields of characteristic 2. The demonstration computations for the two main examples are incorrect, so the paper's advertised results are not reliable as stated.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The Section 4 computation of HS^n(N,F) is false: direct calculation gives HS^1(N,F)=F and HS^2(N,F)=F, contradicting the paper's formula that predicts 0 unless 4 divides n.","rationale":"I chose the false Section 4 computation as the most load-bearing concern because it is a concrete falsification of the paper's main computational claim, not a proof gap that might be repairable. The comparison spectral sequences in Sections 5 are also asserted without proof, and the reader's weakest_assumption focuses on the filtration compatibility in 5.5; that is a real concern, but the one-sentence argument there may be fillable. By contrast, the Section 4 d2 computation is demonstrably wrong: the claim that d2 depends only on the parity of n(n+1)/2 bracket terms confuses the total differential with the spectral sequence differential, and direct low-degree computation refutes the stated formula. Since the abstract promises these spectral sequences as computational tools and Section 4 is the only illustration of the Hochschild-Serre tool, this error alone justifies rejection. I therefore keep the reader's REJECT verdict and note only partial agreement with the reader's stated weakest assumption.","tokens_in":11560,"tokens_out":15278,"duration_ms":148142,"concrete_test":"Compute HS^n(N,F) for n=1,2,3,4 directly from the Chevalley-Eilenberg differential (1.1): take the basis e^i f^{n-i} of S^n(N), write d on each basis cochain, and compute kernel/image. If HS^1 or HS^2 is nonzero, the Section 4 formula is false. This test is immediate and already contradicts the paper's prediction.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The paper's central advertised application of the Hochschild-Serre spectral sequence is the computation of HS^n(N,F) for the 2-dimensional commutative Lie algebra N with basis e,f and [f,f]=e. Section 4 claims d2^{p,q}: S^p(N/h)⊗S^q(h)→S^{p+2}(N/h)⊗S^{q-1}(h) is zero exactly when (p+q)(p+q+1)/2 is even, and uses this to conclude HS^n(N,F)=F^{n+1} if 4|n and 0 otherwise. This is not the correct d2: the second-page differential of the Hochschild-Serre spectral sequence is governed by the extension class of 0→h→g→q→0, not by the parity of the number of bracket terms in the total coboundary of a degree-(p+q) cochain. A direct computation from Definition 2 gives HS^1(N,F)=F (the kernel of d:C^1→C^2 is 1-dimensional, spanned by the cochain vanishing on e) and HS^2(N,F)=F, whereas the paper's formula predicts 0 in both degrees. Thus the paper's flagship illustration of the tool is wrong, so the central claim that these methods provide working computational tools is not supported.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proposes two families of computational tools for the recently introduced commutative cohomology of commutative Lie algebras over fields of characteristic 2. The first part (Sections 1–3) constructs a Hochschild–Serre-type spectral sequence associated to an ideal, with E2-term HS^p(q, HS^q(h, M)) (Corollary 3.2). The second part (Sections 5–6) asserts three comparison spectral sequences relating Chevalley–Eilenberg, commutative, and Leibniz cohomology, and derives vanishing theorems from them (Theorems 5.1–5.3, 6.1–6.4). The methods are illustrated by computations for two-dimensional algebras in Section 4.","tokens_in":11743,"tokens_out":14707,"duration_ms":137803,"significance":"If correct, the Hochschild–Serre spectral sequence would provide a standard tool for computing commutative cohomology, and the comparison spectral sequences would link three cohomology theories in characteristic 2, with potential applications to the classification of simple Lie algebras. The paper explicitly addresses questions raised in the prior literature on commutative cohomology. However, the validity of the main advertised example is contradicted by a direct computation, and the comparison spectral sequences are not actually proved. As it stands, the paper cannot serve as a reliable source for these tools.","major_comments":[{"comment":"The claimed computation HS^n(N,F)=0 for n=1 is false. By Definition 2 with trivial coefficients, d^1 f(x,y)=f([x,y]) for f∈Hom(S^1N,F). Since [f,f]=e and all other brackets of the basis elements vanish, d^1f=0 exactly when f(e)=0, so HS^1(N,F) is 1-dimensional. The spectral sequence bookkeeping in Section 4 is also inconsistent: the term E2^{1,0} cannot be killed because d2^{1,0}:E2^{1,0}→E2^{3,-1} has zero target, and no differential from a valid bidegree lands on E2^{1,0}; hence E3^{1,0}=F. Thus the description of d2 in terms of the parity of n(n+1)/2 is not the Hochschild–Serre differential, and the resulting formula HS^n(N,F)=F^{n+1} if 4|n, and 0 otherwise, is incorrect. This error also undermines the analogous computation for the algebra a later in the same section.","section":"Section 4, Example N"},{"comment":"Theorems 5.1–5.3 are not proved in the manuscript. The text repeatedly asserts that the arguments of Section 2 of [1] go through 'word by word' or 'mutatis mutandis,' but no detailed verification is supplied. In Section 5.5 the compatibility of the filtration F^pCL^n(g,M) with the differential is justified by reference to Equation (1.1), which is the Chevalley–Eilenberg differential; however, the complex being filtered is the relative complex for the inclusion CS^*(g,M)→CL^*(g,M), and the differential on CL^* is the Leibniz differential, not Equation (1.1). The Leibniz differential contains bimodule action terms and bracket terms with the bracket inserted at varying positions, and the symmetry of the bracket alone does not imply that the Leibniz differential preserves the first-p-entry symmetry. Consequently, the asserted compatibility is not established, and Theorems 5.3, 6.3, and 6.4, which depend on it, are unsupported.","section":"Sections 5.3–5.5, comparison spectral sequences"}],"minor_comments":[{"comment":"The degree shift [-2] in the definition of the relative complexes is confusing: the text later says a representative of a class in C^n_{rel,Λ} has n+2 arguments, but after the shift the grading should be adjusted consistently.","section":"Section 5.2"},{"comment":"The Leibniz differential is never written down explicitly. Since the comparison theorems rely on it, a self-contained definition would help the reader verify the claimed inclusions and spectral sequence differentials.","section":"Throughout"},{"comment":"The computation for the algebra a is stated without a derivation; the reader cannot verify the claimed E2-term or the collapse without repeating the same flawed parity argument used for N.","section":"Section 4"},{"comment":"There are numerous typos and grammatical errors, e.g., 'organisors', 'We refrain form stating', 'The spectral sequences has vanishing higher differentials', and double parentheses in 'HS^n(N,F))'.","section":"Throughout"}],"recommendation":"reject","confidential_remarks":"The paper contains a demonstrable false computation in its main example (Section 4), and the central comparison spectral sequences are not actually proved; the reliance on the author's own preprint [1] for 'word by word' transfers is not acceptable for a research article. The Hochschild–Serre-type spectral sequence construction in Sections 1–3 may be salvageable, but the current manuscript requires major corrections and a full proof of the comparison theorems."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The real novelty here is the Hochschild-Serre-type spectral sequence for commutative cohomology in characteristic 2, built in Sections 1–3. That construction is standard in outline and probably correct: the filtration, the E0 identification, and the d1 computation all follow the classical Hochschild-Serre template. If that were all the paper claimed, I would be happy to send it to a referee.\n\nThe problem is Section 4. The paper's computation of HS^n(N,F) for the two-dimensional algebra N with [f,f]=e is wrong. A direct computation from Definition 2 gives HS^1(N,F)=F: the kernel of d1 is spanned by the cochain vanishing on e. The paper's formula predicts 0. That is a load-bearing contradiction, because this computation is the advertised illustration of the spectral sequence in action. I also checked the stress-test's stronger claim that HS^2(N,F)=F; that one does not survive. The kernel of d2 is exactly the image of d1, so HS^2=0, matching the paper's prediction in that degree. But the HS^1 failure is enough to sink the section.\n\nA second real weakness is in Section 5. Theorems 5.1–5.3 are asserted by saying the arguments of [1] go through word by word, but they are not proved here. That would be acceptable for a small adaptation; it is not acceptable for three spectral sequences that mediate between cohomology theories with different symmetry conditions. The filtration compatibility in Section 5.5 is especially delicate: the claim that d preserves symmetry in the first p entries is precisely the kind of thing that can fail without the alternating degeneration the paper sets aside. The author should either prove these transfer arguments or state them as conjectures.\n\nWhat the paper does well: the abstract construction of the HS spectral sequence is real, and the comparison spectral sequences are a sensible ambition. The reliance on the author's earlier work for proof templates is heavy, but the underlying comparison sequence is Pirashvili's, so this is not a hidden circularity.\n\nWho should read this? Specialists in characteristic-2 Lie theory and commutative cohomology. It should not be rejected out of hand, because the machinery might be salvageable and genuinely useful. But as submitted, the central computational result is false, and the comparison theorems need a real proof. I would send it to a serious referee with the expectation of major revision, not desk-reject it.","headline":"The Hochschild-Serre construction for commutative cohomology is plausible, but the paper's flagship computation in Section 4 is plainly wrong and the comparison spectral sequences are deferred rather than proved.","tokens_in":12316,"tokens_out":5749,"would_cite":false,"duration_ms":63448,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["17A32","17B56"],"pacs":[],"model":"deepseek-v4-flash","headline":"New spectral sequences compute commutative Lie algebra cohomology in characteristic 2.","keywords":["commutative cohomology","characteristic 2","spectral sequences","Leibniz cohomology","ordinary Lie algebra cohomology","commutative Lie algebras","relative cohomology","cohomology vanishing"],"falsifier":"On the two-dimensional algebra $N$ with $[f,f]=e$, take a cochain that is symmetric in its first two entries and compute its coboundary term by term; if the coboundary is not symmetric in those two entries, the Section 5.5 filtration is incompatible with the differential and Theorem 5.3 cannot hold as stated.","tokens_in":11244,"feed_emoji":"🧮","tokens_out":13252,"duration_ms":112876,"temperature":0.7,"pith_summary":"The paper builds computational tools for commutative cohomology of commutative Lie algebras over fields of characteristic 2, where the classification of finite-dimensional simple Lie algebras remains open. The first tool is an ideal-filtration spectral sequence: for a short exact sequence $0 \\to h \\to g \\to q \\to 0$ of commutative Lie algebras, the $E_2$ term is $HS^p(q, HS^q(h,M))$ and the sequence converges to $HS^{p+q}(g,M)$. The second class of tools are comparison spectral sequences whose $E_2$ terms pair relative cohomology classes with ordinary, commutative, or Leibniz cohomology, thereby relating the three theories. Two two-dimensional examples are worked out, and a vanishing theorem is proved for cohomology with values in the nontrivial one-dimensional module. These tools are intended to support future computations in the characteristic-2 classification program.","feed_headline":"New spectral sequences compute commutative cohomology in char 2","feed_subtitle":"They split cohomology into ideal and quotient pieces and link it to Leibniz cohomology.","key_machinery":"The machine that carries the first construction is a descending filtration $F^pCS^n(g,M)$ consisting of cochains that vanish whenever $n-p+1$ of their arguments lie in $h$. On the associated graded pieces, the $d_0$ differential identifies with the cohomology differential of $h$ with coefficients in $\\operatorname{Hom}(S^p(g/h),M)$, and the $d_1$ differential with the differential of the quotient $q$; the identity $L_x=d\\circ i_x+i_x\\circ d$ makes the quotient action on $h$-cohomology well defined. For the comparison results, the machinery is the family of cochain complexes quotiented by alternating or symmetric degeneracy relations, filtered by partial alternation or partial symmetry in the first arguments; product maps from the ground-field cochain complex into the dual module $g^*$ produce a relative cochain complex whose cohomology, tensored with the target cohomology, forms the stated $E_2$ terms.","core_discovery":"The paper's central claim is that commutative cohomology in characteristic 2 has the same structural machinery as classical Lie algebra cohomology. In the ideal case, the filtration by number of arguments lying in an ideal $h$ yields a convergent spectral sequence with $E_2^{p,q}=HS^p(q,HS^q(h,M))$, so the cohomology of $g$ is computed from cohomology of the quotient with coefficients in cohomology of the ideal. For comparing theories, the paper asserts three spectral sequences with $E_2$-terms $HR^p_{\\Lambda}(g)\\otimes HL^q(g,M_s)$, $HR^p_{\\Lambda,S}(g)\\otimes HS^q(g,M)$, and $HR^p_S(g)\\otimes HL^q(g,M_s)$, converging to the corresponding relative cohomology groups; the last one is stated for arbitrary commutative Lie algebras. The applications include a vanishing theorem for a one-dimensional ideal acting nontrivially on $F_1$, and computations for two two-dimensional algebras in which $HS^n(g,F)$ is $F^{n+1}$ when $4\\mid n$ and zero otherwise.","pith_inferences":["A testable extension of the paper's comparison claim is to run the Section 5.5 filtration on any symmetric Leibniz algebra rather than only commutative ones, since the stated compatibility argument uses only symmetry of the bracket.","The period-4 pattern found in the two examples suggests a conjecture, not stated by the author, that other commutative Lie algebras with one-dimensional center may exhibit the same periodic vanishing behavior.","If the comparison spectral sequence with $E_2=HR^p_S(g)\\otimes HL^q(g,M_s)$ holds, it makes the relative classes $HR^p_S(g)$ a computable bridge: knowing Leibniz cohomology and the relative cohomology determines commutative cohomology, which the paper does not spell out as a formula."],"forward_implications":["For a commutative Lie algebra $g$ with a one-dimensional ideal $h$ acting nontrivially on $F_1$, $HS^\\bullet(g,F_1)=0$; for Lie algebras this forces $HL^\\bullet(g,F_1)=H^\\bullet(g,F_1)=0$.","Vanishing of ordinary Lie algebra cohomology in degrees up to $n$ forces the same vanishing for commutative and Leibniz cohomology in those degrees, with isomorphisms in degrees $n+1$ and $n+2$; the same transfer holds from commutative to Leibniz cohomology.","The two-dimensional algebras $N$ and $a$ both satisfy $HS^n(g,F)=F^{n+1}$ when $4\\mid n$ and vanish otherwise, so their commutative cohomology is periodic of period 4.","The $E_2$ formula reduces computations of $HS^\\bullet(g,M)$ to the cohomology of an ideal and the cohomology of the quotient, so a full computation can be assembled in two smaller steps."],"supporting_citations":[{"why":"Supplies the Section 2 comparison-spectral-sequence arguments that the paper invokes for Theorems 5.1–5.3.","marker":"[1]"},{"why":"Provides the classical ideal-filtration construction adapted to commutative cohomology in Sections 2–3.","marker":"[2]"},{"why":"Introduces commutative cohomology for commutative Lie algebras in characteristic 2 and lists the questions answered here.","marker":"[5]"},{"why":"Gives the original comparison spectral sequence between ordinary and Leibniz cohomology that the three comparison results adapt.","marker":"[6]"}],"fun_headline_variants":["Split ideal, link Leibniz: spectral sequences in char 2","Ideal splitting and Leibniz links for char-2 commutative cohomology","Spectral sequences split char-2 commutative cohomology by ideals","Hochschild-Serre and comparison sequences for char-2 commutative cohomology"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"Everything in the comparison part rests on the assertion, stated but not proved in Section 5.5, that the Leibniz differential preserves symmetry in the first $p$ entries of a relative cochain; if that compatibility fails, Theorems 5.3, 6.3, and 6.4 lose their support.","fun_headline_variants_meta":{"raw":{"variants":["Split ideal, link Leibniz: spectral sequences in char 2","Ideal splitting and Leibniz links for char-2 commutative cohomology","Spectral sequences split char-2 commutative cohomology by ideals","Hochschild-Serre and comparison sequences for char-2 commutative cohomology"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001073,"raw_usage":{"total_tokens":4432,"prompt_tokens":822,"completion_tokens":3610,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":438,"completion_tokens_details":{"reasoning_tokens":3532}},"tokens_in":438,"tokens_out":3610,"duration_ms":27831,"temperature":1.0,"reasoning_tokens":3532,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T12:39:51.008926+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"On the two-dimensional algebra $N$ with $[f,f]=e$, take a cochain that is symmetric in its first two entries and compute its coboundary term by term; if the coboundary is not symmetric in those two entries, the Section 5.5 filtration is incompatible with the differential and Theorem 5.3 cannot hold as stated.","supporting_citations":[{"cited_title":"On Leibniz cohomology","cited_arxiv_id":"1902.06128","evidence_quote":"Supplies the Section 2 comparison-spectral-sequence arguments that the paper invokes for Theorems 5.1–5.3."},{"cited_title":"Hochschild and J-P","cited_arxiv_id":null,"evidence_quote":"Provides the classical ideal-filtration construction adapted to commutative cohomology in Sections 2–3."},{"cited_title":"Commutative Lie algebras and commutative cohomology in characteristic $2$","cited_arxiv_id":"1907.03690","evidence_quote":"Introduces commutative cohomology for commutative Lie algebras in characteristic 2 and lists the questions answered here."},{"cited_title":"Pirashvili: On Leibniz homology, Ann","cited_arxiv_id":null,"evidence_quote":"Gives the original comparison spectral sequence between ordinary and Leibniz cohomology that the three comparison results adapt."}],"review_version":1}