REVIEW 2 major objections 4 minor 6 references
Spectral Sequences For Commutative Lie Algebras
T0 review · 2 major / 4 minor · reviewed 2026-08-14 · deepseek-v4-flash
Pith's one-line read New spectral sequences compute commutative Lie algebra cohomology in characteristic 2.
desk verdict 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. read the letter →
The pith
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
The reading
What carries the argument
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.
What would settle it
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.
Extended reading notes
Core claim
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.
Load-bearing premise
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.
Editorial extensions
If this is right
- 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.
Reading between the lines
- 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.
Signed reviews
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
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.
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 (2)
- [Section 4, Example N] 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.
- [Sections 5.3–5.5, comparison spectral sequences] 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.
minor comments (4)
- [Section 5.2] 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.
- [Throughout] 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 4] 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.
- [Throughout] 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))'.
Circularity Check
No significant circularity: the spectral sequences are derived from explicit cochain complexes; reliance on author's prior work is self-citation but not a definitional reduction.
full rationale
I find no step in which a claimed prediction or first-principles result is equivalent to its own input by construction. The Hochschild-Serre-type spectral sequence in Sections 2-3 is built from a filtration on the explicitly defined commutative cochain complex, and the E2-term in Corollary 3.2 is derived from the differentials, not assumed. The Section 4 computations use that spectral sequence directly; even if the computation of HS^n(N,F) is challenged on mathematical grounds, an erroneous computation is not a circularity. The comparison spectral sequences in Section 5 are the only place where the paper leans on prior work: Theorems 5.1-5.3 are justified by saying the arguments of Section 2 of [1] go through word by word, and [1] has an overlapping author. This is genuine self-citation, and it is load-bearing for the proofs, so it prevents a score of 0. However, the underlying construction is explicitly attributed to Pirashvili [6] in the Lie-to-Leibniz case, and the theorems are new variants rather than restatements of the cited input. The one-sentence compatibility check in Section 5.5 is a possible gap in the proof of Theorem 5.3, but a missing or terse justification is a correctness risk, not a circular reduction. No parameter is fitted and renamed as a prediction, and no uniqueness theorem is imported from the author's own prior work to force a choice. The paper is therefore not circular in the sense targeted by this review.
Assumptions & free parameters
assumptions (4)
- domain assumption The ground field is of characteristic 2 and the bracket is symmetric; all cohomology complexes are defined over this field.
- standard math The filtration F^p in Section 2 is the standard Hochschild-Serre filtration, read as vanishing when at least n-p+1 elements lie in h, and the spectral sequence of a bounded filtration converges strongly.
- ad hoc to paper The arguments of Section 2 of [1] transfer verbatim to the three comparison filtrations in Theorems 5.1-5.3.
- standard math The Cartan relation Lx = d ix + ix d from Proposition 1.1 implies h acts trivially on its own cohomology, justifying the d1 identification in Lemma 3.1.
Cite this review
Pith. "Pith review of Spectral Sequences For Commutative Lie Algebras." pith.science (2026). https://pith.science/paper/HOTYITEJ
@misc{pith2026190806764,
author = {Pith},
title = {Pith review of: Spectral Sequences For Commutative Lie Algebras},
year = {2026},
howpublished = {\url{https://pith.science/paper/HOTYITEJ}},
note = {Machine review of arXiv:1908.06764}
}
read the original abstract
We construct some spectral sequences as tools for computing commutative cohomology of commutative Lie algebras in characteristic 2. In a first part, we focus on a Hochschild-Serre-type spectral sequence, while in a second part we obtain comparison spectral sequences which mediate between Chevalley-Eilenberg-, commutative-and Leibniz cohomology. These methods are illustrated by a few computations.
Reference graph
Works this paper leans on
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[1]
J. Feldvoss, F. W agemann, On Leibniz cohomology, arXiv:1902.06128
work page Pith review arXiv 1902
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[2]
G. Hochschild and J-P. Serre: Cohomology of Lie algebras , Ann. Math. (2) 57 (1953), no. 3, 591–603. SPECTRAL SEQUENCES FOR COMMUTATIVE LIE ALGEBRAS 13
work page 1953
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[3]
Loday: Une version non commutative des alg` ebres d e Lie: les alg` ebres de Leibniz, Enseign
J.-L. Loday: Une version non commutative des alg` ebres d e Lie: les alg` ebres de Leibniz, Enseign. Math. (2) 39 (1993), no. 3-4, 269–293
work page 1993
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[4]
J.-L. Loday and T. Pirashvili: Universal enveloping alg ebras of Leibniz algebras and (co)homology, Math. Ann. 296 (1993), no. 1, 139–158
work page 1993
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[5]
Commutative Lie algebras and commutative cohomology in characteristic $2$
V. Lopatkin and P. Zusmanovich: Commutative Lie algebra s and commutative cohomology in characteristic 2, arXiv:1907.03690
work page Pith review arXiv 1907
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[6]
Pirashvili: On Leibniz homology, Ann
T. Pirashvili: On Leibniz homology, Ann. Inst. Fourier (Grenoble) 44 (1994), no. 2, 401–411. Laboratoire de math´ematiques Jean Leray, UMR 6629 du CNRS, Universit ´e de Nantes, 2, rue de la Houssini `ere, F-44322 Nantes Cedex 3, France E-mail address : wagemann@math.univ-nantes.fr
work page 1994
Reviewed August 14, 2026 · model on record in the stance chip above.
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