The Chow Ring of the 2-Sylow Subgroup of $\mathrm{GL}(4,2)$

Alexander Ziegler

REVIEW 3 major objections 4 minor 6 references

The Chow Ring of the 2-Sylow Subgroup of $\mathrm{GL}(4,2)$

T0 review · 3 major / 4 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read The mod-2 Chow ring of the 2-Sylow subgroup of GL(4,2) is generated by eight Chern classes with explicit relations in degrees 2, 3, 4, and 6.

desk verdict New mod 2 Chow ring computation for an order-64 group, with solid low-degree work but a real gap in the all-degrees injectivity argument (Lemma 3.2) that the final presentation depends on. read the letter →

arxiv 2506.06512 v1 pith:TCCGSPQX submitted 2025-06-06 math.AG

Alexander Ziegler This is my paper

classification math.AG MSC 14C1514F4320C1555R40

keywords Chowringclassifyingspacemotivicétalefinitegroupcohomologycycleclassmapgamma-filtrationChernclasses2-Sylowsubgroup

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 proves that the mod-2 Chow ring of the motivic étale classifying space of Syl2(GL(4,2)), a group of order 64, has a completely explicit presentation: it is generated by the Chern classes of three named representations, with relations listed in degrees 2, 3, 4, and 6. The proof runs through the injectivity of the mod-2 cycle class map from Chow theory to ordinary mod-2 group cohomology; once that map is injective, the Chow ring is exactly the subring of cohomology generated by Chern classes, and all relations are read off from the known cohomology presentation. This matters because Chow rings of classifying spaces are hard to compute even for small finite groups, and the paper lays out an automated strategy, based on the gamma-filtration on representation rings and on detecting algebraic cycles on elementary abelian subgroups, that can be reused for other p-groups. If correct, the computation supplies a concrete, small example where cohomology enforces all algebraic relations.

What carries the argument

The engine is the gamma-filtration on the complex representation ring, filtered by ideals generated by the operations γ^i(x) = λ^i(x + i − 1), together with the associated graded pieces that feed into Chow ring computations via the motivic Atiyah-Hirzebruch spectral sequence. Universal polynomials take over the symbolic work of expanding Chern classes of tensor products and exterior powers, and the detection theorem from [Tot14] reduces high-degree questions to elementary abelian subgroups. The regularity bound from [Tot14] ensures the generating set is complete in the stated degrees, while Lemma 3.2 converts a no-nilpotents condition on centralizer Chow rings into global injectivity of the cycle class map.

What would settle it

Compute CH*(B C_G(V))/2 explicitly for each noncentral elementary abelian subgroup listed in the proof of Theorem 8.1, namely the copies of L, C2 × Syl2(GL(3,2)), and elementary abelian groups, and look for any non-zero positive-degree nilpotent class; the first such class would violate the hypothesis of Lemma 3.2 and invalidate the global injectivity theorem.

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Extended reading notes

Core claim

The central claim is a complete presentation of CH*(BG)/2, where G = Syl2(GL(4,2)): the ring is the F2-algebra generated by eight Chern classes c1(A), c2(A), c1(B), c2(B), c1(C), c2(C), c3(C), c4(C), subject to an explicit list of relations in degrees 2, 3, 4, and 6. The load-bearing assertion behind the presentation is Theorem 8.1, which states that the mod-2 cycle class map CH*(BG)/2 → $H^{{2*}}$(BG, F2) is injective. Low degrees are computed directly through the gamma-filtration and universal polynomials; injectivity in degrees 2 and 3 is checked by restriction to four copies of Syl2(GL(3,2)) and related subgroups; then a general detection lemma extends injectivity to all degrees because every noncentral elementary abelian subgroup of G has a centralizer whose mod-2 Chow ring contains no nilpotents. Once injectivity is known, the Chow ring is the preimage of the Chern-class subring inside the known F2-cohomology ring, and that preimage is computed to be the listed presentation.

Load-bearing premise

The proof depends on the computer-checked assertion that for every noncentral elementary abelian subgroup V of G, the Chow ring of the centralizer C_G(V) contains no nilpotent elements; if that asserted fact fails, the injectivity argument collapses.

Editorial extensions

If this is right

Cohomology determines this Chow ring: every algebraic relation in CH*(BG)/2 is already forced by relations in ordinary mod-2 group cohomology.
The same automated pipeline should produce explicit Chow-ring presentations for other small p-groups that have a faithful representation of small degree and nilpotent-free centralizer Chow rings.
Truncating Totaro's approximation of BG at finite index yields explicit quasi-projective varieties whose Chow groups agree with the computed ring up to a chosen degree.
The degree-6 relation closes the presentation: no new generators appear above degree 4, so every graded piece of CH*(BG)/2 is finite-dimensional and explicitly known.

Reading between the lines

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

Beyond the paper: a natural test is to run the same gamma-filtration pipeline on the other 64-element 2-groups or on Syl2(GL(5,2)); the first group where the no-nilpotents condition fails would show exactly where this generation strategy stops.
If injectivity of the cycle class map ever fails, the Chern-class subring of cohomology still gives an upper bound, and the discrepancy would measure exotic algebraic cycles invisible to cohomology; this paper shows that discrepancy is zero for G.
Because the no-nilpotents verification is delegated to a computer check not printed in detail, the theorem is contingent on those calculations; an independent hand-check of the centralizer Chow rings listed in the proof of Theorem 8.1 would be a concrete way to confirm the main result.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Dear Colleague,

The main thing you should know: this is the first explicit computation of CH*(BG)/2 for a genuine degree-4 finite 2-group, and the low-degree part is genuinely new and mostly careful. But the central structural claim, Theorem 8.1 (injectivity of the cycle class map in all degrees), is not actually proven, because Lemma 3.2, which is the bridge from degrees ≤3 to all degrees, has a flawed proof.

The paper does well in the low degrees. It computes CH^1, CH^2, and CH^3 in detail, gives explicit restriction maps to four copies of Syl2(GL(3,2)), and checks injectivity of the cycle class map in those degrees. The final presentation in Theorem 8.9 is plausible as the correct answer, and it reproduces Totaro's earlier computation on the restriction to H, which is a genuine external consistency check.

The soft spot is Theorem 8.1. Its proof leans on Lemma 3.2, which asserts injectivity in degrees > n−c assuming centralizers have no nilpotents in their mod p Chow rings. The lemma's proof does not work. In CH*(BV)/p ⊗ CH*(BC_G(V))/p, an element x ⊗ r is not nilpotent just because x has positive degree; it is nilpotent only if r is nilpotent. The paper effectively conflates 'positive degree' with 'nilpotent' in a polynomial ring. The subsequent truncation step for central V also misfires: for k > n−c, the class σ is killed by the truncation CH^{≤n−c}, so the claimed nonzero image does not follow. This is not a minor gap; it is the only argument that takes injectivity from degree ≤3 to all degrees.

There is also a reproducibility issue: the list of centralizers and the 'no nilpotents' checks are asserted from GAP, and the final preimage computation is asserted from Sage, without code or data.

My recommendation: send it to a referee, but the referee should be asked to focus on Lemma 3.2 and to demand either a correct proof or a different argument for all-degrees injectivity. The low-degree computations and the overall strategy are worth preserving. If the gap cannot be fixed, the paper would still be publishable as a computation up to degree 3 with the full presentation stated as a conjecture, but not as a proven theorem.

Referee Report

3 major / 4 minor

Summary. The paper computes the mod 2 Chow ring of the motivic étale classifying space of the 2-Sylow subgroup of GL(4,2), presenting it as an F2-algebra generated by Chern classes of three 2-dimensional representations and one 4-dimensional representation, subject to explicit relations in degrees 2, 3, 4, and 6. The computational strategy follows Totaro's methods: use the γ-filtration and geometric filtration on representation rings, restriction to subgroups isomorphic to Syl2(GL(3,2)) and other centralizers, and the cycle class map to F2-group cohomology. The key structural claim is Theorem 8.1, that the mod 2 cycle class map is injective in all degrees, which together with a known presentation of H*(BG,F2) yields the Chow ring presentation.

Significance. If correct, the paper provides a nontrivial new computation of the mod 2 Chow ring of a classifying space for a group of order 64, and it demonstrates an automated, partially GAP/SAGE-assisted version of Totaro's machinery that could be applied to further examples. The paper includes explicit matrices, restriction maps, and generators/relations, and it checks consistency with Totaro's earlier computation of CH*(BSyl2(GL(3,2)))/2. However, the validity of the central result hinges on a lemma whose proof is seriously flawed, so the significance is conditional on a correct replacement argument for that lemma.

major comments (3)

[Section 3, Lemma 3.2] The proof of Lemma 3.2 contains a false inference: from the fact that an element Σ a_i ⊗ b_i in CH*(BV)/p ⊗ CH*(BC_G(V))/p is nilpotent, and that CH*(BV)/p is a polynomial ring, the author concludes that the b_i paired with positive-degree a_i must be nilpotent. This is incorrect: in F_2[x] ⊗ R with R reduced, the element x ⊗ r is not nilpotent unless r is nilpotent. Positive degree in the polynomial factor does not imply nilpotence of the tensor element. Consequently the deduction that V must be central is unjustified, and the connection to the no-nilpotents assumption on CH*(BC_G(V))/p fails.
[Section 3, Lemma 3.2] The final step of the proof of Lemma 3.2 is also not coherent: for an element σ of degree k > n−c, the quotient CH^{≤n−c}(BG)/p vanishes in degree k, so the claimed nonzero image of σ in CH^{≤n−c}(BG) cannot occur. The argument that the central case violates the original assumptions therefore does not produce a contradiction. In sum, Lemma 3.2 does not establish injectivity of the cycle class map in degrees above n−c.
[Section 8, proof of Theorem 8.1] Theorem 8.1 relies on Lemma 3.2 to pass from the checked injectivity in degrees 1, 2, and 3 (Lemmas 8.4–8.6) to all degrees. Since Lemma 3.2 is unsupported, the all-degrees injectivity of the cycle class map is not proved. Moreover, the no-nilpotents condition for the centralizers (copies of L, C2 × Syl2(GL(3,2)), and elementary abelian groups) is only asserted as 'checked using [GAP4]' with no details, and the proof of Lemma 3.2 gives no way to leverage such a check.

minor comments (4)

[Theorem 2.5 proof] The proof text contains a stray 'This+' instead of 'This is proven'.
[Notation throughout] The paper switches between F_p and Fp, and between Syl_p and Sylp; consistent notation would improve readability.
[Section 7, first paragraph] The phrase 'some index 2 subgroups L ⩽ Syl2(GL(4,2))' in the title of Section 7 is slightly misleading because L is one specific subgroup; the plural could be clarified.
[Appendix F.1] In the restriction of c4(ψ) to C3,G_2, the term X2_1 X2_2 X4_3 appears twice and may contain a typo; please check the arithmetic.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity; the main computation is a theorem-driven consequence of injectivity and external cohomology data.

full rationale

The paper derives the presentation of CH*(BG)/2 from Theorem 8.1, which states that the mod 2 cycle class map is injective. The proof of Theorem 8.1 is not obtained by fitting the Chow ring to the cohomology ring; the low-degree injectivity is established by explicit restriction arguments to the subgroups H0, H∞, I0, I∞ and C2^2, using the previously computed Chow rings of Syl2(GL(3,2)) and L. These prior computations are cross-checked against two independent external sources: Totaro's earlier computation of CH*(BH)/2 and Green--King's cohomology presentations. The final presentation in Theorem 8.9 is formally the preimage of the cohomology relations under the injective cycle class map, together with the generator bound supplied by Totaro's regularity theorem; this is a direct consequence of the proven injectivity, not an input renamed as a prediction. The use of the author's own SAGE package and GAP is computational tooling, and the main structural inputs are the external theorems of Totaro and the external cohomology data of Green--King; these are not circular citations. The no-nilpotents assumption in Lemma 3.2 is a stated hypothesis about centralizers, not a reformulation of the target injectivity, and the paper asserts it is checked by GAP; whether that check is adequately documented is a verification concern, not a circularity. The proof of Lemma 3.2 itself contains an inference that appears mathematically questionable, but a proof gap is not a circular dependence. Overall, the derivation chain is self-contained against external benchmarks and no step reduces by construction to its own input.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

The paper does not introduce new entities, particles, forces, or ad hoc parameters: it is a pure algebra computation with no free parameters fitted to data. The axioms listed are standard background theorems, the external cohomology presentations, the computed representation rings, and the two computational group-theoretic facts that are delegated to GAP without proof. The no-nilpotents condition for centralizers is the only axiom that is explicitly load-bearing and not justified in the text; it is an ad hoc computational input needed for Lemma 3.2.

assumptions (6)

standard math Theorems from [Tot14] (detection theorem 3.1, regularity 3.5, Atiyah-Hirzebruch spectral sequence, and module-generation bound) hold as stated for the motivic etale classifying space.
The entire computation uses Totaro's theorems, which are published and proven in the cited book. They are not re-derived in this paper.
domain assumption The group cohomology presentations for H, L, and G are correctly taken from [GK15] and are complete.
The paper relies on Green-King's data for H^*(BH, F2) etc. It is external and standard, but not machine-checked in this paper.
domain assumption The representation rings and exterior powers computed in the appendices for Syl_p(GL(3,p)), the index-p subgroup L, and Syl_p(GL(4,p)) are correct and complete.
These are derived in the paper via characters and orthogonality, but for p=2 they are also verified by GAP. The paper claims the characters are pairwise orthogonal and number matches conjugacy classes; this is a rigorous argument, but the lengthy computations are not fully reproduced in the main text.
domain assumption The classification of centralizers of non-central elementary abelian subgroups of G (copies of L, C2 x Syl2(GL(3,2)), or elementary abelian) is correct as checked with [GAP4].
This is used in the proof of Theorem 8.1 to apply Lemma 3.2. The paper states it as 'can be checked using [GAP4]' without providing the computation details.
ad hoc to paper The claim that all Chow rings of centralizers of non-central elementary abelian subgroups contain no nilpotents holds.
This is the weakest assumption identified. It is stated in the proof of Theorem 8.1 as a fact 'as can be checked using [GAP4]' but no derivation, argument, or data is provided. The lemma requires this no-nilpotents condition to conclude injectivity of the cycle class map in all degrees. For L, the paper shows CH*(BL)/2 is a subring of cohomology (hence no nilpotents in positive degrees because cohomology of this group has no nilpotents), but for the direct product C2 x Syl2(GL(3,2)) the no-nilpotents property is not justified in the text.
standard math The motivic power operation beta P1 and the identification of its image as the kernel of CH^3(BG) to gr^3_geom R(G) (Lemma 4.11) is applied correctly, with the size of H^3(BG, Z) giving an upper bound on the kernel.
This is a theorem quoted from [Tot14] and used in Lemma 8.6. The computation that H^3(BG, Z) is (Z/2)^4 is attributed to [Ell24] (HAP), an external computational package.

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Pith. "Pith review of The Chow Ring of the 2-Sylow Subgroup of $\mathrm{GL}(4,2)$." pith.science (2026). https://pith.science/paper/TCCGSPQX

@misc{pith2026250606512,
  author       = {Pith},
  title        = {Pith review of: The Chow Ring of the 2-Sylow Subgroup of $\mathrmGL(4,2)$},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/TCCGSPQX}},
  note         = {Machine review of arXiv:2506.06512}
}

abstract

This paper provides a computation of the mod 2 Chow ring of the motivic \'etale classifying space of the finite group $\mathrm{Syl}_2(\mathrm{GL}(4,2))$. It outlines a general computation strategy, adapted from work by Burt Totaro, that has been largely automated by the author. This strategy can be used to compute more examples of Chow rings of motivic \'etale classifying spaces quickly.

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Reference graph

Works this paper leans on

6 extracted references · 5 canonical work pages

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Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.

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Graded Character Rings of Finite Groups

[Che20] Béatrice I. Chetard. “Graded Character Rings of Finite Groups”. In: Journal of Algebra 549 (2020), pp. 291–318. ISSN : 0021-8693. DOI: https : / / doi . org / 10 . 1016 / j . jalgebra . 2019.11.041. [Ell24] Graham Ellis. HAP, Homological Algebra Programming, Version 1.66. https://gap-packages. github.io/hap https://gap-packages.github.io/hap. GAP package

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