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The cohomology of $\mathbb{R}$-motivic $\mathcal{A}(2)$

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Pith's one-line read This paper computes the tri-graded cohomology of R-motivic A(2), the E2-page of a hypothetical Adams spectral sequence for R-motivic modular forms, by running a rho-Bockstein spectral sequence that collapses at E11.

desk verdict A serious rho-Bockstein computation with a clean structural core; the 56-generator answer is plausible and well-constrained, but the completeness of the differential bookkeeping is not independently verifiable from the manuscript alone. read the letter →

arxiv 2509.11266 v2 pith:P6TEHJVC submitted 2025-09-14 math.AT

Konstantin Emming This is my paper · ORCID
classification math.AT MSC 14F4255S1055T15
keywords R-motivichomotopytheoryrho-BocksteinspectralsequencemotivicdualSteenrodalgebraA(2)cohomologyAdamstopologicalmodularformshiddenextensionsExtgroups
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

The paper claims that the rho-Bockstein spectral sequence for R-motivic A(2), the quotient of the R-motivic dual Steenrod algebra dual to the subalgebra generated by Sq^1, Sq^2, and Sq^4, collapses at page 11, with all non-zero differentials on indecomposables exactly those listed in table 7. If true, this determines the additive tri-graded cohomology Ext_{A(2)}(M2,M2): it has 56 indecomposables, listed in table 16. The rho-localized part is identified with a shifted copy of classical A(1) cohomology with τ^8 adjoined. This matters because that Ext group is the E2-page of the Adams spectral sequence for a hypothetical R-motivic modular forms spectrum, and would feed into computations of R-motivic and eventually classical stable homotopy groups.

What carries the argument

The rho-Bockstein spectral sequence, obtained by filtering the cobar complex by powers of the ideal generated by rho. Two tools drive it: rho-localization, which splits the localized Hopf algebroid and determines the rho-free part of the E∞-page explicitly; and the internal coweight s+f−w, a linear combination of degrees preserved by rho-multiplication and by Bockstein differentials, which splits the spectral sequence into smaller charts solvable by comparison with the rho-localized abutment. Hidden extensions on E∞ are analyzed using a short exact sequence relating R-motivic and C-motivic Ext.

What would settle it

A reader could test the collapse claim by independently re-running the internal-coweight check for coweight 5: the grey dots besides h1^5 must all be consumed by differentials, and there is only one pairing that respects the rho-localized abutment. Any alternative pairing, or any non-zero differential on an indecomposable not listed in table 7, would change the 56-generator answer. A direct check would be to verify that every element of table 16 is indecomposable in the abutment; if any is decomposable via a hidden extension not accounted for, the count is wrong.

Watch

Extended reading notes

Core claim

The central discovery is that the rho-Bockstein spectral sequence converging to Ext_{A(2)}(M2,M2) has no differentials beyond d10, giving E11 = E∞. All differentials on indecomposables are tabulated (table 7). Consequently, the cohomology has 56 indecomposables (table 16). The rho-localized part is isomorphic to a shifted classical A(1)_cl cohomology with τ^8 adjoined (corollary 3.5). Under the assumption that an R-motivic modular forms spectrum exists, this Ext group is the Adams E2-page, and the paper also computes the Adams d2-differentials on all indecomposables (table 21).

Load-bearing premise

The tabulation of differentials in tables 7 through 15 is complete; this completeness is checked against supplementary charts not included in the manuscript, presumes the C-motivic input from a cited theorem is complete, and relies on a lemma whose Bockstein computation is deferred to 'arguments similar to' a cited lemma — if any of those checks is wrong, the 56-generator answer changes.

Editorial extensions

If this is right

  • The rho-Bockstein spectral sequence collapses at E11, so all non-zero differentials are d1 through d10 and are exactly those listed in table 7.
  • Ext_{A(2)}(M2,M2) has 56 indecomposables, listed in table 16, fixing the additive tri-graded cohomology of R-motivic A(2).
  • The rho-localized cohomology is a shifted classical A(1)_cl cohomology with τ^8 adjoined, tightly constraining the rho-free part of the answer.
  • A hypothetical R-motivic modular forms spectrum would have Adams E2-page equal to this Ext group, and its Adams d2-differentials on indecomposables are those in table 21.
  • Hidden extensions such as τ^8·h1^4 = ρ^4τ^4P and τ^8·ρ^6g2 = ρ^14∆2 exhibit R-motivic multiplicative relations with no classical or C-motivic analog.

Reading between the lines

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

  • The jump from 16 generators in the C-motivic case to 56 here suggests that an R-motivic modular forms spectrum, if constructed, would have substantially more complicated low-stem homotopy; this is an editorial extrapolation, not a claim of the paper.
  • Table 21 is a testable prediction: any construction of an R-motivic modular forms spectrum with cohomology A//A(2) must reproduce those Adams d2-differentials.
  • The hidden extension τ^8·h1^4 = ρ^4τ^4P propagates into families of hidden multiplications; checking that these families remain consistent under multiplication by powers of g2 could serve as an independent machine verification of the full multiplicative structure.
  • The paper's differential tabulation can be independently checked by re-running the internal-coweight charts: in each coweight, every non-zero module must either be consumed by a differential or survive to the rho-localized abutment; any mismatch changes the final 56-generator count.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

4 major / 5 minor

Summary. The paper computes the tri-graded cohomology Ext_{A(2)}(M2, M2) of the R-motivic quotient algebra A(2), using a ρ-Bockstein spectral sequence whose E1-page is the C-motivic cohomology of A(2) with a freely adjoined ρ. The structural core consists of a ρ-localization computation (Prop. 3.3, Lemma 3.4, Cor. 3.5, Lemma 3.9), an internal-coweight method for organizing differentials, and a page-by-page differential calculation summarized in Table 7. The claimed outcome is that the spectral sequence collapses with E11 ≅ E∞, yielding 56 indecomposables in Ext_{A(2)}(M2, M2) (Table 16). The paper also computes selected hidden extensions and, under the hypothesis of an R-motivic modular forms spectrum, tabulates Adams d2-differentials on all indecomposables.

Significance. If correct, this is a substantial and useful computation: it determines the E2-page of an Adams spectral sequence for a hypothetical R-motivic modular forms spectrum, with potential consequences for R-motivic and classical stable stems. The paper has a clean structural core: the ρ-localized cohomology is derived from Voevodsky's presentation and the classical A(1) computation rather than assumed, and Lemma 3.9 is checkable. The explicit hidden extensions and the Adams d2 table are concrete, falsifiable predictions. However, the central differential tabulation is not independently verifiable from the manuscript alone: many proofs are delegated to charts stored in an external Zenodo repository and to unenumerated 'degree reasons' claims. Since the 56-generator answer and the collapse statement rest on that tabulation, the current level of verifiability is not commensurate with the strength of the claims.

major comments (4)
  1. [§5, Table 7; Prop. 5.57; Table 16] The completeness of Table 7 is load-bearing and is not established within the text. Proofs repeatedly state that 'for degree reasons' only certain indecomposables can support differentials, but the underlying E_r-page class inventories are not included; the reader is referred to the external Zenodo repository [Emm25]. Since E11 ≅ E∞ and the 56 indecomposables of Table 16 depend directly on Table 7, the main computation is not independently checkable. Please include the charts as supplementary material or provide a machine-readable certificate/script that verifies the degree-wise possible sources and targets at each page.
  2. [§3, Prop. 3.3; Table 1; Cor. 3.5] There is an internal grading inconsistency in the ρ-localization statement. Prop. 3.3 says a classical element of degree (s,f) is sent to an element of degree (2s+f, f, s+f). Applying this to classical h1, which has degree (1,1), would place its image in degree (3,1,2), but Cor. 3.5 and Table 1 list h1 as having degree (1,1,1); analogous discrepancies occur for h2, u, and g. As written, the degree-shift formula and the generator table cannot both be correct. This must be fixed or the conventions clarified, because the tri-graded answer is a central deliverable.
  3. [§3, Lemma 3.4] Lemma 3.4 is the key input to the ρ-localized cohomology computation, but its proof defers the essential x-Bockstein differentials to 'arguments similar to [DI17, Lemma 4.2]'. An explicit proof (or a precise statement of the cited result with all degrees and differentials) is needed. As written, the main constraint that drives the differential solving in §5 rests on an unstated computation.
  4. [§5.12, Prop. 5.57; §6] The collapse proof asserts that every E11 indecomposable represents a nonzero element of the ρ-localization, giving only a few examples and 'and so on'. This identification is not fully demonstrated in the text and is intertwined with the hidden-extension analysis of §6, which is explicitly partial: §6.3 treats only coweight 1 mod 8 completely, and the completeness claims in Props. 6.8, 6.9, and 6.11 are again justified by 'degree reasons'. To conclude E11 ≅ E∞, a targeted verification that every E11 class has nonzero ρ-localized image is required, independent of any unproved hidden-extension statements.
minor comments (5)
  1. [Table 16; Table 21] The row for ρ lists weight +1 in both tables, but elsewhere in the paper, including Table 1 and Section 2, ρ has weight −1. The tables should list ρ as (−1, 0, −1).
  2. [§2] The degree notation for ρ, τ, τ_k, and ξ_k uses ordered pairs, e.g. 'the degree of ρ is (−1,−1)'. It would be clearer to state explicitly that these pairs are (stem, weight), with Adams filtration handled separately, since Table 1 uses triples (s,f,w).
  3. [§3.10 vs §7.1] The notation '·' is used both to indicate products on E∞-pages (Notation 3.10) and to denote multiplication in Ext after resolving hidden extensions (Notation 7.1). This overloading can confuse the reader; consider using a different symbol for one of the two.
  4. [Lemma 6.5] In the exact sequence of Ext_{A(2)^R}-modules, the symbols coker(ρ) and ker(ρ) should be defined explicitly as the cokernel and kernel of multiplication by ρ in Ext_{A(2)^R}. Also, the module structure of Ext_{A(2)^C} over Ext_{A(2)^R} is used without being spelled out.
  5. [Table 7] Several table entries have spacing artifacts (e.g., 'τ 3de (31, 8, 15)' and 'τ 5h2 2'). These are presumably typesetting issues but should be cleaned before publication.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: the computation is constrained by external C-motivic/classical inputs and an in-paper ρ-localization theorem, not fitted to the 56-generator output.

full rationale

The derivation chain is not circular in the sense required by the review rules. The E1-page is the C-motivic cohomology Ext_{A(2)^C}(M2^C,M2^C)[ρ], imported from [Isa09, Thm 4.16], and the abutment is Ext_{A(2)}(M2,M2); neither is redefined in terms of the other. The ρ-localized target is computed in Proposition 3.3 and Corollary 3.5 from the Hopf algebroid presentation after inverting ρ and from Ravenel's classical Ext_{A(1)_cl} computation; it is not fitted to the final Table 16. The differential-solving in Section 5 uses that localization as a constraint: for example, Lemma 5.16 excludes d3(τ^3 n2)=τ^2ag because that would leave a ρ-torsion free class in a degree where Corollary 3.5 forbids one. This is constraint-solving against an independently derived object, not reverse-engineering. Lemma 3.4 defers to 'arguments similar to [DI17, Lemma 4.2]', but that is an external published computation, not a self-citation, and it is parameter-free with stated assumptions. The author's own [Emm25] charts are load-bearing in practice—the text says 'they are integral to our computation' (§1.3)—but they are computational data, not a theorem smuggled in as an axiom; their absence from the manuscript is a verifiability/correctness limitation, not a circular reduction. The internal discrepancies noted by the skeptic (ρ weight in Table 16 vs. Section 2 and Table 1, and the degree-shift formula in Proposition 3.3 vs. Table 1) are bookkeeping inconsistencies, not definitional circularities. No equation or fitted parameter is renamed as a prediction; no uniqueness theorem from the same authors is invoked to force a choice. The central claim therefore retains independent mathematical content.

Assumptions & free parameters 0 free parameters · 6 assumptions · 1 invented entities

The central claim rests on standard spectral sequence facts and three external computational inputs: Voevodsky's presentation of the R-motivic dual Steenrod algebra (section 2), Isaksen's complete computation of C-motivic A(2) cohomology used as the E1-page (section 5), and Ravenel's classical A(1)_cl computation (Cor 3.5). Lemma 3.4, needed for the ρ-localization, is a cited variant of [DI17, Lemma 4.2] with a two-sentence proof. No numerical parameters are fitted; the only hand choices are representational, e.g. the naming of E∞ indecomposables (section 5.13). Section 7 additionally assumes an unconstructed mmf_R, which the author flags. The computation is therefore anchored to external benchmarks rather than circular, but its verification substrate (charts) is the author's own deposit.

assumptions (6)
  • domain assumption Voevodsky presentation of the R-motivic dual Steenrod algebra: A ≅ M2[τ0,τ1,...,ξ0,ξ1,...] with τ_k^2 = τξ_{k+1} + ρτ_{k+1} + ρτ0ξ_{k+1}; A(2) is the quotient setting τ3 = τ4 = ... = 0, ξ1^4 = 0, ξ2^2 = 0.
    Used throughout section 2 as the algebraic starting point; cited to [Voe03], [Voe10], not re-derived.
  • domain assumption C-motivic A(2) cohomology Ext_{A(2)_C}(M2^C, M2^C) is exactly as computed by Isaksen [Isa09, Thm 4.16 and Table 7], and forms the entire E1-page input of the ρ-Bockstein spectral sequence.
    Invoked in section 3 (signature of the SS) and section 5 ('we will freely make use of [Isa09, Theorem 4.16]'). All differential targets are read off this page.
  • domain assumption Classical A(1)_cl cohomology has the presentation of Ravenel [Rav86, 3.1.25 Theorem]; its four generators map degree-shifted to h1, h2, u, g.
    Used in Corollary 3.5 to identify the ρ-localized cohomology of A(2).
  • domain assumption Lemma 3.4: the cohomology of (F2[t], F2[t,x]/(x^8)) is F2[t^8], with x-Bockstein differentials d1(t)=h0, d2(t^2)=h1, d4(t^4)=h2.
    Load-bearing for 3.3/3.5 and hence for the constraint driving all differential computations; the proof is a two-sentence sketch 'similar to [DI17, Lemma 4.2]'.
  • standard math Localization at ρ commutes with cohomology, and the ρ-Bockstein spectral sequence converges to Ext_{A(2)}(M2, M2).
    Standard exactness/spectral sequence facts, used in Props 3.2, 3.3 and throughout section 5.
  • ad hoc to paper A hypothetical R-motivic modular forms spectrum mmf_R exists with mod 2 cohomology A//A(2), making Ext_{A(2)}(M2,M2) the E2-page of its Adams spectral sequence.
    Assumed only in section 7; the author explicitly flags that mmf_R has not been constructed (section 1.1, Problem 1.1). All Adams d2 results are conditional on this.
invented entities (1)
  • mmf_R, hypothetical R-motivic modular forms spectrum
    purpose: Section 7 computes Adams d2-differentials for a hypothetical object whose mod 2 cohomology is A//A(2); it would give Ext_{A(2)}(M2,M2) its stated role as an Adams E2-page.
    No construction or falsifiable prediction is provided; the author states it does not exist yet (Problem 1.1). Section 7 is explicitly conditional on it.

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Pith. "Pith review of The cohomology of $\mathbb{R}$-motivic $\mathcal{A}(2)$." pith.science (2026). https://pith.science/paper/P6TEHJVC

@misc{pith2026250911266,
  author       = {Pith},
  title        = {Pith review of: The cohomology of $\mathbbR$-motivic $\mathcalA(2)$},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/P6TEHJVC}},
  note         = {Machine review of arXiv:2509.11266}
}
abstract

We compute the cohomology of the quotient algebra $\mathcal{A}(2)$ of the $\mathbb{R}$-motivic dual Steenrod algebra. We do so by running a $\rho$-Bockstein spectral sequence whose input is the cohomology of $\mathbb{C}$-motivic $\mathcal{A}(2)$. The purpose of our computation is that the cohomology of $\mathcal{A}(2)$ is the input to an Adams spectral sequence of a hypothetical $\mathbb{R}$-motivic modular forms spectrum. This Adams spectral sequence computes the homotopy groups of such an $\mathbb{R}$-motivic modular forms spectrum, which in turn can be used to make inferences about the homotopy groups of the $\mathbb{R}$-motivic sphere spectrum and eventually about the classical stable stems.

Figures

Figures reproduced from arXiv: 2509.11266 by the authors.

Figure 1
Figure 1. ρ-Bockstein E2-page ρ-free quotient, internal coweight 5 figure 1 the horizontal axis represents the stem s and the vertical axis represents the Adams filtration f. Each dot represents a free F2[ρ]-module with internal coweight 5. The gray [PITH_FULL_IMAGE:figures/full_fig_p017_1.png] view at source ↗
Figure 2
Figure 2. ρ-Bockstein E2-page ρ-free quotient, internal coweight 5, with differentials describes their behavior in the ρ-Bockstein spectral sequence. The arrow on h 5 1 means that [PITH_FULL_IMAGE:figures/full_fig_p018_2.png] view at source ↗

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