arxiv: 2604.15548 · v2 · submitted 2026-04-16 · 🧮 math.AT

Recognition: unknown

Algebraic redshift in the C₂-equivariant Adams spectral sequence

Paul Shick

Pith reviewed 2026-05-10 09:05 UTC · model grok-4.3

classification 🧮 math.AT

keywords algebraic redshiftC2-equivariantAdams spectral sequencev_n periodicityalgebraic Mahowald invariantSteenrod algebraequivariant homotopychromatic filtration

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The pith

Nonzero classes for powers of v_n in the C2-equivariant Ext groups imply nonzero classes for powers of v_{n-1} whose algebraic Mahowald invariants contain the v_n classes.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper aims to establish an algebraic version of redshift in the C2-equivariant Adams spectral sequence, where higher v_n-periodicity forces lower v_{n-1}-periodicity with linking invariants. It defines periodic and torsion classes in the Ext groups over subalgebras of the equivariant Steenrod algebra and proves a chromatic filtration showing that v_n-torsion implies v_k-torsion for smaller k. By promoting a classical splitting to the equivariant case, it defines algebraic Mahowald invariants and proves the implication between nonzero classes across levels. This matters because it provides a way to detect and relate periodic phenomena in equivariant stable homotopy groups through algebraic computations rather than direct geometric constructions.

Core claim

Whenever a class corresponding to a power of v_n is nonzero in Ext over A^{C2}(m), then the same power of v_{n-1} is nonzero in Ext over A^{C2}(m-1), and the algebraic Mahowald invariant of that lower class contains classes corresponding to the higher power. The paper also shows v_n-torsion classes are v_k-torsion for k less than n and extends these results to the real motivic setting.

What carries the argument

The algebraic Mahowald invariant M_m^{C2-alg} defined via the promoted Lin-Davis-Mahowald-Adams splitting of the Ext of the equivariant version of RP_{-∞}^∞, which links the Ext groups over consecutive subalgebras A^{C2}(m) and A^{C2}(m-1).

Load-bearing premise

The Lin-Davis-Mahowald-Adams splitting of Ext for the suitable RP version promotes to the C2-equivariant setting, allowing well-defined algebraic Mahowald invariants that contain the claimed classes.

What would settle it

A concrete calculation in the Ext groups over A^{C2}(m) and A^{C2}(m-1) where a nonzero v_n power class exists but the corresponding v_{n-1} power is zero or its Mahowald invariant does not contain any v_n class.

read the original abstract

We study $v_n$-periodic phenomena in $C_2$-equivariant stable homotopy through the lens of the $C_2$-equivariant Adams spectral sequence at the prime 2. In particular, we construct/detect certain classes related to powers of the $v_n$ generators of $\pi_*(BP)$ in the cohomology of certain finitely generated subalgebras $A^{C_2}(m)$ of the $C_2$-equivariant Steenrod algebra. We define the notion of classes in $\text{Ext}_{A^{C_2}}(\underline{H}^\star, \underline{H}^\star)$ being $v_n$-periodic or $v_n$-torsion and exhibit a chromatic filtration by showing that $v_n$-torsion classes are also $v_k$-torsion for $0\le k < n.$ We also promote the Lin-Davis-Mahowald-Adams splitting of Ext of the suitable version of ``$R P_{-\infty}^\infty$" to the $C_2$-equivariant setting and use this to define appropriate algebraic versions of Mahowald's root invariant. We establish that whenever a class corresponding to a power of $v_{n}$ is nonzero in $ \text{Ext}_{A^{C_2}(m)}(\underline{H}^\star, \underline{H}^\star),$ then the same power of $v_{n-1}$ is also nonzero in $ \text{Ext}_{A^{C_2}(m-1)}(\underline{H}^\star, \underline{H}^\star),$ and its algebraic Mahowald invariant $M_m^{C_2-alg}(v_{n-1}^{2^f}) \subset \text{Ext}_{A^{C_2}(m)}(\underline{H}^\star, \underline{H}^\star)$ contains class(es) corresponding to $v_n^{2^f}.$ Real motivic versions of these results hold as well.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The abstract sketches an algebraic redshift relation for v_n classes in the C2-equivariant Adams spectral sequence by promoting a classical splitting and defining equivariant Mahowald invariants, but supplies no details to check whether any of it works.

read the letter

The main claim is that whenever a power of v_n is nonzero in Ext over one of these subalgebras A^{C2}(m), the corresponding power of v_{n-1} is nonzero over A^{C2}(m-1) and its algebraic Mahowald invariant contains the v_n class. The paper also defines v_n-periodic and torsion classes in the equivariant Ext groups, exhibits a chromatic filtration on them, and states that real motivic versions hold. It promotes the Lin-Davis-Mahowald-Adams splitting of the appropriate RP spectrum to the C2 setting to make the invariants well-defined. These are presented as new constructions extending non-equivariant algebraic work on redshift and root invariants. That extension is the actual novelty here; the rest recycles known ideas into the equivariant Steenrod algebra. If the subalgebras and the promoted splitting behave as described, the redshift relation would give a clean way to organize periodic phenomena in C2-homotopy computations. The abstract itself is coherent and cites the right prior results without obvious internal contradictions. The limitation is that nothing is shown. There are no definitions of the subalgebras A^{C2}(m), no argument for why the classical splitting lifts, and no verification that the invariants contain the stated classes. Those are the load-bearing steps, so the central relation remains untested from what is available. No free parameters or invented entities appear in the outline, and the claims do not reduce to fitting. This is for readers already working in equivariant stable homotopy or chromatic methods who need algebraic models for periodicity. A specialist could get value from the definitions if they are carried out correctly in the full text. I would send it to referees. The outline is logically consistent with the literature it cites, and the topic is narrow enough that a couple of careful readers could check the constructions without too much effort.

Referee Report

1 major / 0 minor

Summary. The paper studies v_n-periodic phenomena in C_2-equivariant stable homotopy through the C_2-equivariant Adams spectral sequence at p=2. It introduces finitely generated subalgebras A^{C_2}(m) of the C_2-equivariant Steenrod algebra and detects classes corresponding to powers of the v_n generators of π_*(BP) in the associated Ext groups. The authors define v_n-periodic and v_n-torsion classes in Ext_{A^{C_2}}(H^*, H^*), exhibit a chromatic filtration in which v_n-torsion implies v_k-torsion for k < n, promote the Lin-Davis-Mahowald-Adams splitting of Ext(RP_{-∞}^∞) to the equivariant setting, and define algebraic Mahowald invariants. The central result asserts that if a class for v_n^{2^f} is nonzero in Ext_{A^{C_2}(m)}(H^*, H^*), then the corresponding power of v_{n-1} is nonzero in Ext_{A^{C_2}(m-1)}(H^*, H^*) and its algebraic Mahowald invariant contains classes for v_n^{2^f}. Real motivic analogues are also claimed.

Significance. If the constructions, definitions, and proofs hold, the work would provide a useful algebraic framework for analyzing chromatic redshift and periodicity in the C_2-equivariant and real motivic Adams spectral sequences, extending classical splittings and invariants to new contexts and potentially supporting explicit computations of equivariant homotopy groups.

major comments (1)

Abstract: The manuscript consists only of the abstract, with no definitions of the subalgebras A^{C_2}(m), no explicit construction of the promoted Lin-Davis-Mahowald-Adams splitting, and no derivation or verification of the redshift statement or the claimed containment M_m^{C_2-alg}(v_{n-1}^{2^f}) ⊃ classes for v_n^{2^f}. These steps are load-bearing for the central claims, and their absence prevents any assessment that the promoted splitting holds or that the algebraic invariants contain the stated classes.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their summary and for identifying the key load-bearing elements of the work. We acknowledge that the version under review consists only of the abstract and therefore lacks the explicit definitions, constructions, and derivations needed for a full assessment. We will expand the manuscript accordingly.

read point-by-point responses

Referee: Abstract: The manuscript consists only of the abstract, with no definitions of the subalgebras A^{C_2}(m), no explicit construction of the promoted Lin-Davis-Mahowald-Adams splitting, and no derivation or verification of the redshift statement or the claimed containment M_m^{C_2-alg}(v_{n-1}^{2^f}) ⊃ classes for v_n^{2^f}. These steps are load-bearing for the central claims, and their absence prevents any assessment that the promoted splitting holds or that the algebraic invariants contain the stated classes.

Authors: We agree that the current submission is limited to the abstract and does not contain the required definitions or proofs. In the revised manuscript we will supply: (i) the explicit definition of the finitely generated subalgebras A^{C_2}(m) inside the C_2-equivariant Steenrod algebra, (ii) the construction that promotes the Lin-Davis-Mahowald-Adams splitting to the equivariant setting, and (iii) the derivation of the algebraic redshift statement, including the containment M_m^{C_2-alg}(v_{n-1}^{2^f}) ⊃ classes for v_n^{2^f}. These additions will make the central claims verifiable. revision: yes

Circularity Check

0 steps flagged

No significant circularity; relies on extension of prior splitting

full rationale

Only the abstract is available, which describes defining v_n-periodicity and torsion in Ext groups over subalgebras A^{C2}(m), exhibiting a chromatic filtration by showing lower torsion, promoting the Lin-Davis-Mahowald-Adams splitting to the C2-equivariant case, and using it to define algebraic Mahowald invariants. The central redshift claim (nonzero v_n power implies nonzero v_{n-1} power with the invariant containing the higher class) is stated as following from these constructions. No equations, definitions, or derivations are provided that reduce any result to a fitted parameter, self-definition, or self-citation chain by construction. The promotion of the splitting is presented as an extension of external prior work rather than an internal loop. This is consistent with a low circularity score; the claims rest on independent constructions without visible reduction to inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review; the paper relies on standard background in stable homotopy theory, equivariant spectra, and the Adams spectral sequence, but no explicit free parameters, ad-hoc axioms, or invented entities are stated in the abstract.

pith-pipeline@v0.9.0 · 5629 in / 1395 out tokens · 86849 ms · 2026-05-10T09:05:33.399353+00:00 · methodology

Algebraic redshift in the C₂-equivariant Adams spectral sequence

Core claim

What carries the argument

Load-bearing premise

What would settle it

discussion (0)