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A homological approach to chromatic complexity of algebraic K-theory

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Pith's one-line read The paper proves that topological periodic cyclic homology raises the chromatic height of every Thom spectrum y(n) by one.

desk verdict A genuinely new TP red-shift family at all heights, but the main theorem's bridge from continuous to ordinary Morava K-theory rests on an HF_p-nilpotent completeness hypothesis the paper never verifies. read the letter →

arxiv 1908.09164 v3 pith:L6XG52VN submitted 2019-08-24 math.AT math.KT

Gabriel Angelini-Knoll , J.D. Quigley This is my paper · ORCID
classification math.ATmath.KT MSC 55P4255P4355T9919D55
keywords chromaticred-shiftconjecturetopologicalperiodiccyclichomologyMoravaK-theoryThomspectray(n)homologicalTatespectralsequenceMargolisalgebraic
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 studies the chromatic complexity of algebraic K-theory through its approximations, topological periodic cyclic homology and topological negative cyclic homology. For the family of Thom spectra y(n) that interpolate between the sphere spectrum and the mod-2 Eilenberg–MacLane spectrum, it proves that the m-th Morava K-theory of TP(y(n)) vanishes for every 1≤m≤n, so TP raises the chromatic height by at least one at every level of the family. This is intended as uniform evidence for the chromatic red-shift conjecture, stated in terms of Morava K-theory vanishing rather than finite-presentation height. The method is homological: a Tate spectral sequence computes the continuous homology of TP(y(n)), Margolis homology detects the vanishing, and a comparison step passes from continuous to ordinary Morava K-theory.

What carries the argument

The load-bearing machinery is the homological Tate spectral sequence, which computes the continuous homology H^c_*(TP(R)) = lim_i H_*(TP(R)[i]) from the filtration defining it. Its E2-page is P($t^{{±1}}$) ⊗ H_*(THH(R)), and its differentials are governed by the circle action; for R=y(n), those differentials are evaluated using the spectral sequence computing H_*(THH(y(n))) and a precise description of the map THH(y(n))→THH(HF2). The key identification is H^c_*(TP(y(n))) ≅ P($t^{{±1}}$) ⊗ H_*(z(n)/v_n) as comodules over the subalgebra generated by the Milnor primitives, which transfers the problem to a spectrum whose Margolis homology is computed by elementary chain complexes. The localized Adams spectral sequence then converts Margolis homology vanishing into continuous Morava K-theory vanishing, and a theorem on commuting unbounded homotopy limits with Morava K-theory converts that into ordinary vanishing.

What would settle it

Test the missing hypothesis directly: for each n and i, check whether TP(y(n))[i] is nilpotently complete with respect to HF_2 (for instance by computing the relevant completion map or $lim^{1}$ term). A single failure would remove the bridge from continuous to ordinary Morava K-theory, leaving Theorem 1.2 without its final step; alternatively, compute K(m)_*(TP(y(n))) for m=n in a small case such as n=1 with an independent method and look for a nonvanishing class.

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Extended reading notes

Core claim

The central discovery is a computation of the continuous homology of TP(y(n)): it is isomorphic, as a comodule over the subalgebra generated by Milnor primitives, to P($t^{{±1}}$) ⊗ H_*(z(n)/v_n), where z(n) is an integral analog of y(n) and z(n)/v_n is the cofiber of a v_n self-map. Because the Margolis homology of z(n)/v_n vanishes for Q_1,...,Q_n, the localized Adams spectral sequence gives K(m)^c_*(TP(y(n))) = 0 for 1≤m≤n. A technical comparison then upgrades this to ordinary K(m)_*(TP(y(n))) = 0, proving the main theorem. The same machinery yields vanishing for relative topological cyclic homology and for relative algebraic K-theory K(y(n), HF2) in the range 0≤m≤n−1, and it reveals a nonvanishing obstruction in the top continuous Morava K-theory of TC^-(y(n)) that makes TP look like the better red-shift detector.

Load-bearing premise

The argument that continuous vanishing of Morava K-theory implies ordinary vanishing depends on each spectrum TP(y(n))[i] being complete with respect to mod-2 homology in a strong technical sense; the paper never proves this, so if one of these completeness conditions fails, the main theorem does not follow from the given proof.

Editorial extensions

If this is right

  • If correct, the main theorem makes TP(y(n)) K(m)-acyclic for every 1≤m≤n, so its chromatic complexity is at least n+1 while y(n) itself has type n: an explicit height shift at every level of the family.
  • Relative algebraic K-theory K(y(n), HF2) is K(m)-acyclic for 0≤m≤n−1, so algebraic K-theory of this family at least preserves chromatic complexity.
  • The same method proves vanishing for relative topological cyclic homology TC(y(n), HF2) in the range 0≤m≤n−1, via the fiber sequence relating TC, TC^-, and TP together with the standard equivalence between relative TC and relative K-theory.
  • Negative cyclic homology is predicted not to shift in general: for n=1 the continuous K(1)-homology of TC^-(y(1)) is nonzero, so TC^- can fail to raise height even where TP succeeds.
  • The theorem gives evidence for the chromatic red-shift conjecture in a setting where the input spectrum is not finitely presented, so pure fp height cannot even be defined.

Reading between the lines

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

  • If the missing nilpotent-completeness condition for TP(y(n))[i] is verified, the same continuous-to-ordinary comparison would likely apply to any E1 ring spectrum whose topological Hochschild homology is understood as a comodule algebra.
  • Conjecture 4.11, that TP(y(n)) is equivalent to the Tate construction on z(n)/v_n after completion, would turn TP computations into Tate constructions of finite-type spectra; proving it would give a general red-shift machine for inputs without finite presentation.
  • The paper's TC^- obstruction at the top continuous Morava K-theory suggests an explicit test: compute the analogous T0 contribution for higher n and see whether it always obstructs K(n)-vanishing of TC^-(y(n)).
  • The z(n) spectra and their v_n-cofibers carry exactly the information needed for these TP calculations; they may be useful as integral building blocks for other chromatic-shift questions.
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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

3 major / 4 minor

Summary. This paper studies the chromatic complexity—defined via the vanishing range of Morava K-theory—of the trace invariants TP, TC^-, TC, and algebraic K-theory of the Mahowald Thom spectra y(n), which interpolate between the sphere spectrum and HF_2 and satisfy K(m)_*(y(n)) = 0 for 0 ≤ m < n while K(n)_*(y(n)) ≠ 0. The main theorem (Theorem 1.2) asserts that K(m)_*(TP(y(n))) = 0 for 1 ≤ m ≤ n, so topological periodic cyclic homology raises the Morava K-theory vanishing threshold by one for every member of the y(n) family, giving evidence for a TP-version of the Ausoni–Rognes red-shift conjecture at all chromatic heights. The paper also proves that the relative algebraic K-theory K(y(n), HF_2) is K(m)-acyclic for 0 ≤ m ≤ n−1 (Theorem 6.8).

Significance. If the main theorem holds, the paper is a significant step: it gives the first family of ring spectra indexed by arbitrary chromatic height for which TP is shown to increase the Morava K-theory vanishing range by one, supporting the red-shift program at all heights. The methodology—using the homological Tate spectral sequence and Margolis homology in place of unavailable homotopy-group computations of THH(y(n))—is new and likely to have further applications, and the identification of H_*^c(TP(y(n))) with P(t^{±1}) ⊗ H_*(z(n)/v_n) as E_*-comodules (Proposition 4.9) is a clean structural result. The paper is careful in its claims: it explicitly corrects its earlier TC^- statement (Remark 5.9), separates proven statements from conjectures (Conjecture 4.11, footnote 3), and provides a fully worked example showing K(1)^c_*(TC^-(y(1))) ≠ 0. These features indicate careful scholarship. The main deficit, detailed below, is that the bridge from continuous to ordinary Morava K-theory via [5, Thm 3.5] rests on an unverified nilpotent-completeness hypothesis and on the companion preprint [5]; closing that gap is required before the main theorem can be considered established.

major comments (3)
  1. [§6.1, Theorem 6.5 (alleged application of [5, Thm 3.5])] The proof of Theorem 6.5 applies Theorem 6.1 of [5] to conclude K(m)_*(TP(y(n))) = 0 from the continuous vanishing computed in Theorem 4.15. The quoted Theorem 6.1 explicitly lists as a hypothesis that each spectrum Y[i] in the sequence be HF_p-nilpotently complete. The verification in the proof of Theorem 6.5 refers only to Lemma 6.2 (bounded below finite type), Corollary 6.3 (finite generation), Lemma 6.4 (primitive degree bounds), and Theorem 4.15 (vanishing of the limit of Margolis homologies); none of these statements states or proves that TP(y(n))[i] is HF_2-nilpotently complete for each i. Since this condition is precisely what justifies passing from lim_i K(m)_*(TP(y(n))[i]) = 0 to K(m)_*(holim_i TP(y(n))[i]) = 0, the isomorphism (11), and with it Theorem 1.2, is conditional on an unverified hypothesis. The authors should either prove nilpotent completeness directly or supply a precise citation of a Bousfield- or Miller-type convergence theorem that applies to the bounded-below finite-type spectra TP(y(n))[i], and they should state the direction of the limit in the Greenlees tower so that it matches the sequence indexed by increasing i in Theorem 6.1.
  2. [§6.1, Lemma 6.4] Lemma 6.4 supplies the uniform bound M = 2^{n+1} on the degree of comodule primitives that is required by Theorem 6.1, but the proof is not rigorous as written. First, the displayed equality '2^{n+1}−1 = |∏_{j=1}^n σξ_j|' is arithmetically incorrect: |σξ_j| = 2^j, so |∏_{j=1}^n σξ_j| = ∑_{j=1}^n 2^j = 2^{n+1}−2; the claimed strict bound M = 2^{n+1} still holds, but the displayed equality should be corrected. Second, the final paragraph of the proof ('we need to choose a final sequence such that elements that are not divisible by t always appear, but this is possible') is not an argument, and the claim that coactions on t^a x for a > 0 cannot be canceled by adding higher-filtration terms needs a complete justification. Since the primitive-degree bound is an explicit hypothesis of [5, Thm 3.5], the proof should be expanded or replaced by a precise reference.
  3. [§6, Theorems 6.1 and 6.5 (dependency on companion preprint)] The passage from continuous to ordinary Morava K-theory—the only route to the main theorem—goes through Theorem 3.5 of the companion preprint [5] (Angelini-Knoll–Salch, arXiv:2003.03510), which is quoted verbatim but not proved in the paper. Because Theorem 1.2 rests on this external result, the paper cannot be fully evaluated unless [5] is stable, publicly available, and its hypotheses match the statement quoted here. The authors should either include the proof of the quoted theorem in an appendix or confirm the publication status of [5] so that the referee can verify the argument.
minor comments (4)
  1. [§6.1, Theorem 6.1 (statement)] There is a typo in the quoted hypothesis list: 'HFp-nilpotently compete' should read 'HF_p-nilpotently complete'.
  2. [§4.4, proof of Theorem 4.15] In the second paragraph of the proof, 'H(V(i);Q_n)' should be 'H(V(i);Q_m)', since m is the index of the Morava K-theory in the statement.
  3. [§1, Abstract and Question 1.1] The abstract states that TP(y(n)) 'has chromatic complexity n+1', but the results prove only the vanishing statements K(m)_*(TP(y(n))) = 0 for 1 ≤ m ≤ n; nonvanishing of K(n+1)_*(TP(y(n))) is not established, so the phrasing should be 'chromatic complexity at least n+1' or 'raises the vanishing threshold by one'.
  4. [§4.2, Definition 4.2 and Theorem 4.15] The direction of the limit 'lim_i' in the definition of continuous homology and in Theorems 4.15 and 5.8 is not pinned down explicitly; since [5, Thm 3.5] is stated for a sequence Y[2]→Y[1]→Y[0], the authors should specify the indexing convention for the Greenlees tower so that the match between the two towers is unambiguous.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the spectral sequence computations are self-contained, and the key cited theorem [5] is a general statement rather than a fitted or self-defined prediction.

full rationale

The derivation chain is not circular. The chromatic-complexity inputs for y(n), z(n), and z(n)/v_n are computed internally from the Q_m-action on the dual Steenrod algebra (Lemmas 2.15-2.21), not imported from the target result. H_*(THH(y(n))) is computed from the Bokstedt spectral sequence (Proposition 3.2), and the map to THH(HF_2) is analyzed directly (Proposition 3.8). The continuous homology of TP(y(n)) and TC^-(y(n)) is then computed from the homological Tate and homotopy fixed point spectral sequences (Propositions 4.5 and 5.3), with the collapse verified against the benchmark HF_2 case; no term of the abutment is used as an input to the differentials. Continuous Morava K-theory vanishing (Theorems 4.15 and 5.8) is obtained from the localized Adams spectral sequence and the independently computed Margolis homologies. The final passage from continuous to ordinary K(m)_* uses [5, Thm. 3.5], a general theorem about commuting homotopy limits with Morava K-theory. Although that citation shares an author, it is not a uniqueness claim and its hypotheses do not include the conclusion K(m)_*(TP(y(n)))=0; it is therefore independent support rather than a self-citation chain. The unverified HF_p-nilpotent completeness hypothesis is a genuine correctness risk, but a missing hypothesis is not circularity. Accordingly no circular step is identified.

Assumptions & free parameters 0 free parameters · 8 assumptions · 0 invented entities

No numbers are fitted. The central computations are mostly self-contained, but several external theorems are imported; one imported condition, HF_p-nilpotent completeness of TP(y(n))[i], appears unverified.

assumptions (8)
  • standard math For an E1 ring spectrum R, the homological Tate spectral sequence satisfies d2(x) = t σ(x) for x in H_*(THH(R)).
    Imported from Bruner and Rognes [14, Prop 3.2]; used in Propositions 4.5, 5.3 and Corollary 5.4 to compute E3 pages.
  • domain assumption The homological Tate and homotopy fixed point spectral sequences converge to the continuous homology of TP(y(n)) and TC^-(y(n)).
    Convergence is asserted in Definition 4.2 and Section 5.1; the proof depends on the Greenlees filtration and inverse limits.
  • domain assumption The inverse limit Adams spectral sequence converges to π_*(holim_i X_i) for bounded below finite type spectra X_i.
    Used in Sections 4.3 and 5.2 to compute k(m)^c and continuous Morava K-theory; cited to Lunoe-Nielsen-Rognes [28].
  • domain assumption The localized Adams spectral sequence of Miller and Ravenel converges to K(m)_*(X) under the Margolis homology vanishing and vanishing-line conditions.
    Corollary 2.14, following [30,35], is applied throughout Sections 2, 4 and 5.
  • ad hoc to paper The spectra TP(y(n))[i] are HF_p-nilpotently complete for each i, as required by [5, Thm 3.5].
    This condition is stated in Theorem 6.1 but not checked in Section 6.1; Lemma 6.2, Corollary 6.3 and Lemma 6.4 do not mention nilpotent completeness.
  • standard math Dundas-Goodwillie-McCarthy gives an equivalence TC(A,B) ≃ K(A,B) for maps of connective E1 ring spectra after p-completion.
    Used in Theorems 6.7 and 6.8 to transfer TC vanishing to relative algebraic K-theory.
  • standard math Nikolaus-Scholze gives a fiber sequence TC(A,B) → TC^-(A,B) → TP(A,B).
    Used in Theorem 6.7 to derive TC vanishing from the TC^- and TP vanishing results.
  • standard math The Mahowald-Ravenel-Shick computation of the localized Adams E2-page for y(n), and the existence of the v_n self-map for z(n), are correct.
    These prior results are used in Proposition 2.22 and Lemma 2.19; not rederived in this paper.

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Pith. "Pith review of A homological approach to chromatic complexity of algebraic K-theory." pith.science (2026). https://pith.science/paper/L6XG52VN

@misc{pith2026190809164,
  author       = {Pith},
  title        = {Pith review of: A homological approach to chromatic complexity of algebraic K-theory},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/L6XG52VN}},
  note         = {Machine review of arXiv:1908.09164}
}
abstract

The family of Thom spectra $y(n)$ interpolates between the sphere spectrum and the mod two Eilenberg--MacLane spectrum. Computations of Mahowald, Ravenel, Shick, and the authors show that the associative ring spectrum $y(n)$ has type $n$. Using trace methods, we give evidence that algebraic K-theory preserves this chromatic complexity. Our approach sheds light on the chromatic complexity of topological negative cyclic homology and topological periodic cyclic homology, which approximate algebraic K-theory and are of independent interest. Our main contribution is a homological approach that can be applied in great generality, such as to associative ring spectra $R$ without additional structure whose coefficient rings are not completely understood.

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