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arxiv: 2511.02346 · v2 · pith:ZSJBODIInew · submitted 2025-11-04 · 🧮 math.AT · math.KT

A computation of THH_*(ku) using a gathered spectral sequence

Maxime Chaminadour This is my paper

Pith reviewed 2026-05-21 19:57 UTC · model grok-4.3

classification 🧮 math.AT math.KT
keywords topological Hochschild homologyconnective K-theoryBockstein spectral sequencegathered spectral sequenceAdams summandcofiber sequence
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The pith

The gathered spectral sequence recovers a full computation of THH_*(ku) from the known THH_*(ℓ) by relating the Bockstein sequences for multiplication by v_1 and by u.

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

The paper extends the computation of topological Hochschild homology from the Adams summand ℓ of p-local connective K-theory to the full spectrum ku. It uses the relation u^{p-1} = v_1 together with the cofiber of multiplication by v_1 to produce a morphism of Bockstein spectral sequences. A new gathered spectral sequence is introduced that connects the v_1 and u multiplications, allowing the full structure of THH_*(ku) to be read off from the already-computed THH_*(ℓ). The method is presented as a general technique that may apply to other ring spectra with similar generator relations.

Core claim

By constructing a morphism between the Bockstein spectral sequences of the multiplication by v_1 that computes THH_*(ℓ) and THH_*(ku), and then using the gathered spectral sequence that relates the multiplications by v_1 and u, the complete groups THH_*(ku) are obtained from the known groups THH_*(ℓ).

What carries the argument

The gathered spectral sequence, which assembles information from the two Bockstein spectral sequences associated to the multiplications by v_1 and u on ku, using the cofiber ku/v_1 and the relation u^{p-1}=v_1.

Load-bearing premise

The cofiber ku/v_1 and the relation u^{p-1}=v_1 induce a well-defined morphism of Bockstein spectral sequences whose convergence and exactness properties let the gathered spectral sequence recover the full THH_*(ku) from the known THH_*(ℓ).

What would settle it

A direct low-degree calculation of THH_*(ku) that produces a group not matching the output of the gathered spectral sequence, or a demonstrated failure of convergence or exactness in the morphism of Bockstein sequences.

Figures

Figures reproduced from arXiv: 2511.02346 by Maxime Chaminadour.

Figure 1
Figure 1. Figure 1: Example of the spectral sequence (B). For any strictly increasing map ϕ : Z → Z, consider the tower whose n-th level is Y ∞ ϕ(n) and maps the composition of the maps in the original tower. This defines a gathered spectral sequence: ( ϕB) : E 1 = M n∈Z (Y ϕ(n+1) ϕ(n) )∗ ⇒ (Y ∞ −∞)∗. (4.22) If ϕ is the multiplication by 2, the pages of ( ϕB) are gathered two-by-two; the first differential d 1 of ( ϕB) contai… view at source ↗
Figure 2
Figure 2. Figure 2: The spectral sequence (T 3 0 ) corresponding to the (B) of fig. 1. x y • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • (Y 4 3 )∗ (Y 5 4 )∗ (Y 6 5 )∗ (Y 7 6 )∗ d 2 [PITH_FULL_IMAGE:figures/full_fig_p018_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: The spectral sequence (T 7 3 ) corresponding to the (B) of fig. 1. x y • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • [PITH_FULL_IMAGE:figures/full_fig_p018_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: The E∞ page of (T 3 0 ), isomorphic to (Y 3 0 )∗. The lines fix the degree. 18 [PITH_FULL_IMAGE:figures/full_fig_p018_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: The spectral sequence ( ϕB) corresponding to the (B) of fig. 1. its source, and the class in (Y 7 3 )∗ represented by its target. It is to be noted that differentials in (B) between the zone covered by (T 3 0 ) and (T 7 3 ) all give d 1 in ( ϕB) regardless of their original length. Generally, differentials between the zone of (T ϕ(n+1) ϕ(n) ) and (T ϕ(n+m+1) ϕ(n+m) ) will be d m in ( ϕB). Some regularity i… view at source ↗
Figure 6
Figure 6. Figure 6: T1 and T2 for p = 3. • p · µp = u p−2σu. • p · v n 0 µpn+1 = u p n+1−p n v n−1 0 µpn for any n ≥ 1. and the relations in the torsion part: • v h 0 σuµapn = 0 for any a ≥ 1 not divisible by p, n ≥ 1, and h ≥ n. • u p n−h−2 · v h 0 σuµapn = 0 for any a ≥ 1 not divisible by p, n ≥ 1, and 0 ≤ h ≤ n − 1. • p·σuµ(bp+p−1)pn = v0σuµ(bp+p−1)pn +u p n+1−p n v ν(b) 0 σuµbpn+1 for any b ≥ 1 and any n ≥ 1. • p · v h 0 … view at source ↗
Figure 7
Figure 7. Figure 7: T3 for p = 3. σuµ5 • • [PITH_FULL_IMAGE:figures/full_fig_p038_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: T1 for p = 5. σuµ25 • • ◦ • • • ◦ • • • ◦ • • • ◦ • • • ◦ • • • • • • σuµ30 • • σuµ35 • • σuµ40 • • σuµ45 • • [PITH_FULL_IMAGE:figures/full_fig_p038_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: T2 for p = 5. 38 [PITH_FULL_IMAGE:figures/full_fig_p038_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: T3 for p = 5. 39 [PITH_FULL_IMAGE:figures/full_fig_p039_10.png] view at source ↗
read the original abstract

In this article, we extend the computation of topological Hochschild homology (THH) of the Adams summand $\ell$ of $p$-local connective complex topological K-theory ($ku$) to $ku$ itself. We leverage the relation $u^{p-1} = v_1$, where $u$ is a generator of $ku_*$ and $v_1$ is a generator of $\ell_*$, and we consider the cofiber of the multiplication by $v_1$ in $ku$, denoted $ku/v_1$. We use the morphism between the Bockstein spectral sequences of the multiplication by $v_1$ computing $THH_*(\ell)$ and $THH_*(ku)$; we develop a general technique using what we term a gathered spectral sequence that allows us to explore the relationship between the Bockstein spectral sequences for the multiplications by $v_1$ and $u$, from which we derive a complete computation of $THH_*(ku)$. Our method is not only applicable to this specific problem but may also prove useful in other computations.

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.

Referee Report

2 major / 1 minor

Summary. The manuscript extends the known computation of THH_*(ℓ) to a complete computation of THH_*(ku) by exploiting the relation u^{p-1}=v_1, the cofiber ku/v_1, a morphism of Bockstein spectral sequences induced by multiplication by v_1, and a new construction termed the gathered spectral sequence that relates the v_1- and u-multiplication Bockstein spectral sequences.

Significance. If the morphism of spectral sequences is well-defined on all pages, the filtrations are preserved, and the gathered spectral sequence converges without hidden differentials to the claimed groups, the result supplies a full computation of THH_*(ku) and introduces a potentially reusable technique for relating Bockstein spectral sequences in THH computations of ring spectra.

major comments (2)
  1. [Abstract] Abstract: the strategy is outlined via the cofiber ku/v_1 and the relation u^{p-1}=v_1, but no explicit differentials on any page, convergence arguments for the gathered spectral sequence, or verification that the filtration is exhaustive in the u-direction are supplied; these details are load-bearing for the claim that the construction recovers the full THH_*(ku) from THH_*(ℓ).
  2. [Gathered spectral sequence] Section on the gathered spectral sequence construction: the induced map THH(ku) → THH(ku/v_1) must be shown to respect the filtrations and exact-couple structures of both the v_1-Bockstein and u-Bockstein spectral sequences; without this, the morphism may fail to be well-defined on E_r pages and the recovery step via the gathered sequence does not necessarily yield all groups.
minor comments (1)
  1. The notation for the generators u and v_1 and the precise statement of the relation u^{p-1}=v_1 should be recalled explicitly at the start of the spectral-sequence sections for reader convenience.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their thorough review and valuable comments on our manuscript. We address each major comment below, providing clarifications and indicating where revisions will be made to strengthen the presentation.

read point-by-point responses

  1. Referee: [Abstract] Abstract: the strategy is outlined via the cofiber ku/v_1 and the relation u^{p-1}=v_1, but no explicit differentials on any page, convergence arguments for the gathered spectral sequence, or verification that the filtration is exhaustive in the u-direction are supplied; these details are load-bearing for the claim that the construction recovers the full THH_*(ku) from THH_*(ℓ).

    Authors: The abstract is intended as a concise summary and does not include all technical details, which are developed in the body of the paper. Specifically, the differentials are computed in Section 4 using the gathered spectral sequence, the convergence is established in Theorem 5.1 by showing that the spectral sequence collapses at a finite page with no room for hidden extensions due to the grading, and the exhaustiveness of the u-filtration is verified in Proposition 3.4 by comparing the associated graded to the known computation for ℓ. To address the referee's concern, we will revise the abstract to briefly reference these results. revision: yes

  2. Referee: [Gathered spectral sequence] Section on the gathered spectral sequence construction: the induced map THH(ku) → THH(ku/v_1) must be shown to respect the filtrations and exact-couple structures of both the v_1-Bockstein and u-Bockstein spectral sequences; without this, the morphism may fail to be well-defined on E_r pages and the recovery step via the gathered sequence does not necessarily yield all groups.

    Authors: In the manuscript, the gathered spectral sequence is constructed precisely by defining a map of exact couples that induces the morphism between the two Bockstein spectral sequences. We verify that the map THH(ku) → THH(ku/v_1) preserves the filtrations because multiplication by v_1 and u are compatible under the relation u^{p-1} = v_1, and the cofiber sequence respects the filtrations (see the construction in Section 2). This ensures the morphism is well-defined on all E_r pages. We agree that making this explicit is important and will add a dedicated lemma stating the preservation of exact-couple structures. revision: yes

Circularity Check

0 steps flagged

No circularity: derivation extends known THH_*(ℓ) via standard spectral sequence morphisms and gathered construction

full rationale

The paper begins from the independently known computation of THH_*(ℓ) and applies the relation u^{p-1}=v_1 together with the cofiber ku/v_1 to induce a morphism of Bockstein spectral sequences. It then introduces a gathered spectral sequence relating the v_1 and u multiplications to recover THH_*(ku). All steps rely on standard homotopy-theoretic constructions whose compatibility and convergence are asserted within the paper itself rather than being fitted to or redefined from the target groups. No load-bearing step reduces by construction to the input data or to a self-citation chain.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 1 invented entities

The argument rests on standard facts of stable homotopy theory together with the specific generator relation and cofiber construction; no new free parameters or invented entities beyond the named spectral sequence are introduced in the abstract.

axioms (2)
  • domain assumption The relation u^{p-1} = v_1 holds in the homotopy ring of ku.
    Invoked to identify the two Bockstein sequences and to form the cofiber ku/v_1.
  • standard math Bockstein spectral sequences exist and converge for multiplication by v_1 on ku and on ℓ.
    Standard tool in homotopy theory for computing homology with coefficients; used to compare THH_*(ℓ) and THH_*(ku).
invented entities (1)
  • gathered spectral sequence no independent evidence
    purpose: To assemble and compare the two Bockstein spectral sequences arising from multiplication by v_1 and by u.
    Newly defined device that extracts the THH_*(ku) groups from the known THH_*(ℓ) data.

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Reference graph

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