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On the Residue Sequence in Logarithmic Topological Cyclic Homology

T0 review · 0 major / 4 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read Logarithmic THH is the localization cofiber, with matching residue maps.

desk verdict Resolves Rognes's conjecture with a careful, internally coherent proof; the only real risk is the heavy dependence on same-week preprints. read the letter →

arxiv 2506.14545 v1 pith:X7OIWWBS submitted 2025-06-17 math.AT math.AGmath.KT

Tommy Lundemo This is my paper · ORCID
classification math.ATmath.AGmath.KT MSC 19D5555P4255P91
keywords logarithmictopologicalHochschildhomologycyclotomicspectralocalizationsequencesresidueevenE2-ringscycliclogdifferentialgradedringsrepletebarconstruction
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

Topological Hochschild homology (THH) is an invariant of rings that takes values in cyclotomic spectra, spectra with Frobenius-like maps that feed topological cyclic homology. Because THH is a localizing invariant, localization of rings yields cofiber sequences, but the cofiber term THH(A|x) is usually just a module and is hard to compute. Logarithmic THH THH(A,) is a richer construction with a multiplication and a residue sequence. The paper proves that these two objects are the same: an equivalence of THH(A)-modules in cyclotomic spectra, with the residue boundary map identified with the localization boundary map. This resolves a conjecture and shows that logarithmic THH, TR, and TC extend older localization constructions without needing the regularity hypotheses classically used for devissage.

What carries the argument

The load-bearing object is the replete bar construction $B^{\mathrm{rep}}(M)$, the pullback of the cyclic bar construction on a commutative monoid $M$ with the cyclic bar construction on its group completion $M^{\mathrm{gp}}$; it supplies the logarithmic term $\mathrm{THH}(A,\langle x\rangle)$ by replacing the cyclic bar construction in the definition of $\mathrm{THH}$. The proof works in graded cyclotomic spectra, where objects carry an integer weight. The central technical control is Lemma 2.7: mapping spectra from objects concentrated in non-negative weights to objects concentrated in non-positive weights forget the module structure, so module-level maps can be read off from weight-zero data. The universal case is the free even $\mathbb{E}_2$-ring $S[t_{2d}]$ with its class $t_{2d}$; the paper identifies the two constructions there as inclusions of the non-negative weight part, and then base-changes along $S[t_{2d}]\to A$.

What would settle it

The central claim would fail if, for some even $\mathbb{E}_2$-ring $A$ and class $x\in\pi_{2d}(A)$, the spectra $\mathrm{THH}(A,\langle x\rangle)$ and $\mathrm{THH}(A|x)$ had different homotopy groups, or if the two boundary maps disagreed after composition with the equivalence; computing $\pi_1$ for a concrete even ring outside the paper's examples and comparing the residue map to the predicted log differential quotient would settle the matter.

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Extended reading notes

Core claim

The central claim is that two a priori different constructions of logarithmic topological Hochschild homology coincide. For an even $\mathbb{E}_2$-ring $A$ (a homotopy-coherently commutative ring spectrum with homotopy groups concentrated in even degrees) and a class $x\in\pi_{2d}(A)$, the paper proves an equivalence $$\varphi\colon \mathrm{THH}(A,\langle x\rangle)\xrightarrow{\simeq}\mathrm{THH}(A|x)$$ of $\mathrm{THH}(A)$-modules in cyclotomic spectra, and it proves that the residue boundary map $\partial_{\mathrm{rep}}$ of the logarithmic residue sequence is homotopic to $\partial\circ\varphi$, where $\partial$ is the boundary map of the cofiber sequence defining $\mathrm{THH}(A|x)$. For ordinary commutative rings this is Theorem 1.1 with $x$ a non-zero divisor, and the regularity hypotheses classically needed for devissage are not required. The equivalence is first proven on the universal example $\mathrm{THH}(S[t_{2d}])$ using the weight grading, then base-changed to $A$.

Load-bearing premise

The load-bearing premise is that the ring spectrum $A$ is even, meaning its homotopy groups are concentrated in even degrees, because that is what supplies the map from the universal graded ring $S[t_{2d}]$ to $A$ on which the whole base-change strategy depends.

Editorial extensions

If this is right

  • For discrete rings, $\mathrm{THH}(A,\langle x\rangle)$ is a multiplicative replacement for the cofiber $\mathrm{THH}(A|x)$, and the logarithmic residue sequence is literally the localization cofiber sequence.
  • There exists a stable $\infty$-category $\mathrm{Perf}(A,\langle x\rangle)$ whose THH is $\mathrm{THH}(A,\langle x\rangle)$, so logarithmic THH, TR, and TC are realized as values of localizing invariants for discrete valuation rings, connective complex $K$-theory, and truncated Brown-Peterson spectra.
  • For discrete valuation rings, the logarithmic coefficient ring is isomorphic to the earlier Waldhausen-category construction as a log differential graded ring, so known computations of topological cyclic homology transfer to this construction.
  • The module-level identification of the residue maps is exactly the compatibility needed to compute TC through localization sequences, not just at the level of underlying spectra.
  • The conjecture that the realizing category can be chosen symmetrically monoidal, with evidence for discrete valuation rings, would make the multiplicative structure of logarithmic THH categorical rather than an extra add-on.

Reading between the lines

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

  • If the conjecture on symmetric monoidal realizing categories extends beyond the discrete valuation ring case, logarithmic THH, TR, and TC would become fully fledged localizing invariants with multiplicative structure, potentially yielding new trace maps from logarithmic $K$-theory.
  • The weight-graded strategy suggests a general template for proving 'residue equals cofiber' identifications: formalize a theory in graded cyclotomic spectra, identify maps from weight-zero information, then base-change; the same pattern may apply to higher logarithmic structures beyond a single cone.
  • The paper's observation that $K(\mathrm{Perf}(BP\langle n\rangle,\langle v_n\rangle))$ differs from $K(E(n))$ suggests that logarithmic THH tracks a nilpotent or ramified variant of $K$-theory rather than ordinary localization, and comparing TC along this difference could yield new chromatic filtrations of TC.
  • A concrete testable next step is to compute $\pi_*\mathrm{THH}(A,\langle x\rangle)$ for an even ring beyond the examples treated in the paper and verify that the identified log differential graded ring structure agrees with the cofiber model.
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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

0 major / 4 minor

Summary. The paper proves a conjecture of Rognes relating the logarithmic topological Hochschild homology THH(A,⟨x⟩) of an even E2-ring A at an element x ∈ π_{2d}(A) to the cofiber THH(A|x) of THH(A/x) → THH(A), as THH(A)-modules in cyclotomic spectra, with the residue maps identified (Theorem 1.9). For discrete commutative rings this gives Theorem 1.1, and via the Ramzi–Sosnilo–Winges construction of localizing motives it yields stable ∞-categories Perf(A,⟨x⟩) realizing the logarithmic term (Theorem 1.3). The paper also proves Theorem 1.7, that the isomorphism of Theorem 1.1 is one of log differential graded rings in the case of complete DVRs, and discusses applications to localization sequences for TR and TC without the usual regularity hypotheses.

Significance. If correct, the main theorem provides a long-sought bridge between two independent constructions of logarithmic THH and gives a clean proof of localization sequences for TC in settings where dévissage fails. The proof is detailed and makes systematic use of graded cyclotomic spectra, including an effective mapping-space lemma (Lemma 2.7) that is used to upgrade equivalences to module-level equivalences. The paper is unusually candid about its limitations: it does not prove an E∞-refinement (Remark 4.15), the (ko,w) case is conditional on constructing a cyclotomic structure (Remark 3.32), and several key inputs are very recent preprints (RSS25, RSW25). These are stated limitations rather than hidden gaps, and they are weighed in the assessment below.

minor comments (4)
  1. [§4.16] The assertion that the map (4.6) is an equivalence should be justified explicitly: both THH(OK|K) and THH(OK|π) are cofibers of the same transfer map THH(k)→THH(OK), but the compatibility of (4.6) with those cofiber sequences is not spelled out, even though the phrase 'explicit equivalence' in the first paragraph of §4.16 depends on it.
  2. [§3.12] In the proof of Proposition 3.13, the statement that THH(S[t±1])_{<0} contains no cyclotomic summand equivalent to S^{triv} because the p-typical Frobenius multiplies weights by p is plausible but is not justified in detail; a one-sentence argument or a reference would suffice.
  3. [§2.8] The definition of THH(S[t2d],⟨t2d⟩) as the 'weight-connective cover' of THH(S[t±1 2d]) could be made more precise by explicitly stating the splitting of (2.6) and noting that the resulting object is independent of the chosen presentation of the weight grading.
  4. [Global] There are a few typographical errors (e.g. 'inolving' near the end of the first page and 'theright-hand' in §4.13) and some sentences in the introduction are overly compressed; a careful proofread is recommended.

Circularity Check

0 steps flagged · score 2.0 of 10

No significant circularity: the main equivalence is derived from the RSS25/RSW25 inputs rather than assumed.

full rationale

The central theorem (Theorem 1.9) compares two independently defined objects: the logarithmic term THH(A,<x>) from RSS25 and the cofiber term THH(A|x). The proof is a genuine comparison: it proves the S[t2d]-level equivalence via an explicit splitting (Corollary 3.3, Proposition 3.13), lifts it to a module-level statement in graded cyclotomic spectra (Construction 3.18, Propositions 3.19 and 3.22), and identifies the residue maps through a mapping-spectrum computation (Theorem 3.24, Corollary 3.28). None of these steps invokes Theorem 1.9 or assumes the target equivalence. The main external inputs, RSS25 and RSW25, are prior constructions used as tools, not as hidden versions of the conclusion. The author's own earlier work (BLPØ23a/b, Lun21) appears mainly in supporting roles, such as the log differential graded ring structure, and is not used to force the main equivalence. Theorem 1.3 is explicitly an existence result obtained from the universal localizing invariant of RSW25, and is not presented as a predictive derivation. Remark 4.15 explicitly leaves the E-infinity refinement open, which is a stated limitation rather than a circular assumption. No fitted parameters appear, and no equation reduces by construction to its own input. The only minor point is the presence of several self-citations, but they are not load-bearing for the central claim.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

The central claim depends on the recent RSS25 construction of log THH and on standard infinity-category machinery. No free parameters or invented entities are used.

assumptions (5)
  • standard math There is a well-behaved infinity-category Cat^{perf}_infinity of small idempotent-complete stable infinity-categories with localizing invariants as functors to spectra.
    The paper works throughout with the framework of BGT13 and RSW25, e.g., Section 1.2.
  • domain assumption The universal localizing invariant Uloc is a Dwyer-Kan localization (RSW25, Theorem 1.1).
    Used to define Perf(A,<x>) as a cofiber in localizing motives in Section 1.2 and proof of Theorem 1.3.
  • domain assumption The model for logarithmic THH of RSS25, including THH(S[t2d], <t2d>) as a weight-connective cover and the base-change definition THH(A, <x>) := THH(A) tensor_{THH(S[t2d])} THH(S[t2d], <t2d>), is correct.
    Section 2.8 defines the main object; the paper does not redefine it but relies on the RSS25 construction including its cyclotomic structure.
  • domain assumption A is an even E2-ring and x is a degree 2d element, guaranteeing the existence of an E2-ring map S[t2d] -> A sending t2d to x.
    Theorem 1.9 and Section 2.8: 'We here crucially use that A is assumed to be even.' This is the key structural restriction.
  • standard math The p-typical cyclotomic Frobenius multiplies weights by p, so no negative-weight factor of THH(S[t±1_{2d}]) can be equivalent to S^{triv}.
    Used in Proposition 3.13 to identify the map f as an inclusion of summands.

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Cite this review

Pith. "Pith review of On the Residue Sequence in Logarithmic Topological Cyclic Homology." pith.science (2026). https://pith.science/paper/X7OIWWBS

@misc{pith2026250614545,
  author       = {Pith},
  title        = {Pith review of: On the Residue Sequence in Logarithmic Topological Cyclic Homology},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/X7OIWWBS}},
  note         = {Machine review of arXiv:2506.14545}
}
read the original abstract

As a localizing invariant, THH participates in localization sequences of cyclotomic spectra. We resolve a conjecture of Rognes by relating these to residue sequences in logarithmic THH. Consequently, logarithmic THH, TR, and TC serve as strict generalizations of the constructions of Hesselholt--Madsen and Blumberg--Mandell, which moreover enjoy localization sequences without the regularity hypotheses usually required for d\'evissage. Combined with work of Ramzi--Sosnilo--Winges, our results imply that there exists a stable infinity-category C such that THH(C), TR(C), and TC(C) realize the relevant logarithmic term for specific log structures, such as the natural ones on discrete valuation rings, connective complex K-theory, and truncated Brown--Peterson spectra. Finally, we conjecture that the category C can be chosen to reflect the additional structure present on the logarithmic terms, and we give evidence for this in the case of discrete valuation rings.

Discussion (0). Continue with ORCID to comment.

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Works this paper leans on

2 extracted references · 2 canonical work pages

  1. [2024]

    [BLPØ23a] Federico Binda, Tommy Lundemo, Doosung Park, and Paul Arne Østvær, A Hochschild-Kostant-Rosenberg theorem and residue sequences for logarithmic Hochschild homology, Adv

    arXiv:2408.15627. [BLPØ23a] Federico Binda, Tommy Lundemo, Doosung Park, and Paul Arne Østvær, A Hochschild-Kostant-Rosenberg theorem and residue sequences for logarithmic Hochschild homology, Adv. Math. 435 (2023), Paper No. 109354, 66, DOI 10.1016/j.aim.2023.109354. MR4659233 [BLPØ23b] , Logarithmic prismatic cohomology via logarithmic THH, Int. Math. R...

    work page arXiv 2023

  2. [2025]

    Localization sequences for logarithmic topological cyclic homology

    arXiv:2506.08492. [RSW25] M. Ramzi, V. Sosnilo, and C. Winges,Localizing motives as a localization of cate- gories., 2025. arXiv:2503.11338. [SS03] Stefan Schwede and Brooke Shipley,Stable model categories are categories of mod- ules, Topology 42 (2003), no. 1, 103–153, DOI 10.1016/S0040-9383(02)00006-X. MR1928647 [Sto20] Bruno Stonek, Higher topological ...

    work page Pith review arXiv 2003

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