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TR with logarithmic poles and the de Rham-Witt complex

T0 review · 5 major / 5 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read For p-completely smooth log formal schemes over the p-adic complex numbers, log $\mathrm{TR}^r$ is filtered by Nygaard-graded log prismatic cohomology, whose cohomology is the log de Rham–Witt complex.

desk verdict Plausible and honestly conditional, but the main theorem is not proved: the odd homotopy groups are explicitly dropped, and the étale vanishing meant to replace them is asserted without a real proof. read the letter →

arxiv 2412.00929 v1 pith:JVWASHL7 submitted 2024-12-01 math.AG math.KT

Faidon Andriopoulos This is my paper · ORCID
classification math.AGmath.KT MSC 14F3014F4219D55
keywords logtopologicalrestrictionhomologydeRham–Wittcomplexprismaticcohomologyr-Nygaardfiltrationmotivicp-adicformalschemesperfectoidringsHochschildwithpoles
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 aims to prove, for p-completely smooth affine p-adic formal schemes over the field of p-adic complex numbers, a conjecture relating topological restriction homology with logarithmic poles ($\mathrm{TR}^r$) to the absolute log de Rham–Witt complex. The central claim is that, étale locally, $\mathrm{TR}^r$ carries a motivic filtration whose graded pieces are the graded pieces of the $r$-Nygaard filtration on log prismatic cohomology, identified with $\tau_{\leq n}A_\Omega/\xi^r$, and that the cohomology of these pieces is the relative log de Rham–Witt cohomology $W_r\Omega^{n,\mathrm{cont}}_{(-,-)/\mathcal{O}_C}$. If this holds, the associated spectral sequence degenerates on the second page and computes log $\mathrm{TR}^r$ completely, supplying the predicted de Rham–Witt description over $\mathcal{O}_C$. The proof is conditional on an explicit assumption, stated as 'as expected', that the algebraic and homotopy-theoretic constructions of completed log prismatic cohomology over a perfectoid base agree.

What carries the argument

The load-bearing object is the log $r$-Nygaard filtration $N^{\geq n}_r\widehat{\Delta}_{(S,Q)/\mathbb{A}_{\mathrm{inf}}}\{n\}$, an $r$-fold iterated pullback of the ordinary Nygaard filtration on completed log prismatic cohomology. In the smooth case this filtration is identified with the décalage filtration $L\eta^{\geq n}_{\xi^r} A_\Omega$, and it is what the even parts of the motivic filtration on log $\mathrm{TR}^r$ and on its $S^1$-fixed points are shown to be. The motivic filtration itself is produced by log quasisyntomic sheafification of the double-speed Postnikov filtration from the log quasiregular-semiperfectoid case, which transfers homotopy-theoretic information about $\mathrm{THH}$ and its Frobenius into an algebraic filtration. The de Rham–Witt comparison then runs through the conjugate filtration on log $r$-Hodge–Tate cohomology and general properties of the décalage functor.

What would settle it

A concrete test would be to compute the $n$-th graded piece of the motivic filtration on log $\mathrm{TR}^r$ for a specific log smooth base over $\mathcal{O}_C$, such as the canonical log structure on the $p$-adic affine line, once through the algebraic log $A_\Omega$/log prismatic construction and once through the homotopy-theoretic log prismatic construction; if the two $r$-Nygaard filtered complexes differ, or if the spectral sequence of Theorem 1.1 fails to degenerate on $E_2$, the central claim is false.

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Extended reading notes

Core claim

The paper's central result is that for a p-completely smooth affine p-adic formal scheme $X=\operatorname{spf} S$ over $\operatorname{spf} \mathcal{O}_C$, equipped with the canonical log structure from its generic fibre, the spectrum $\mathrm{TR}^r((X,\mathcal{M}_X);\mathbb{Z}_p)$ admits, étale locally, a complete, exhaustive, descending, multiplicative motivic filtration. The $n$-th graded piece is identified with the $n$-th graded piece of the $r$-Nygaard filtration on log prismatic cohomology, equivalently with $\tau_{\leq n} A_\Omega/\xi^r$, and the resulting spectral sequence degenerates on the second page. Taking cohomology of the graded pieces gives the comparison $H^n(N^n_r\widehat{\Delta}_{(-,-)/\mathbb{A}_{\mathrm{inf}}}) \simeq W_r\Omega^{n,\mathrm{cont}}_{(-,-)/\mathcal{O}_C}$ with the relative log de Rham–Witt complex, the relation predicted by the conjectures. The argument builds a motivic filtration on log $\mathrm{TR}^r$ and its $S^1$-fixed points by log quasisyntomic sheafification of double-speed Postnikov filtrations, identifies the even pieces with an $r$-fold iterated pullback version of the Nygaard filtration, and then uses the log $A_\Omega$ and log de Rham–Witt comparisons.

Load-bearing premise

The whole comparison rests on the assumption, stated in the introduction as 'as expected', that the algebraic and homotopy-theoretic constructions of completed log prismatic cohomology over a perfectoid base coincide; this identification is needed to replace the $r$-Nygaard filtration by the décalage filtration $L\eta^{\geq n}_{\xi^r} A_\Omega$, and if it fails the main theorem's identifications do not go through.

Editorial extensions

If this is right

  • The motivic filtration spectral sequence for log $\mathrm{TR}^r$ degenerates on $E_2$, so the homotopy groups of log $\mathrm{TR}^r$ are determined by the cohomology of the $r$-Nygaard filtered log prismatic complex.
  • Étale locally, the graded pieces of log $\mathrm{TR}^r$ coincide with truncated log $A_\Omega$-cohomology $\tau_{\leq n}A_\Omega/\xi^r$, connecting topological restriction homology to the log $A_\Omega$ formalism.
  • The comparison $H^n(N^n_r\widehat{\Delta}_{(-,-)/\mathbb{A}_{\mathrm{inf}}}) \simeq W_r\Omega^{n,\mathrm{cont}}_{(-,-)/\mathcal{O}_C}$ yields the conjectured relation between log $\mathrm{TR}^r$ and the relative log de Rham–Witt complex.
  • The odd parts of the motivic filtration vanish étale locally for smooth $X$, so no odd homotopy-group terms obstruct the log comparison in the étale topology.
  • Passing to the inverse limit over restriction maps extends these descriptions to $\mathrm{TR}$ and its $S^1$-fixed points through the $r=\infty$ Nygaard filtrations.

Reading between the lines

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

  • Beyond the paper: since the key identifications are étale local and the motivic filtration is built by log quasisyntomic descent, the affine restriction in Theorem 1.1 should extend to arbitrary p-completely smooth formal schemes over $\mathcal{O}_C$ by gluing.
  • Beyond the paper: the assumed agreement between algebraic and homotopy-theoretic log prismatic cohomology could be tested directly on log quasiregular-semiperfectoid rings, where both sides are computable; a mismatch there would refute the conditional theorem.
  • Beyond the paper: if the comparison holds, the localization fibre sequences relating algebraic $K$-theory to log $\mathrm{TR}$ would make the log de Rham–Witt complex a computational input for the $K$-theory of semistable $p$-adic schemes, a consequence the paper gestures toward but does not develop.
  • Beyond the paper: the paper leaves open whether the odd parts vanish in the full log quasisyntomic topology rather than only étale locally; checking this on non-smooth log quasisyntomic bases would clarify how much of Theorem 1.2 survives without the smoothness assumption.
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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

5 major / 5 minor

Summary. The paper claims that, étale locally for X = spf S a p-completely smooth affine p-adic formal scheme over spf O_C, the log topological restriction homology TR^r((X,M_X); Z_p) carries a complete, exhaustive, descending, multiplicative motivic filtration whose graded pieces identify with r-Nygaard filtered log prismatic cohomology, and that the cohomology of these graded pieces computes Matsuue's p-complete relative log de Rham–Witt complex. It also claims a spectral sequence converging to TR^r that degenerates on E2. The work builds on the author's preceding non-logarithmic framework [And24a] and on the logarithmic motivic filtration of Binda–et al. [Bin+23], and it is explicitly conditional on the as-yet-unproved identification of algebraic and homotopy-theoretic approaches to completed log prismatic cohomology over perfectoid bases.

Significance. If established, the main theorem would prove a form of Hesselholt's conjectures over algebraically closed p-adic fields, extending the non-logarithmic results of Bhatt–Morrow–Scholze and the author's previous work to the logarithmic setting. The paper is honest about its conditional assumptions and clearly identifies the parts of the argument that are deferred to other preprints. The potential significance is high, and the geometric setup is natural. However, the central claims rest on two load-bearing gaps: the odd homotopy groups of the motivic filtration are explicitly not handled in the proof of Theorem 1.2, and the étale odd vanishing of Corollary 4.4 is only asserted rather than proved. The paper is therefore best viewed as a research announcement whose full theorems are not yet established.

major comments (5)
  1. [Section 3, proof of Theorem 1.2] Theorem 1.2 is stated for the full motivic filtration of TR^r((S,Q);Z_p) and of its S^1-homotopy fixed points, with associated graded identified with the two-term complexes τ^{[2n-1,2n]}. The proof, however, ends with: 'the odd homotopy groups are junk terms... We do not know how to do this in the log setting, therefore, we simply ignore the terms.' The odd homotopy groups are part of the two-term complexes and of the claimed complete filtration, so Theorem 1.2 is not proven as stated. Since Theorem 1.1(1) invokes Theorem 1.2 for the graded pieces of log TR^r, this is a load-bearing gap; the odd vanishing must be supplied or the theorem statements weakened accordingly.
  2. [Section 4.2, proof of Corollary 4.4] The entire proof of étale local odd vanishing is the sentence: 'Therefore, in the first two columns in the above diagram above, this filtration on algebraic K-theory is étale locally identified with that coming from TC, under the trace map. Hence, the same happens for the third column, as well.' This requires compatibility of the motivic filtration with the Hesselholt–Madsen fibre sequence, strictness of the trace map, and control of the boundary map's effect on odd homotopy groups; none of these is shown. The appeal to [And24a, Thm. 1.5] does not transfer because the non-log odd vanishing of [BS22] is exactly what is unavailable in the log setting. Corollary 4.4 is essential for Theorem 1.1(1), so this gap is load-bearing.
  3. [Introduction and Section 4] The paper states in the Introduction that it works 'under the additional assumption that the algebraic approach [ČK19; Kos20; KY23] and the homotopy theoretic approach [Bin+23] to completed log prismatic cohomology over a perfectoid base coincide (as expected)', and Section 4 says this identification is used 'freely'. Construction 4.1 relies on this identification to replace N_r^n Δhat with the décalage filtration Lη_{ξ^r}^{≥n} A_Ω, and this replacement underpins both the graded-piece identifications and the de Rham–Witt comparison in Theorem 1.1. Since the equivalence is not proved, Theorem 1.1 is conditional on a nontrivial conjecture; the condition should be stated as part of the theorem or eliminated.
  4. [Proposition 3.4] Proposition 3.4 is a key technical step used in the proof of Theorem 1.2 to pass between TR^r and its S^1-homotopy fixed points, but it is stated without proof or reference. The displayed formula involving 'v_r' is also notationally unclear (the quotient by v_r is not defined in the text). This proposition is load-bearing for the identification of the even homotopy groups of TR^r with the r-Nygaard filtered pieces, so a proof or a precise citation is needed.
  5. [Proposition 4.2 and Theorem 1.1(2)] Proposition 4.2 establishes the relative log de Rham–Witt comparison for semistable p-adic formal schemes spf S, as in [ČK19]. Theorem 1.1, however, is stated for arbitrary p-completely smooth affine p-adic formal schemes over spf O_C. The paper does not explain how the semistable case is extended to the general smooth case, e.g., by log smooth descent, deformation, or a reduction argument. Without such an argument, Theorem 1.1(2) is not justified.
minor comments (5)
  1. [Throughout] There are numerous typographical errors, including 'conj ectures', 'comple x', 'homotopy theoretic approach t o', and inconsistent capitalization of 'r-nygaard'. A thorough proofreading pass is needed.
  2. [Proposition 3.4 statement] The statement of Proposition 3.4 is garbled; the chain of equivalences is not typeset correctly and the notation 'v_r' is not defined. Please clarify what is being asserted.
  3. [Theorem 1.1(2) display] The Frobenius equalizer sequence in Theorem 1.1(2) is displayed only partially as 'τ≤n i_* j^* μ^{⊗n}_{p^v} i_* N^n_∞ Δhat / p^v i_* N^n_∞ Δhat / p^v 1−F'; this is incomplete and should be written out fully.
  4. [References] Reference [Hes05] is listed as 'The absolute de Rham-Witt complex' with no venue or preprint number; for a published item, full bibliographic data should be provided. Several other references are to preprints with only arXiv numbers, which is acceptable but should be checked for consistency.
  5. [Section 4] The paper repeatedly calls its arguments 'proof sketches'. Given that the main theorems are conditional on unproved assumptions and on vanishing results that are not established, the paper should clearly label the results as conditional or as sketches in the theorems themselves, not only in the surrounding text.

Circularity Check

0 steps flagged · score 2.0 of 10

No circular reduction: the main gaps are an unproved étale odd-vanishing assertion and an assumed algebraic/homotopy comparison, not circularity.

full rationale

I find no step in the paper where a claimed output is equivalent to an input by construction, where a fitted parameter is renamed as a prediction, or where the central result is forced by a self-citation chain. The graded-piece identification in Theorem 1.1(1) rests on Theorem 1.2 and on the log prismatic/Nygaard machinery imported from [Bin+23] and [And24a]; the r-Nygaard filtration is obtained by an iterated pullback from the non-r Nygaard filtration and then compared to TR^r even homotopy groups, rather than being simply defined as the TR^r filtration. The de Rham–Witt comparison in Theorem 1.1(2) is delegated to [BMS18, Sec. 11], [Aok23, Sec. 4], and [Kos20, Ex. 1.5], which are external inputs, not the paper's own conclusion. The self-citations to [And24a] supply non-log analogies and arguments, but they do not assume the log theorem being proved. The explicit limitation in the proof of Theorem 1.2 — 'We do not know how to do this in the log setting, therefore, we simply ignore the terms' — and the sketch in Corollary 4.4 asserting that the third column of the Hesselholt–Madsen fibre sequence behaves like the first two are genuine proof gaps: the étale odd vanishing needed for Theorem 1.1 is not established. Likewise, Section 4 'freely use[s] the fact that the algebraic approach and the homotopy theoretic approach to log prismatic cohomology coincide,' and the results are explicitly conditional on that unproved identification. These are completeness and correctness risks, not circular reasoning. I set the score at 2 to reflect the load-bearing reliance on the author's companion preprint and the unproved compatibility/vanishing assumptions, while emphasizing that no circular reduction is present.

Assumptions & free parameters 0 free parameters · 3 assumptions · 0 invented entities

No free parameters or invented entities. The argument rests on the cited log THH and prismatic machinery, one explicitly assumed identification of two log prismatic approaches, and an unproved odd-vanishing statement. These are the main unpaid inputs.

assumptions (3)
  • domain assumption Algebraic and homotopy-theoretic log prismatic cohomology over a perfectoid base coincide.
    Stated in the Introduction as 'as expected' and used freely in Section 4; not proved or cited.
  • ad hoc to paper Odd homotopy groups of the motivic filtration of log TR^r vanish locally (étale or quasisyntomic, depending on the setting).
    Section 3 admits this is not known in the log setting ('we simply ignore the terms'); Corollary 4.4 asserts the étale version by referring to [And24a, Thm. 1.5].
  • domain assumption Affine reduction: X = spf S with S the p-adic completion of a smooth O_C-algebra suffices for the theorem by étale descent.
    The Introduction restricts to the affine case 'for matters of simplicity'; the theorems are claimed étale locally, which would need descent.

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Pith. "Pith review of TR with logarithmic poles and the de Rham-Witt complex." pith.science (2026). https://pith.science/paper/JVWASHL7

@misc{pith2026241200929,
  author       = {Pith},
  title        = {Pith review of: TR with logarithmic poles and the de Rham-Witt complex},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/JVWASHL7}},
  note         = {Machine review of arXiv:2412.00929}
}
abstract

In the article of Hesselholt [Hes05], a set of conjectures is laid out. Given a smooth scheme $X$ over the ring of integers $\mathcal{O}_K$ of a $p$-adic field $K$, these conjectures concern the expected relation between log topological restriction homology $\mathrm{TR}^r (X,M_X)$ and the absolute log de Rham--Witt complex $W_r\Omega_{(X,M_X)}$. In this note, which is companion to [And24a], we discuss the case of a $p$-completely smooth $p$-adic formal scheme $X$ over $\mathrm{spf} \mathcal{O}_C$, where $C$ is the field of $p$-adic complex numbers. Along the way, we study the motivic filtration of log $\mathrm{TR}^r$ and its $S^1$-homotopy fixed points, following ideas of [Bin+23].

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

5 extracted references · 4 canonical work pages

  1. [1]

    TR and the r-Nygaard filtered prismatic cohomology

    [And24a] Faidon Andriopoulos. “ TR and the r-Nygaard filtered prismatic cohomology”. In: Preprint (2024). 10 REFERENCES [And24b] Faidon Andriopoulos. “On the Motivic Filtration o f TR”. PhD thesis. The University of Chicago, https://knowledge.uchicago.edu/record/12395,

    work page 2024

  2. [5]

    Arbeitsgemeins chaft: Topological cyclic homology

    2004, pp. 1–43. [HS19] Lars Hesselholt and Peter Scholze. “Arbeitsgemeins chaft: Topological cyclic homology”. In: Oberwolfach Reports 15.2 (2019), pp. 805–940. [Kos20] Teruhisa Koshikawa. “Logarithmic Prismatic Cohom ology I”. In: arXiv:2007.14037 (2020). [KY23] Teruhisa Koshikawa and Zijian Yao. “Logarithmic pri smatic cohomology II”. In: arXiv:2306.003...

    arXiv 2019

  3. [11]

    Logarithmic A$_{\rm inf}$-cohomology

    Cambridge University Press. 2023, e1. [BMS18] Bhargav Bhatt, Matthew Morrow, and Peter Scholze. “ Integral p-adic Hodge theory”. In: Publications mathématiques de l’IHÉS 128.1 (2018), pp. 219–397. [BMS19] Bhargav Bhatt, Matthew Morrow, and Peter Scholze. “ Topological Hochschild homology and integral p-adic Hodge theory”. In: Publications mathématiques de...

    work page Pith review arXiv 2018

  4. [2005]

    On the K-theory of local fields

    [HM03] Lars Hesselholt and Ib Madsen. “On the K-theory of local fields”. In: Annals of mathematics 158.1 (2003), pp. 1–113. [HM04] Lars Hesselholt and Ib Madsen. “On the de Rham–Witt co mplex in mixed characteristic”. In: Annales scientifiques de l’Ecole normale supérieure . Vol

    work page 2003

  5. [2024]

    A p-adic Cartier isomorphism between the A_inf-cohomology and de Rham-Witt complexes for semistable formal schemes

    [Aok23] Kensuke Aoki. “A p-adic Cartier isomorphism between the Ainf -cohomology and de Rham- Witt complexes for semistable formal schemes”. In: arXiv:2303.10492 (2023). [BM23] Bhargav Bhatt and Akhil Mathew. “Syntomic complexes and p-adic étale Tate twists”. In: Forum of Mathematics, Pi . Vol

    work page Pith review arXiv 2023

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