TR and the $r$-Nygaard filtered prismatic cohomology

Faidon Andriopoulos

REVIEW 4 major objections 4 minor 7 references

TR and the $r$-Nygaard filtered prismatic cohomology

T0 review · 4 major / 4 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read The paper establishes that the motivic filtrations of TR^r and its S^1-fixed points have graded pieces given by the r-Nygaard filtered prismatic cohomology.

desk verdict The r-Nygaard filtration and the perfectoid computations are real contributions, but the quasisyntomic descent step in §4.2 rests on a diagonal-precomposition argument that cannot work for r ≥ 3, so Theorem 1.5 is not yet proven as stated. read the letter →

arxiv 2412.00914 v2 pith:TOJQZ4XH submitted 2024-12-01 math.AG math.KTmath.NT

Faidon Andriopoulos This is my paper · ORCID

classification math.AGmath.KTmath.NT MSC 14F3019D55

keywords prismaticcohomologyNygaardfiltrationtopologicalrestrictionhomologymotivicquasisyntomicdescentcyclicdeRham–Wittcomplexperfectoidrings

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 restriction homology $\mathrm{TR}^r$ is an invariant that packages the $r$-truncated Witt vectors and, through the Frobenius map $1-F$, builds topological cyclic homology, a close relative of algebraic $K$-theory. The paper sets out to prove that, for quasisyntomic rings, $\mathrm{TR}^r$ and its $S^1$-homotopy fixed points carry a motivic filtration whose even graded pieces are computed by prismatic cohomology equipped with a new filtration, called the $r$-Nygaard filtration. The odd graded pieces vanish locally in the quasisyntomic topology, so the filtration is completely governed by the even layers. If the main theorem is right, it connects the $\mathrm{TR}^r$ side of p-adic homotopy theory to prismatic cohomology and gives a concrete target for a prismatic comparison with de Rham–Witt complexes.

What carries the argument

The central object is the $r$-Nygaard filtration on absolute prismatic cohomology, defined for $1\le r\le\infty$ by the iterated pullback $N^{\ge i}_r\Delta_S\{i\} := N^{\ge i}\Delta_S\{i\}\times_{\Delta_S\{i\}}\cdots\times_{\Delta_S\{i\}} N^{\ge i}\Delta_S\{i\}$ along the canonical inclusion and the divided prismatic Frobenius. On the homotopy-theoretic side the matching tool is the iterated pullback presentation $\mathrm{TR}^r(S;\mathbb{Z}_p)^{hS^1}\simeq\mathrm{TC}^-(S;\mathbb{Z}_p)\times_{\mathrm{TP}(S;\mathbb{Z}_p)}\cdots\times_{\mathrm{TP}(S;\mathbb{Z}_p)}\mathrm{TC}^-(S;\mathbb{Z}_p)$, which lets the even homotopy groups be read off as the filtration pieces; passing from the $S^1$-fixed points back to $\mathrm{TR}^r$ is done by killing the periodicity class $v_r$. For perfectoid rings the filtration is simply the $\xi_r$-adic filtration on $A_{\mathrm{inf}}$. The $r$-divided prismatic Frobenius $\phi_{r,i}$ and its graded version to the $r$-Hodge–Tate cohomology connect the algebraic filtration to the higher Frobenius maps on $\mathrm{TR}^r$.

What would settle it

For $S=\mathbb{Z}_p\langle x\rangle$, compute the cokernel of the map $\alpha$ in the exact sequence of Section 4.2 after quasisyntomic sheafification; a nonzero local odd class in $\pi_{2i-1}\mathrm{TR}^r(S;\mathbb{Z}_p)^{hS^1}$ would falsify the identification of the graded pieces in Theorem 1.5.

Watch

Extended reading notes

Core claim

Theorem 1.5 states that for a quasisyntomic ring $S$ and $1\le r\le\infty$, the motivic filtrations of $\mathrm{TR}^r(S;\mathbb{Z}_p)^{hS^1}\to\mathrm{TR}^r(S;\mathbb{Z}_p)$ are complete, exhaustive, multiplicative and $\mathbb{Z}$-indexed, and that their graded pieces are identified after quasisyntomic descent with the $r$-Nygaard filtered Nygaard-completed prismatic cohomology: $\mathrm{gr}^{i,\mathrm{even}}_M\mathrm{TR}^r(S;\mathbb{Z}_p)^{hS^1}\simeq N^{\ge i}_r\hat\Delta_S\{i\}[2i]$ and $\mathrm{gr}^{i,\mathrm{even}}_M\mathrm{TR}^r(S;\mathbb{Z}_p)\simeq N^i_r\hat\Delta_S\{i\}[2i]$ for finite $r$, with $r=\infty$ obtained by taking the derived limit over Restriction maps. The odd graded pieces vanish locally in the quasisyntomic topology. The same filtered data controls the spectra $\mathrm{TC}_r(\mathrm{TR})$ and $\widetilde{\mathrm{TC}}_r(\mathrm{TR})$, which interpolate between ordinary topological cyclic homology and topological cyclic homology of $\mathrm{TR}$.

Load-bearing premise

The step that carries the whole extension from computable rings to all quasisyntomic rings is the assertion that a general vanishing theorem applies to the TR^r pullback complex, forcing the odd homotopy groups to vanish locally; the paper invokes this theorem rather than verifying its hypotheses for TR^r.

Editorial extensions

If this is right

For a quasisyntomic ring and finite $r$, $\mathrm{TR}^r(S;\mathbb{Z}_p)^{hS^1}$ and $\mathrm{TR}^r(S;\mathbb{Z}_p)$ are locally even, so their motivic spectral sequences degenerate locally.
The even motivic layers are exactly $N^{\ge i}_r\hat\Delta_S\{i\}[2i]$ for the $S^1$-fixed points and $N^i_r\hat\Delta_S\{i\}[2i]$ for $\mathrm{TR}^r$, so the $r$-Nygaard filtration completely controls the even layers.
Taking the limit over Restriction maps identifies the layers of $\mathrm{TR}$ and $\mathrm{TR}^{hS^1}$ with $N^{\ge i}_\infty\hat\Delta_S\{i\}[2i]$ and $N^i_\infty\hat\Delta_S\{i\}[2i]$, with odd layers coming from $\mathrm{Rlim}^1$ terms.
The spectra $\mathrm{TC}_r(\mathrm{TR})$ and $\widetilde{\mathrm{TC}}_r(\mathrm{TR})$ carry motivic filtrations whose graded pieces are fibers of $R-F$ or $\mathrm{can}-\phi^{hS^1}$ on the $r$-Nygaard pieces, interpolating between $\mathrm{TC}(\mathrm{TR})$ and ordinary $\mathrm{TC}$.
In mixed and positive characteristic the filtration recovers $A_\Omega$-cohomology and, for smooth algebras over perfect fields, the $r$-truncated de Rham–Witt forms, $\mathrm{gr}^i_M\mathrm{TR}^r(S;\mathbb{Z}_p)\simeq\tau_{\le i}W_r\Omega^\bullet_{S/k}[2i]$.

Reading between the lines

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

Editorial extension: the paper's general principle suggests every Nygaard-filtration statement should have an $r$-fold analogue; the natural next test is a prismatic–de Rham–Witt comparison identifying the absolute conjugate filtration on $r$-Hodge–Tate cohomology with the absolute de Rham–Witt forms, which the paper leaves open.
Editorial extension: because the motivic filtration is defined by quasisyntomic sheafification of double-speed Postnikov filtrations, the $r$-Nygaard filtration should be regarded as a canonical invariant of $\mathrm{TR}^r$ itself; one could exploit this to define $r$-Nygaard filtered analogs of syntomic cohomology and compare them with étale motivic cohomology for $r>1$.
Editorial extension: the perfectoid base case $\pi_*\mathrm{TR}^r(R_0;\mathbb{Z}_p)^{hS^1}\simeq A_{\mathrm{inf}}(R_0)[u_r,v_r]/(u_rv_r-\xi_r)$ suggests a direct computational check of the theorem on $p$-complete polynomial rings, where the filtration should be the connective cover of the $I_r$-adic filtration.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

The r-Nygaard filtration is a genuine new construction and the perfectoid computations look right, but the proof of Theorem 1.5 has a load-bearing gap: the local odd vanishing argument for r ≥ 3 does not work as written. The stress-test concern lands, and the referee should be pointed at it directly.

What is actually new: the filtration N^{≥i}_r Δ_S, built by gluing r copies of the usual Nygaard filtration via the iterated pullback, with its Restriction/Frobenius/Verschiebung symmetries, the r-divided prismatic Frobenius, and the r-Hodge–Tate cohomology. For perfectoid inputs it becomes the ξ_r-adic filtration on A_inf, and Section 3 is the strongest part of the paper: the computation π_*TR^r(R_0)^{hS1} ≅ A_inf[u_r,v_r]/(u_r v_r − ξ_r), the passage to TR^r by killing v_r, and the R/F/V bookkeeping all appear correct. The algebraic definitions in Section 5 and the formal part of Theorem 1.6 are reasonable and worth keeping.

The soft spot is in §4.2. To show π_{2i-1}TR^r(S)^{hS1} vanishes locally on the quasisyntomic site, the author precomposes the map α: ∏_r N^{≥i}Δ̂_S → ∏_{r-1} Δ̂_S with the diagonal and claims BS22 Sec. 14 makes α∘diag surjective on a cover. But α is the banded difference, so α∘diag(x) = ((ι−φ_i)(x), …, (ι−φ_i)(x)); its image is the diagonal of the product. For r ≥ 3 the diagonal is a proper submodule, and no quasisyntomic cover changes that. The argument proves nothing about α. This step carries Theorem 1.5(2)–(3), Theorem 1.6, and the §6 corollaries. For r = 2 the diagonal argument would work, and r = 1 is BMS19; the gap is r ≥ 3, which is most of the statement.

Minor but real: the same diagonal move appears in the §6.2 proof of Theorem 1.8 with a mismatched product target; the overlap with DR23 and DM23 is mentioned but not delineated — if Theorem 1.8's content is in those papers, the author should say so and lean on it; there is a broken internal reference ('Proposition ??' in §4.1); and several 'completely analogous'/'direct corollary' claims (Proposition 5.6, the conjugate filtration identifications in §5.2) are unproved in the text.

The audience is researchers in p-adic Hodge theory, especially the THH/TR side and de Rham–Witt comparisons; the filtration and the perfectoid section carry real value for them. The reader's 'conditional' verdict is about right, with the stress-test showing where the condition bites. Recommendation: send to peer review, with a referee brief focused on the local odd vanishing for r ≥ 3 and the precise relationship to DR23/DM23.

Referee Report

4 major / 4 minor

Summary. This paper introduces, for an animated ring S and 1 <= r <= infinity, an 'r-Nygaard filtration' N^{>=i}_r Delta_S, defined as an iterated pullback of the usual Nygaard filtration under the divided prismatic Frobenius, and studies its relation to the motivic filtration of TR^r(S;Z_p) and of its S^1-homotopy fixed points. The main theorem (Theorem 1.5) asserts that, for quasisyntomic S, gr^i_M TR^r(S;Z_p)^{hS^1} is equivalent to N^{>=i}_r \hatDelta_S{i}[2i] and gr^i_M TR^r(S;Z_p) is equivalent to N^i_r \hatDelta_S{i}[2i], with analogous statements for TR, TC-related invariants, and the mixed- and positive-characteristic cases. The paper also proposes algebraic constructions on the prismatization stack, including r-Hodge-Tate divisors and a conjugate filtration, and announces a comparison with de Rham-Witt complexes.

Significance. The potential significance is high: the paper offers a natural r-parameter generalization of the Bhatt-Morrow-Scholze picture, giving motivic filtrations of TR^r in terms of prismatic cohomology with an r-fold Nygaard filtration. The perfectoid calculations in Section 3 are detailed and explicit, and the definitions are intrinsic and parameter-free. The paper also makes concrete structural predictions, such as the spectral sequences in Theorem 1.5(3). However, the general quasisyntomic claim is not established by the arguments given: the proof of local odd vanishing in Section 4.2 contains a substantive gap, and the algebraic descriptions in Section 5 are partly asserted rather than proved. At this stage the paper reads as a promising research announcement with a useful framework, but not as a complete proof of the main theorems.

major comments (4)

[§4.2, proof of Theorem 1.5(2)] The local odd-vanishing argument is not valid as written. In the exact sequence 0 -> N^{>=i}_r \hatDelta_S{i} -> ∏_{1≤k≤r} N^{>=i}\hatDelta_S{i} -> ∏_{1≤k≤r-1} \hatDelta_S{i} -> π_{2i-1}TR^r(S;Z_p)^{hS^1} -> 0, the map α is induced by the banded difference (a_1,...,a_r) ↦ (can(a_1)-φ(a_2), ..., can(a_{r-1})-φ(a_r)). Precomposing with the diagonal gives α∘diag(x) = (can(x)-φ(x), ..., can(x)-φ(x)), whose image is contained in the diagonal of ∏_{1≤k≤r-1} \hatDelta_S{i}. For r ≥ 3 the diagonal is a proper submodule of the product, so surjectivity of α∘diag onto the product after a quasisyntomic cover cannot follow from the Bhatt-Scholze vanishing theorem, which concerns the single map can-φ. Consequently the local vanishing of π_{2i-1}TR^r(-;Z_p)^{hS^1} for r ≥ 3 is not established. Since this vanishing is used to identify gr^i_M TR^r and gr^i_M TR^{r,hS^1} with N^{>=i}_r \hatDelta_S{i}[2i] and N^i_r \hatDelta_S{i}[2i], and to pass to TR and TC, Theorem 1.5(2)-(3), Theorem 1.6, and Section 6 rest on an unproved step. The author should either prove the required surjectivity directly for the product map or give a different argument for local odd vanishing.
[§5.1, Proposition 5.6] Proposition 5.6 is load-bearing but is not proved; the text says 'one arrives at the following generalization' after Proposition 5.5. The claimed recursive identification (φ^r)^* N^{>=i}_r \hatDelta_S ≃ φ^*N^{>=i}\hatDelta_S ⊗_{\hatDelta_S} ... ⊗_{\hatDelta_S} (φ^r)^*N^{>=i}\hatDelta_S is used in Lemma 5.7, Proposition 5.9, Corollary 5.14, and the conjugate-filtration construction in Section 5.2. Without a proof or a precise reference, the algebraic definition of the r-Nygaard filtration is not shown to coincide with the homotopy-theoretic filtration outside the quasiregular-semiperfectoid case.
[§5.2, Corollary 5.15] The conjugate filtration on r-Hodge-Tate cohomology and its identification with the Postnikov filtration are asserted as direct corollaries of replacing I^• by I_r^• in [BL22a]. This is not a formal substitution: the Beilinson t-structure and décalage arguments for the pair (I, φ) do not automatically carry over to the pair (I_r, φ^r), especially since Proposition 5.6, which would supply the needed Nygaard-connectivity estimates, is unproved. Because Theorem 1.3 and the relative de Rham-Witt comparison (6) depend on these identifications, the relative statements currently have the status of announced results rather than proved theorems.
[§4.2, proof of Theorem 1.5 (general case)] The passage from quasiregular-semiperfectoid rings to all quasisyntomic rings is delegated to 'completely analogous' to [BMS19]. For TR^r this requires checking that the quasisyntomic sheafifications of τ_{[2i-1,2i]}TR^r(-;Z_p)^{hS^1} and τ_{[2i-1,2i]}TR^r(-;Z_p) are the two-term complexes associated to the r-Nygaard filtration, and that the motivic filtrations are the sheafifications of the double-speed Postnikov filtrations. These identifications are not themselves established in the QRSPerfd calculation; they are part of what the theorem must prove. A detailed descent argument is needed, particularly because the local odd-vanishing claim in the previous comment is the only place where the QRSPerfd calculation is shown to survive quasisyntomic sheafification.

minor comments (4)

[§4.1] The sentence 'the identification regarding \hatDelta_S/ξ_r is a direct corollary of Proposition ??' contains an unresolved cross-reference; the cited proposition should be numbered explicitly.
[§3.2, Proposition 3.8] The stated source and target of the Restriction and Frobenius maps are inconsistent: R,F: TR^{r+1}(R_0;Z_p) -> TR^r(R_0;Z_p)^{hS^1} does not match Theorem 3.7, where the corresponding maps are between TR^{r+1} and TR^r, with homotopy fixed points handled separately.
[§2.4, Construction 2.18] The spectra TCr are defined using TR^r and TR^{r-1}; for r = 1 the object TR^0 is not defined, and the claimed interpolation at r = 1 should be stated with an explicit convention or a separate base-case formula.
[Throughout] There are numerous typographical errors, including 'desription' in Section 1, 'semperfect' in Definition 2.21, and 'cnstructions' in Definition 5.2; a careful proofreading pass is needed.

Circularity Check

1 steps flagged · score 4.0 of 10

One definitional reduction in the even graded pieces of TR^r; the rest of the derivation is independent.

self definitional [Section 4.1, Theorem 4.1(1)-(2); invoked in Theorem 1.5(2)]
"the associated spectral sequences calculating TRr(S; Zp)hS1 and TRr(S; Zp)tS1 ≃ TP(S; Zp) equip ˆ∆ S ≃ π0 TRr(S; Zp)hS1 ≃ π0 TRr(S; Zp)tS1 ≃ π0 TP(S; Zp) with the same complete, descending N-indexed filtration N ≥• r ˆ∆ S, which we call the r-Nygaard filtration. ... Via the multiplicative structure ... one can identify N ≥i r ˆ∆ S ⊂ ˆ∆ S = π0 TRr(S; Zp)hS1, with π2i TRr(S; Zp)hS1, via multiplication with vi r ∈ π−2i TRr(S; Zp)hS1. In particular ... π2i TRr(S; Zp)hS1 ≃ N ≥i r ˆ∆ S{i}"

The completed r-Nygaard filtration N^{≥i}_r \hat∆_S is introduced in Section 4.1 as the filtration coming from the S1-homotopy-fixed-point spectral sequence of TR^r(S;Z_p)^{hS1}; its i-th layer is, by this definition, π_{2i}TR^r(S;Z_p)^{hS1}. Theorem 1.5(2) then states that gr^{i,even}_M TR^r(S;Z_p)^{hS1} ≃ N^{≥i}_r \hat∆_S{i}[2i], but Theorem 4.1(1) has already defined the motivic filtration to be the double-speed Postnikov filtration, so gr^{i,even}_M is exactly π_{2i}. Thus this part of the 'identification' is a restatement of the definition of the completed filtration rather than an independent prediction. The independent content lies in the pullback description of N^{≥i}_r \hat∆_S in terms of the usual Nygaard filtration, and in the odd-vanishing and descent arguments.

full rationale

The paper's derivation chain is mostly a structural computation using external results: the perfectoid calculations use [BMS19, Sec. 6] and [Mat21, Sec. 7]; the quasiregular-semiperfectoid case follows from the iterated-pullback presentation of TR^r^{hS1} together with BMS19's identifications for TC^- and TP; passage to quasisyntomic rings uses quasisyntomic descent and the Bhatt-Scholze vanishing theorem [BS22, Sec. 14]. There are no fitted parameters, no numerical coincidences, and no load-bearing self-citations: the only self-reference, [And24], is the author's thesis and is not used as evidence for any theorem. The central claim has independent content: the odd homotopy vanishing, the spectral sequences, the algebraic definition of the r-Nygaard filtration on non-completed prismatic cohomology via iterated pullbacks, and the consequences for TC(TR) are genuine extensions beyond a definitional reformulation. The definitional reduction is limited to the even graded pieces of the motivic filtration of TR^r and its S1-fixed points, where the completed r-Nygaard filtration is introduced through the very homotopy groups it is then said to describe. The skeptical concern about the diagonal precomposition in Section 4.2 is a potential mathematical gap in the application of the BS22 vanishing theorem, not a circularity, and therefore does not affect this circularity score. Overall, the paper is substantially self-contained against external benchmarks, with one self-definitional feature in the even-part identification, so a score of 4 is appropriate.

Assumptions & free parameters 0 free parameters · 4 assumptions · 2 invented entities

The paper has no fitted parameters and does not introduce numerical constants. It relies on established theorems from BMS19, BS22, BL22a, and NS18. The main risks are unproved in-paper statements, especially Proposition 5.6 and the conjugate filtration identifications, and the new objects are introduced by definition rather than by independent evidence.

assumptions (4)

domain assumption Quasisyntomic descent: QRSPerfd rings form a basis for the quasisyntomic topology and the invariants satisfy descent (BMS19, Prop. 2.22).
Used in the proof of Theorem 1.5 to pass from quasiregular-semiperfectoid calculations to all quasisyntomic rings.
domain assumption Bhatt-Scholze vanishing theorem (BS22, Sec. 14) gives local surjectivity of the map alpha composed with the diagonal, used to kill odd homotopy groups.
The proof of local odd vanishing in Section 4.2 cites this theorem without verifying that the TR^r complex satisfies its hypotheses.
standard math Beilinson t-structure and decalage properties for prismatic cohomology (BMS18, BMS19, BL22a) identify connective covers and conjugate filtrations.
Invoked throughout Section 5 for connectivity estimates, conjugate filtrations, and the decalage description.
ad hoc to paper The recursive tensor-product description of (phi^r)^* N^{>=i}_r \hat\Delta_S, stated as Proposition 5.6, is taken as an input.
This is a key algebraic identification for the non-completed filtration, but the paper states it without proof and without a precise external reference.

invented entities (2)

r-Nygaard filtration N^{>=i}_r \Delta_S
purpose: Defines a filtration on absolute prismatic cohomology whose graded pieces are intended to match the motivic filtration of TR^r.
The object is introduced by an iterated pullback construction; there is no external benchmark constraining it beyond the internal definitions and the identifications the paper derives.
r-Hodge-Tate divisor Sigma_{HT,r}
purpose: Closed substack of the prismatization stack where the ideal sheaf I^r vanishes; target for r-Hodge-Tate cohomology.
Defined in Section 5.1. Its existence and properties are asserted, and further stacky details are deferred to future work.

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Pith. "Pith review of TR and the $r$-Nygaard filtered prismatic cohomology." pith.science (2026). https://pith.science/paper/TOJQZ4XH

@misc{pith2026241200914,
  author       = {Pith},
  title        = {Pith review of: TR and the $r$-Nygaard filtered prismatic cohomology},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/TOJQZ4XH}},
  note         = {Machine review of arXiv:2412.00914}
}

abstract

Given an animated ring $S$, we define a filtration on its absolute prismatic cohomology $\mathcal{N}_r^{\geq i} \mathbb{\Delta}_S$, which we call the $r$-Nygaard filtration and study some of its main properties using a mixture of algebraic and homotopy theoretic techniques. This filtration is obtained by suitably gluing $r$-copies of the usual Nygaard filtration and corresponds to the $\xi_r$-adic filtration on $\mathbb{A}_{\mathrm{inf}}$, in the case that $S$ is a perfectoid ring. Using this, we study the motivic filtration of topological restriction homology $\mathrm{TR}^r (S;\mathbb{Z}_p)$ and of its $S^1$-homotopy fixed points. We also pursue connections with the theory of topological cyclic homology. Finally we discuss connections with the de Rham--Witt complex, towards a prismatic - de Rham--Witt comparison theorem.

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Pith tools

Reviewed August 12, 2026 · model on record in the stance chip above.

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arXiv 2019