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This paper computes the syntomic cohomology of every E1 MU-algebra form of the truncated Brown–Peterson spectrum BP⟨n⟩, and from the explicit formula derives the redshift, telescope, and Lichtenbaum–Quillen properties for the algebraic K-th

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load-bearing objection A genuinely new computation of syntomic cohomology for all E1 MU-algebra forms of BP<n>, with the right consequences if the imported differentials from the unpublished AKHW24 preprint hold up. the 4 major comments →

arxiv 2602.14380 v3 pith:C46JLG7O submitted 2026-02-16 math.KT math.AT

Syntomic cohomology of truncated Brown--Peterson spectra

Gabriel Angelini-Knoll This is my paper
classification math.KT math.AT MSC 19D5514F3055N22
keywords syntomic cohomologytruncated Brown–Peterson spectrumalgebraic K-theoryredshifttelescope conjectureLichtenbaum–Quillenmotivic filtrationtopological Hochschild homology
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

This paper proves an explicit formula for the syntomic cohomology of every E1 MU-algebra form of the truncated Brown–Peterson spectrum BP⟨n⟩, modulo (p, v_1, …, v_{n+1}). The formula — a finite bigraded F_p-vector space with exterior generators ∂, λ_1,…,λ_{n+1} and correction classes Ξ_{j,d} — is uniform across all primes and heights. From it the author derives the redshift, telescope, and Lichtenbaum–Quillen statements for the algebraic K-theory K(BP⟨n⟩): the K-theory has height exactly n+1, the telescope localization is an equivalence, and the Lichtenbaum–Quillen map has bounded-above fiber. This extends previous results from E3 MU-algebra forms to all E1 forms, the generality needed because several natural forms provably admit no E2 ring structure. The same machine also yields new explicit computations of TC and algebraic K-theory for BP⟨2⟩ at primes p≥5 and for height-3 BP⟨3⟩ at primes p≥7.

Core claim

On the paper's own terms, the central discovery is Theorem 5.5: for any E1 MU-algebra form of BP⟨n⟩, the mod (p, v_1, …, v_{n+1}) syntomic cohomology is isomorphic, as a bigraded F_p-vector space, to F_p⟨∂, λ_1,…,λ_{n+1}⟩ ⊕ ⊕_{j=1}^{n+1} F_p⟨λ_s : s≠j⟩{Ξ_{j,d} : 0<d<p}, with bidegrees ∥∂∥=(−1,1), ∥λ_i∥=(2p^i−1,1), ∥Ξ_{j,d}∥=(2p^j−1−2dp^{j−1},1), and the v_{n+1}-Bockstein spectral sequence collapses. In particular, the answer is finite, of F_p-dimension 2^{n+2}+2n(n+1)(p−1). Corollaries 6.5–6.7 convert this into the qualitative statement that K(BP⟨n⟩) is an fp-spectrum of type n+1, so the Lichtenbaum–Quillen map has bounded-above fiber, the telescope conjecture holds for K(BP⟨n⟩), and redshif

What carries the argument

The argument runs through the MU-based motivic filtration on topological Hochschild homology. The graded pieces for THH(BP⟨n⟩;F_p) are computed as F_p⟨λ_1,…,λ_{n+1}⟩[μ^{p^{n+1}}], and the comparison map to THH(F_p) is shown injective modulo (λ_1,…,λ_{n+1}). The prismatic-cohomology step is the periodic t-Bockstein spectral sequence, whose d_1-differential is d_1(ε_{n+1})=t μ^{p^{n+1}} (coming from the null-homotopy of v_{n+1} acting trivially) and whose higher differentials d_{p^m}(t^{p^m−1}λ_S)=t^{p^m+p^{m−1}}λ_m λ_S are transferred from the BP spectral sequence via that injective comparison. Careful bookkeeping of these differentials splits the TC−-page into four summands and yields the sy

Load-bearing premise

The load-bearing assumption is that the periodic t-Bockstein differentials d_{p^m}(t^{p^m−1}λ_S) = t^{p^m+p^{m−1}}λ_mλ_S hold for BP⟨n⟩; the paper transfers them from the BP spectral sequence and from the injectivity of the comparison map into the F_p spectral sequence modulo (λ_1,…,λ_{n+1}), and if either of those inputs fails, Theorem 5.5 and all qualitative consequences collapse.

What would settle it

Compute the periodic t-Bockstein spectral sequence for a specific E1 MU-algebra form of BP⟨2⟩ at p=5 (or BP⟨3⟩ at p=7) and compare the bidegrees of surviving classes with Theorem 5.5: the theorem predicts exactly 2^{n+2}+2n(n+1)(p−1) classes in the stated bidegrees, so a single unexpected differential — for instance d_{p^m}(t^{p^m−1}λ_S) differing by a unit or a λ_m appearing where the formula says none — would show up as an extra or missing class and falsify the transfer.

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

  • K(BP⟨n⟩) is an fp-spectrum of type n+1, so the Lichtenbaum–Quillen map K(BP⟨n⟩)_{(p)} → L^f_{n+1}K(BP⟨n⟩)_{(p)} has bounded-above fiber.
  • The telescope localization L^f_{n+1}K(BP⟨n⟩) → L_{n+1}K(BP⟨n⟩) is an equivalence; the telescope conjecture holds for the algebraic K-theory of BP⟨n⟩.
  • Algebraic K-theory of BP⟨n⟩ has chromatic height exactly n+1: redshift holds, in particular T(n+1)⊗K(BP⟨n⟩) is nonzero.
  • For n=2 and p≥5, TC(BP⟨2⟩)/(p,v_1,v_2) is computed explicitly as F_p[v_3]⟨∂,λ_1,λ_2,λ_3⟩ ⊕ ⊕_{j=1}^3 F_p[v_3]⟨λ_s:s≠j⟩{Ξ_{j,d}}, and K(BP⟨2⟩)/(p,v_1,v_2) has the corresponding description up to a small correction; this is new at p=5 and for all E1 forms.
  • For n=3 and p≥7, the paper provides the first explicit mod (p,v_1,v_2,v_3) K-theory computation for an E1-ring of chromatic height 3.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • If the transferred t-Bockstein differentials of Proposition 4.2 are correct, the same computation should apply to any E1 MU-algebra quotient of BP whose homotopy is a polynomial ring on v_1,…,v_n truncated at v_{n+1}; the explicit formula of Theorem 5.5 would then be the generic answer for all height-n truncations. A direct check at n=2, p=5 on a form that provably admits no E2 structure would be
  • The bidegree pattern of the classes Ξ_{j,d} suggests a duality with the exterior classes λ_i; the rotational symmetry the paper points out in Figure 1 hints at a higher-height Tate duality. Testing the same formula for BP⟨3⟩ at primes p=5 and p=7 would show whether this pattern persists beyond the lowest nontrivial height.
  • Because the only imported ingredient is the BP t-Bockstein differential computation, a fully self-contained proof would follow from a direct derivation of those differentials. A sceptical reader could attempt that derivation from the published BP syntomic computation alone.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

4 major / 4 minor

Summary. The paper computes, for every prime p and every n ≥ −1, the mod (p, v_1, …, v_{n+1}) MU-based syntomic cohomology of an arbitrary E_1 MU-algebra form of the truncated Brown–Peterson spectrum BP⟨n⟩. The stated result (Theorem 5.5) is an explicit exterior/polynomial description with generators λ_1,…,λ_{n+1}, ∂, and Ξ_{j,d}. From this computation the paper derives qualitative consequences for algebraic K-theory: K(BP⟨n⟩) has fp-type n+1 (Theorem 6.4/Corollary 6.5), the telescope localization map L^f_{n+1}K(BP⟨n⟩) → L_{n+1}K(BP⟨n⟩) is an equivalence (Corollary 6.6), and K(BP⟨n⟩) has redshift height exactly n+1 (Corollary 6.7). It also claims an explicit calculation of TC(BP⟨2⟩)/(p,v_1,v_2) for all p ≥ 5, extending previous work and covering arbitrary E_1 MU-algebra forms (Theorem 6.1, Corollary 6.2). The technical route goes through the motivic filtration, Hochschild homology, Hodge–Tate cohomology, and a periodic t-Bockstein spectral sequence for prismatic cohomology, with several inputs imported from the unpublished preprint [AKHW24] by the same author group.

Significance. If correct, the main computation is a substantial advance: it removes the E_3/higher-structure hypothesis from the Hahn–Wilson redshift results and gives the first explicit height-3 algebraic K-theory computation. The extension to arbitrary E_1 MU-algebra forms is conceptually important because many natural forms, such as those from Araki or Hazewinkel generators, admit only E_1 structures. The paper also states the resulting Lichtenbaum–Quillen, telescope, and redshift conclusions cleanly. However, the central computation and the telescope corollary depend at load-bearing points on [AKHW24], an unpublished preprint by the same author group. The present manuscript does not re-derive the needed differentials or fully verify the imported hypotheses. The qualitative consequences are therefore conditional on results not available to the reader. This does not make the claims obviously wrong, but it does mean the current version is not self-contained enough for the central theorems to be checked.

major comments (4)
  1. [§4, Proposition 4.2] The periodic t-Bockstein differentials d_{p^m}(t^{p^m−1}λ_S) = t^{p^m+p^{m−1}}λ_m λ_S are asserted to follow from [AKHW24, Proposition 3.2.9], but that result is not reproduced and no proof of the transfer from BP to BP⟨n⟩ is given. In particular, the manuscript does not prove that these differentials remain valid after quotienting by (λ_1,…,λ_{n+1}), nor that the comparison map of Theorem 3.1 remains injective after t-localization and passage to higher pages. These differentials determine the E_{p^{n+2}}-page, the A_{11} summand in Proposition 5.2, and ultimately the entire description (5.6). Since [AKHW24] is an unpublished preprint by the same author group, Theorem 5.5 and all of its consequences are conditional on an external, not-yet-verifiable input.
  2. [§6.3, Corollary 6.6] The telescope equivalence is deduced from [AKHW24, Proposition 4.3.2], another unpublished same-author criterion. The proof lists several properties — finite generation, bounded below THH, bounded motivic cohomological dimension, evenness of THH(BP⟨n⟩/MU) — but it does not explicitly check every hypothesis of [AKHW24, Prop. 4.3.2] for the arbitrary E_1 MU-algebra form under consideration. If that criterion requires more than the stated E_1 conditions, the telescope conclusion is unsupported even assuming Theorem 5.5. The authors should either state and prove the criterion in this paper or verify its hypotheses in detail.
  3. [§6.1, Theorem 6.1] The proof of the p ≥ 5 computation of TC(BP⟨2⟩)/(p,v_1,v_2) tells the reader to 'extrapolate the necessary information from this picture' (Figure 1) and then asserts, based on degree gaps, that certain motivic differentials cannot occur. This is not a complete spectral sequence argument. In particular, the exclusion of differentials from the 0-line to the 3-line and from the 1-line to the 4-line is only sketched, with no systematic analysis of all bidegrees or differentials involving the Ξ-classes. Since Theorem 6.1 and Corollary 6.2 are presented as new explicit computations at p = 5, this needs to be turned into a rigorous argument.
  4. [§2.3, Proposition 2.12] The proof of the Hochschild homology computation asserts that the E_0 algebra structure 'implies' the differentials d_{2p^i−2}(μ^{p^i}) = σv_i and d_{2p^j−2}(μ^{p^j}) = 0, and then states 'we observe that there is no room for hidden E_0 THH(BP)-algebra extensions' without further justification. These assertions are load-bearing because Proposition 2.12 feeds directly into even flatness (Proposition 2.13) and the injectivity statement in Theorem 2.17. Please provide the missing details or explicit references to proofs of these extension and differential assertions.
minor comments (4)
  1. [Abstract] The abstract says the computation is 'mod (p,v_1,…,v_n)', while Theorem B and the body compute mod (p,v_1,…,v_{n+1}). Please align the wording.
  2. [Throughout] Several typos: 'the the motivic spectral sequence' (Introduction), 'monomomorphism' (Theorem 3.1 proof), 'Lichtenbaum–Qullen' (Remark 6.1), and 'A dditionally' in the abstract.
  3. [Theorem 6.1] The notation 'F_p[v^d_3]' is awkward and potentially ambiguous; it should be clearly defined as the subring F_p[v_3^d] generated by the d-th power of v_3. This occurs in both Theorem 6.1 and Corollary 6.2.
  4. [§3, Theorem 3.1] The diagram in Theorem 3.1 is missing descriptions of the vertical maps on the right-hand side; the sentence 'the vertrical maps are the tensor product...' has a typo ('vertrical') and the Frobenius maps on the right are not explicitly named. Please clarify.

Circularity Check

2 steps flagged

Main computation imports load-bearing t-Bockstein differentials and telescope criterion from unpublished same-author [AKHW24]; no definitional circularity, but the derivation chain is not self-contained.

specific steps
  1. self citation load bearing [Section 4, Proposition 4.2 (proof)]
    "The differentials d_{p^m}(t^{p^m−1}λ_S)=t^{p^m+p^{m−1}}λ_mλ_S, for all S⊂{1,···,n+1} with m∉S for each m≥1 follow from the action of the spectral sequence for BP and the differentials in that spectral sequence, computed in [AKHW24, Proposition 3.2.9]."

    These d_{p^m} t-Bockstein differentials are the exact input that determines the higher pages, the A_{11} summand (Proposition 5.2), and therefore the closed-form answer in Theorem 5.5. They are not derived or checked in this manuscript; they are imported wholesale from [AKHW24], an unpublished preprint coauthored by the present author. If [AKHW24, Prop. 3.2.9] or its transfer to arbitrary E_1 MU-algebra forms of BP⟨n⟩ fails, Theorem 5.5 and all qualitative corollaries collapse. This is not a definitional equivalence, because the target BP⟨n⟩ computation is not itself stated in [AKHW24], but it makes the central derivation chain depend on an unverified same-author citation.

  2. self citation load bearing [Section 6.3, Corollary 6.6 (proof)]
    "We computed that BP⟨n⟩∧_p has bounded below topological Hochschild homology in Proposition 2.12, the THH(MU)-module THH(BP⟨n⟩) has bounded MU-based motivic cohomological dimension in Proposition 2.14 and that THH(BP⟨n⟩/MU)=THH(R)⊗_{THH(MU)}MU is even in Proposition 2.13. Therefore, the result follows from [AKHW24, Proposition 4.3.2]."

    The telescope equivalence L^f_{n+1}K(BP⟨n⟩)→L_{n+1}K(BP⟨n⟩) is one of the paper's main qualitative conclusions (Theorem A(2)). The proof verifies a list of hypotheses but then invokes the decisive criterion entirely from [AKHW24], an unpublished preprint by the same author. The criterion is not stated or proved in this paper, and the hypotheses are not shown to imply the criterion by any external theorem. Thus this consequence is supported by the same-author citation chain rather than by a self-contained argument.

full rationale

Score 4. The central computation of Theorem 5.5 is not assumed in the cited [AKHW24] and is not obtained by renaming data or by fitting parameters, so no definitional circularity is present. In particular, the A00⊕A10⊕A01⊕A11 decomposition and the can−φ kernel/cokernel computation are internal to this paper once the differentials are granted. However, the decisive t-Bockstein differentials of Proposition 4.2—the input that shapes the E_{p^{n+2}}-page, A_{11}, and the final closed form—are imported verbatim from [AKHW24, Prop. 3.2.9], an unpublished preprint by the present author (with Hahn and Wilson), and are not re-derived or independently checked here. Likewise the telescope conclusion Corollary 6.6 depends on [AKHW24, Prop. 4.3.2] for its key criterion. These are load-bearing same-author citations rather than reductions to the theorem's own statement; the target BP⟨n⟩ computation has independent content. The paper's special-case benchmarks (LW22, HRW22, AKAC+25) provide external confidence and keep the score below 6.

Axiom & Free-Parameter Ledger

1 free parameters · 8 axioms · 0 invented entities

No data were fitted. The only undetermined numeric parameter is the exponent d in the v_3^d-module statements for BP⟨2⟩. The paper's effective assumptions are the cited external theorems, most critically the unpublished [AKHW24] spectral-sequence inputs and the conditional existing Smith–Toda complexes. No new particles, forces, or invented algebraic objects are introduced beyond the standard E1 MU-algebra-form convention.

free parameters (1)
  • d (v_3^d self-map exponent) = unspecified integer ≥ 1
    Theorem 6.1 and Corollary 6.2 identify TC/(p,v1,v2) and K/(p,v1,v2) only as F_p[v_3^d]-modules for some integer d≥1. No method is given to determine d; it is an undetermined parameter in the headline BP⟨2⟩ computation.
axioms (8)
  • domain assumption Quillen idempotent defines an E2-ring map BP→MU_(p) and BP⟨n⟩ is an E1 BP-algebra.
    Section 2.1 cites [CM15, Cor 1.3, Thm 1.2]; this gives the E1 MU-algebra structure on BP⟨n⟩ and the π_*-surjection from MU_(p).
  • domain assumption Hahn–Wilson existence of E3 MU-algebra forms of BP⟨n⟩ for all primes p and heights n.
    Section 1 cites [HW22]; the paper's conventions for arbitrary E1 forms depend on this existence result.
  • domain assumption The motivic filtration relative to MU is defined for E1 MU-algebras and has the stated E0 algebra properties.
    Recollection 2.4 cites [HRW22, Definition 4.2.1]; this is the scaffold for all gr^∗_mot computations.
  • domain assumption The BP-based periodic t-Bockstein differentials and permanent-cycle statements from [AKHW24, Prop 3.2.9, Cor 3.2.6].
    Used in Proposition 4.2 and Theorem 3.1. This is an unpublished preprint by the same author group and is load-bearing for the transfer to BP⟨n⟩.
  • domain assumption Pstrągowski's even flatness criterion applies to THH(BP⟨n⟩;F_p) over THH(MU;F_p).
    Proposition 2.13 invokes [Pst23, Prop 4.20, Thm 4.21] to conclude evenness and reduce the motivic computation.
  • standard math Dundas–Goodwillie–McCarthy fiber sequence K(R)→TC(R)→Σ^{-1}Z(R).
    Used in Corollary 6.2 and Theorem 6.4 via [DGM13, Theorem 3.1.14].
  • standard math Mahowald–Rezk fp-type criterion connecting finite TC to bounded above fibers.
    Corollary 6.5 uses [MR99, Theorem 8.2(2)] as in [HW22, Theorem 3.1.3].
  • domain assumption Existence of generalized Smith–Toda complexes S/(p^{i0}, v_1^{i1}, ..., v_{n+1}^{i_{n+1}}) for the chosen exponents.
    The definition of T(n) and Theorem 6.4 explicitly condition on existence; Corollary 6.6 needs these complexes for the telescope statement. Existence of such complexes is not known in full generality, so this is a flagged domain assumption.

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read the original abstract

We compute the $\mathrm{MU}$-based syntomic cohomologies, mod $(p,v_1,\cdots,v_n)$, of all $\mathbb{E}_1$ $\mathrm{MU}$-algebra forms of the truncated Brown--Peterson spectrum $\mathrm{BP}\langle n\rangle$. As qualitative consequences, we resolve the Lichtenbaum--Quillen, telescope, and redshift questions for the algebraic K-theories of all $\mathbb{E}_{1}$ $\mathrm{MU}$-algebra forms of $\mathrm{BP} \langle n\rangle$. This extends work of the Hahn and Wilson. We also explicitly compute the algebraic K-theory of arbitrary $\mathbb{E}_{1}$ $\mathrm{MU}$-algebra forms of $\mathrm{BP}\langle 2\rangle$ at all primes $p\ge 5$ extending previous work of the author, Ausoni, Culver, H\"oning, and Rognes.A dditionally, we present a new computation of mod $(p, v_1, v_2, v_3)$ algebraic K-theory of arbitrary $\mathbb{E}_1$ $\mathrm{MU}$-algebra forms of $\mathrm{BP}\langle 3\rangle$ at all primes $p\ge 7$, the first explicit computation of algebraic K-theory of an $\mathbb{E}_1$-ring of height $3$.

Figures

Figures reproduced from arXiv: 2602.14380 by Gabriel Angelini-Knoll.

Figure 1
Figure 1. Figure 1: The mod (2, v1, v2, v3)-syntomic cohomology of BP⟨2⟩ When n = 2 and p ≥ 5, we argue that the the motivic spectral sequence collapses and we can prove the following. Theorem C (Theorem 6.1 and Corollary 6.2). Let p ≥ 5 and let BP⟨2⟩ be an arbitrary E1 MU-algebra form of the second truncated Brown–Peterson spectra at the prime p, see Definition 2.1. We can identify TC∗(BP⟨2⟩)/(p, v1, v2) = Fp[v3]⟨∂, λ1, λ2, … view at source ↗

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Reference graph

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