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arxiv: 1906.12267 · v1 · pith:ZIY3OH7Enew · submitted 2019-06-28 · 🧮 math.AG

Derived invariants from topological Hochschild homology

Benjamin Antieau , Daniel Bragg This is my paper

Pith reviewed 2026-05-25 13:29 UTC · model grok-4.3

classification 🧮 math.AG
keywords derived invariantstopological Hochschild homologyHodge-Witt cohomologycrystalline cohomologyslope spectral sequencepositive characteristicHodge numbersderived equivalence
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The pith

Topological Hochschild homology makes slope numbers, domino numbers, and Hodge-Witt numbers into derived invariants.

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

This paper establishes that p-adic quantities tied to Hodge-Witt and crystalline cohomology, such as slope numbers, domino numbers, and Hodge-Witt numbers, remain the same for derived equivalent varieties in positive characteristic. The argument relies on topological Hochschild homology combined with the slope spectral sequence to track these quantities across equivalences. A reader would care because it imposes concrete limits on the possible Hodge numbers of such varieties, extending earlier work from characteristic zero. These invariants can therefore be used to distinguish derived equivalence classes in new ways.

Core claim

We show that various p-adic quantities related to Hodge-Witt and crystalline cohomology groups, including slope numbers, domino numbers, and Hodge-Witt numbers, behave as derived invariants. This follows from examining their behavior under derived equivalences using the slope spectral sequence theory. Consequently, derived equivalent varieties satisfy the same restrictions on their Hodge numbers, partially extending results from characteristic zero to positive characteristic.

What carries the argument

topological Hochschild homology together with the slope spectral sequence for extracting p-adic invariants

If this is right

  • Slope numbers are preserved by derived equivalences.
  • Domino numbers are preserved by derived equivalences.
  • Hodge-Witt numbers are preserved by derived equivalences.
  • Derived equivalent varieties in positive characteristic share the same possible Hodge numbers up to the derived invariant constraints.

Where Pith is reading between the lines

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

  • One could test these invariants on explicit examples like elliptic curves or K3 surfaces in positive characteristic to see the restrictions in action.
  • These results might connect to questions about when two varieties are derived equivalent by providing additional numerical obstructions.
  • Extending the method to other cohomology theories could yield more derived invariants.

Load-bearing premise

The slope spectral sequence theory applies directly to the topological Hochschild homology of derived equivalent varieties without additional obstructions.

What would settle it

Observation of two derived equivalent varieties over a positive characteristic field with different slope numbers would disprove the invariance.

Figures

Figures reproduced from arXiv: 1906.12267 by Benjamin Antieau, Daniel Bragg.

Figure 1
Figure 1. Figure 1: Four spectral sequences associated to a smooth prope [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Four spectral sequences associated to a smooth prope [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: A portion of the E2-page of the Tate spectral sequence (6) computing TP. The Tate spectral sequence is 2-periodic in the columns and for C smooth and proper over a perfect field of characteristic p it is bounded in the rows by [AN18, Corollary 5]. We will need the following proposition. Proposition 3.8. If X is smooth and proper over a perfect field of positive characteristic p, then the four spectral sequ… view at source ↗
Figure 4
Figure 4. Figure 4: F V = V F = p F a = σ(a)F for all a ∈ W V σ(a) = aV for all a ∈ W da = ad for all a ∈ W d 2 = 0 F dV = d [PITH_FULL_IMAGE:figures/full_fig_p009_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: The E1-page of the slope spectral sequence (8) (with horizontal differentials) and the E2-page of the descent spectral sequence for TR (5) (with diagonal differentials). Let F⋆TR(X) = RΓ(X, τ>⋆TR(OX)) be the filtration giving rise to the descent spectral sequence. The differentials in the slope and descent spectral sequences are compatible in the following sense. Lemma 4.3. Let X be a smooth proper variety… view at source ↗
Figure 6
Figure 6. Figure 6: The E1-page of the slope spectral sequence and the E2-page of the descent spectral sequence for TR of a smooth proper surface X over k. The horizontal arrow is the only possibly non-zero differential on the first page of the slope spectral sequence. All differentials on all pages of the descent spectral sequence vanish (see Lemma 5.4). The red box (the small 1 × 1 box in the upper left) indicates the sourc… view at source ↗
Figure 7
Figure 7. Figure 7: The E1-page of the slope spectral sequence and the E2-page of the descent spectral sequence for TR of a smooth proper threefold X over k. The horizontal arrows are all possibly non-zero differentials on the first page of the slope spectral sequence. The diagonal arrows are all possibly non-zero differentials on the second page of the descent spectral sequence (see Lemma 5.4). The descent spectral sequence … view at source ↗
Figure 8
Figure 8. Figure 8: The E2 and E3 = E∞ pages of the descent spectral sequence for (X, α). The horizontal arrows are the maps induced by the differentials appearing on the E1 page of slope spectral sequence for X. The term ord(α)W is the kernel of the non-zero differential, which is generated by ord(α) ∈ W, where ord(α) = ord(dlog(α)) is the order of α [PITH_FULL_IMAGE:figures/full_fig_p030_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: A portion of the E2-page of the Tate spectral sequence for (X, α). · · · TR0(X, α) 0 TR0(X, α) 0 TR0(X, α) · · · · · · 0 0 0 0 0 · · · · · · K(X, α) 0 K(X, α) 0 K(X, α) · · · [PITH_FULL_IMAGE:figures/full_fig_p031_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: A portion of the E3 = E∞-page of the Tate spectral sequence for (X, α). We therefore find short exact sequences 0 → TR0(X, α) → TPi(X, α) → K(X, α) → 0 (46) [PITH_FULL_IMAGE:figures/full_fig_p031_10.png] view at source ↗
read the original abstract

We consider derived invariants of varieties in positive characteristic arising from topological Hochschild homology. Using theory developed by Ekedahl and Illusie-Raynaud in their study of the slope spectral sequence, we examine the behavior under derived equivalences of various $p$-adic quantities related to Hodge-Witt and crystalline cohomology groups, including slope numbers, domino numbers, and Hodge-Witt numbers. As a consequence, we obtain restrictions on the Hodge numbers of derived equivalent varieties, partially extending results of Popa-Schell to positive characteristic.

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 paper claims that various p-adic quantities (slope numbers, domino numbers, and Hodge-Witt numbers) extracted from Hodge-Witt and crystalline cohomology via the slope spectral sequence are derived invariants because they are determined by topological Hochschild homology (THH). As a consequence, it obtains restrictions on the Hodge numbers of derived-equivalent varieties in positive characteristic, partially extending Popa-Schell.

Significance. If the central claims are established, the work supplies new derived invariants in positive characteristic and extends known Hodge-number restrictions, which would be a useful contribution to the study of derived equivalences for varieties over fields of positive characteristic.

major comments (2)
  1. The central claim requires that the slope numbers, domino numbers, and Hodge-Witt numbers extracted via the Ekedahl-Illusie-Raynaud slope spectral sequence are recovered from THH and preserved under derived equivalences. The manuscript must supply explicit comparison maps or functoriality statements showing that the relevant pages of the slope spectral sequence (or the slope filtration on crystalline cohomology) are determined by THH; without these, the invariance statements do not follow from the abstract invocation of the existing theory.
  2. The extension of Popa-Schell results to positive characteristic rests on the same comparison; any gap in the THH-to-slope-spectral-sequence link directly affects the Hodge-number restrictions stated in the final section.
minor comments (1)
  1. Clarify the precise relationship between the homotopy groups of THH and the graded pieces of the de Rham-Witt complex used in the slope spectral sequence.

Simulated Author's Rebuttal

2 responses · 0 unresolved

Thank you for the detailed report. We address the major comments below and will revise the manuscript accordingly to strengthen the exposition of the comparisons.

read point-by-point responses

  1. Referee: The central claim requires that the slope numbers, domino numbers, and Hodge-Witt numbers extracted via the Ekedahl-Illusie-Raynaud slope spectral sequence are recovered from THH and preserved under derived equivalences. The manuscript must supply explicit comparison maps or functoriality statements showing that the relevant pages of the slope spectral sequence (or the slope filtration on crystalline cohomology) are determined by THH; without these, the invariance statements do not follow from the abstract invocation of the existing theory.

    Authors: We acknowledge that the current manuscript relies on the established theory of Ekedahl-Illusie-Raynaud without spelling out the functoriality in detail. In the revision, we will add explicit statements and, where necessary, construct the relevant comparison maps between THH and the slope spectral sequence. This will clarify how the slope numbers etc. are recovered from THH and hence invariant under derived equivalences. We believe this addresses the concern directly. revision: yes

  2. Referee: The extension of Popa-Schell results to positive characteristic rests on the same comparison; any gap in the THH-to-slope-spectral-sequence link directly affects the Hodge-number restrictions stated in the final section.

    Authors: As the Hodge number restrictions derive from the invariance of the Hodge-Witt numbers (which are part of the slope spectral sequence data), the added explicit comparisons in the revision will also solidify the extension to positive characteristic. We will update the final section to reference the new details. revision: yes

Circularity Check

0 steps flagged

No circularity; derivation relies on external Ekedahl-Illusie-Raynaud theory

full rationale

The paper invokes the slope spectral sequence theory of Ekedahl and Illusie-Raynaud (external authors) to analyze p-adic quantities from THH under derived equivalences. No self-citation chains, fitted parameters renamed as predictions, self-definitional steps, or ansatzes smuggled via the authors' prior work appear in the abstract or described derivation. The central claim extracts invariants from THH and applies established prior results; the applicability question raised by the skeptic is a correctness concern, not a reduction of the argument to its own inputs by construction. The derivation is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the applicability of the Ekedahl-Illusie-Raynaud slope spectral sequence theory to the THH setting for derived equivalences; no free parameters or invented entities are mentioned in the abstract.

axioms (1)
  • domain assumption Slope spectral sequence theory of Ekedahl and Illusie-Raynaud applies to topological Hochschild homology invariants under derived equivalences.
    Invoked to examine behavior of p-adic quantities.

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