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arxiv: 2607.05360 · v1 · pith:ADDTADFF · submitted 2026-07-06 · math.AT · math.AG· math.KT· math.SG

Noncommutative Cartier Formulae

Semon Rezchikov This is my paper

pith:ADDTADFFreviewed 2026-07-07 14:47 UTCmodel glm-5.2open to challenge →

classification math.AT math.AGmath.KTmath.SG
keywords mathbbsymplecticformulaquantumanalogcartiercurvatureproduct
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The pith

Cap product meets cyclotomic structure in noncommutative Cartier formula

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

The paper proves that for every E_1-algebra A, the cap product action of topological Hochschild cohomology (THC) on topological Hochschild homology (THH) is compatible with the cyclotomic structure map on THH. This compatibility is expressed through a commutative square relating the ordinary cap product, the p-fold cap product via the Hill-Hopkins-Ravenel norm, and the cyclotomic structure map. The author extends this to a relative setting over a base ring R, and shows that when specialized to R = F_p, the formula recovers and generalizes the classical Cartier formula describing how the Cartier isomorphism conjugates interior products on differential forms. The paper then applies this algebraic machinery to compute the p-curvature of the Getzler-Gauss-Manin connection on periodic cyclic homology as an equivariant p-fold cap product with the Kodaira-Spencer class. Finally, under standard assumptions on the Fukaya category, the author shows that for Calabi-Yau symplectic manifolds with rational symplectic form, the p-curvature of the quantum connection computes the Quantum Steenrod operations, giving an algebro-geometric interpretation of these equivariant Gromov-Witten invariants.

Core claim

The central mechanism is a commutative square (Theorem 1.5) showing that the cyclotomic structure map on THH intertwines the cap product with the p-fold cap product via the HHR diagonal. This square is proved by constructing the cap product using the prismatic subdivision of McClure-Smith and verifying that it commutes with the cyclotomic structure defined through the norm, reducing to a combinatorial check. The relative version (Theorem 1.15) identifies the domain of the relative cyclotomic structure map with classical Hochschild invariants when the base is F_p, using the key diagram relating Z_p, THH(F_p), and F_p via Tate constructions. This identification, combined with the comparison of

What carries the argument

The key objects are: (1) the p-fold cap product, constructed using the edgewise subdivision and the Hill-Hopkins-Ravenel norm; (2) the HHR diagonal, which is a point-set homeomorphism on cofibrant objects; (3) the relative Tate diagonal, which serves as a noncommutative analog of the Cartier isomorphism; (4) the Kodaira-Spencer class, whose cap product computes the Getzler-Gauss-Manin connection; and (5) the inverse Cartier transform, whose p-curvature is computed and matched with the p-curvature of the GGM connection via the cyclotomic structure.

Load-bearing premise

The symplectic application (Theorem 1.27) depends on Assumptions A-G, particularly that the open-closed map is an isomorphism (nondegeneracy), which is a geometric condition requiring enough Lagrangian submanifolds, and on equivariant open-closed comparison results whose foundations are still being developed.

What would settle it

Find a smooth proper dg category C over Z[1/N][[x]] where the p-curvature of the Getzler-Gauss-Manin connection provably disagrees with the equivariant p-fold cap product formula of Theorem 1.23, or exhibit a Calabi-Yau symplectic manifold satisfying Assumptions A-G where the p-curvature of the quantum connection disagrees with the Quantum Steenrod operations.

Figures

Figures reproduced from arXiv: 2607.05360 by Semon Rezchikov.

Figure 1
Figure 1. Figure 1: The Cyclotomic Deligne Conjecture. It is well known that THC(A) is an E2 algebra (i.e. an algebra over the little disks operad, upper left), while T HH(A) is a genuine S 1 -spectrum (i.e. it is acted on by the circle, thought of as a sphere with one input, one output, and an ‘angle’ marking.) Moreover, T HH(A) is a module over T HC(A) via the cap product; the spaces of collections of embedded disks into th… view at source ↗
Figure 2
Figure 2. Figure 2: The ‘geometric proof ’ of the noncommutative Cartier formula. The cap product action of T HC(A) on T HH(A) corresponds to the picture on the left. The picture on the right is the p-fold cover along the central axis of the picture on the left (here p = 2). The intuition is that the ‘geometric fixed points’ of the operation on the right are exactly the picture on the left. More formally, the cyclotomic Delig… view at source ↗
Figure 3
Figure 3. Figure 3: Cut and paste interpretations of decompositions of simplices. On the left, we describe the canonical decomposition of ∆a × ∆b into simplices ∆a+b indexed by shuffles. On the right, we describe the prismatic subdivision on the simplex into products of simplices. On both sides we identify ∆n with decompositions of an interval into n + 1-subintervals. Taken together, these diagrams show that the cap product i… view at source ↗
Figure 4
Figure 4. Figure 4: Commutative diagram proving T HH(R)-linearity of the cap product ∩ ′ R. Here ∩ ′ R is actually the map used to define ∩ ′ R on the corresponding piece of the prismatic subdivision, and f0 is induced from the corresponding automorphism of T HH(R). The dotted arrows for m actually denote maps defined in terms of fµ,ν going the other way; these maps are used to define the module structure m of T HH(A) over T … view at source ↗
read the original abstract

We prove, for every $\mathbb{E}_1$ algebra $A$, a formula describing the interaction of the action of the cap product on topological Hochschild homology of $A$ with the cyclotomic structure map, as well as a variant of this result relative to a ring $R$. Specializing to $R = \mathbb{F}_p$ gives a noncommutative analog of a formula of Cartier which describes the conjugation of interior product action on differential forms by the Cartier isomorphism, and which computes the $p$-curvature of the Getzler-Gauss-Manin connection in terms of an equivariant cap product. The motivation for this formula comes from symplectic geometry, where (in the case $R=\mathbb{F}_p$ or a Novikov analog) the symplectic analog of this formula explains the interaction between the cyclotomic structure on symplectic cohomology and the quantum Steenrod operations. We prove, under standard transversality and nondegeneracy assumptions on the Fukaya category, that for a Calabi-Yau symplectic manifold with rational symplectic form, the $p$-curvature of the quantum connection computes the Quantum Steenrod operations. In particular, the $p$-curvature of the quantum connections of projective Calabi-Yau hypersurfaces, and many other examples in mirror symmetry, can be interpreted in terms of $\mathbb{Z}/p\mathbb{Z}$-equivariant genus zero Gromov-Witten invariants.

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

3 major / 7 minor

Summary. The paper proves a noncommutative analog of Cartier's formula relating the cap product on topological Hochschild (co)homology to the cyclotomic structure map. The main algebraic results (Theorems 1.5 and 1.15) establish commutative squares in the ∞-category of spectra for arbitrary E1-algebras A, using point-set models in equivariant orthogonal spectra with the convenient model structures of [20]. The relative version (Theorem 1.15) extends this to algebras over a base R. Specializing to R = F_p and using relative Tate diagonals, the author connects the spectral formula to the classical Cartier isomorphism (Theorem 7.14) and computes the p-curvature of the Getzler-Gauss-Manin connection (Theorem 1.23). The paper then applies these results to symplectic geometry (Theorem 1.27), showing that under Assumptions A–G on the Fukaya category, the p-curvature of the quantum connection computes the Quantum Steenrod operations for Calabi-Yau symplectic manifolds. The proofs of Theorems 1.5 and 1.15 proceed via explicit simplicial/cosimplicial constructions using the prismatic subdivision of McClure-Smith, reducing to combinatorial verification. The bridge from the spectral to the algebraic setting (Theorem 8.4) requires an independent reproof of a variant of Petrov-Vaintrob-Vologodsky [93] because no comparison between Kaledin's algebraic Cartier map and the spectral construction is available.

Significance. The paper makes a substantial contribution by providing a uniform spectral-algebraic framework connecting cyclotomic structure on THH with the Cartier isomorphism in characteristic p. The core algebraic theorems (1.5, 1.15) are clean and their proofs are direct: the key diagram (4.5) commutes 'essentially by construction,' and homotopical control is handled by Proposition 2.9 and Lemma 4.10, which are sound. The explicit point-set construction of the cap product via prismatic subdivision (Section 3) and the verification that it agrees with the derived cap product (Lemma 3.23) are valuable. The application to symplectic geometry (Theorem 1.27) is clearly conditional on Assumptions A–G, which the author transparently flags as consequences of the state of Fukaya-categorical foundations. Theorem 1.23 gives a falsifiable, computable prediction for p-curvature. The paper ships a concrete, checkable construction rather than a purely abstract existence result.

major comments (3)
  1. Theorem 8.4 is the load-bearing step connecting the spectral noncommutative Cartier formula (Theorem 6.17) to the algebraic inverse Cartier transform. Its proof constructs diagram (8.8) and identifies its top row with (1⊗F'⊗1)∘F*(κ')⊗1, then invokes Lemma 8.1 to conclude this is the Cartier transform. This identification depends on the commutativity of diagram (8.6), a large diagram involving E∞-ring maps between HH(R/F_p), HH(R_{Z_p}/Z_p), THH(R̃), their Tate constructions, and Ω^1. The author states this commutativity 'follows from Theorem 7.14 and the diagrams (6.13) and (6.14),' but the verification requires checking compatibility of multiple E∞-ring maps and module structures across a diagram with roughly a dozen objects. The text does not trace through this verification in sufficient detail for the reader to confirm it. Since this is the single step that cannot be shortcut by a文献引用
  2. Section 8.2, proof of Theorem 8.4: the identification of the top row of (8.8) with the map (1⊗F'⊗1)∘F*(κ')⊗1 is stated to follow from 'the outer square (consisting of rows 2 and 4 mapping to rows 1 and 5) of the diagram (8.6), together with the factorization claim about F' in Theorem 7.14.' The reader needs to see how the module structures over the various E∞-rings in (8.6) are transported through (8.8), particularly how the T HH(R)-module structure on the domain of ϕ_R interacts with the R_{Z_p}-module structure appearing in the top row of (8.8). A more explicit verification, or at minimum a lemma isolating the compatibility of module structures across (8.6), would strengthen this critical step.
  3. Theorem 1.23 requires that HH*(C/R) is p-torsion-free, which restricts to p > f(m,n,N,r). The proof of Theorem 1.23 also cannot use Petrov-Vaintrob-Vologodsky [93] directly because 'there is no comparison between Kaledin's noncommutative Cartier map and corresponding spectral constructions available in the literature' (Section 8). This necessitating an independent reproof. While the reproof via Theorem 8.4 is a reasonable strategy, the p-torsion-free hypothesis is essential for the identification in Theorem 8.4 (via Proposition 6.16, which requires dualizability and thus the lift to a standard cyclotomic base). The author should clarify whether the p-torsion-free condition enters only through the applicability of Proposition 6.16, or whether it is also needed for the commutativity of (8.6) itself.
minor comments (7)
  1. The paper would benefit from a notation table. The proliferation of symbols (R, R̃, R_{Z_p}, R_p, ˆΩ^1, etc.) across Sections 6–8 makes it difficult to track which ring is the base at each step.
  2. In diagram (6.7), the statement that 'all rectangles commute except for the right-most rectangle involving THH(F_p) and F_p' is important but easy to miss. Consider highlighting this more prominently, as it is the key reason F_p is not a standard cyclotomic base.
  3. Section 7.3.2: the claim that HH(k[x]/k) is the trivial square-zero extension k[x]⊕Ω^1_{k[x]/k}[1] as E∞-algebras when the S^1-action is forgotten relies on degree considerations and [76, 7.4.1.18]. A brief indication of why the extension is n-small would help the reader.
  4. Remark 7.20 discusses completed tensor products and their homotopical meaning. The reference to Appendix I for the condensed mathematics approach is noted, but the main text could briefly state whether the results of Theorem 7.16 for R=k[[x]], k((x)) depend on any properties specific to the condensed framework beyond what is in Lemma 10.19.
  5. The cyclotomic Deligne conjecture (Appendix A) is stated as a conjecture and not proven in full generality. The paper proves it 'for one particular point in O_∩(p).' This is sufficient for the main theorems but should perhaps be stated more explicitly in the introduction to manage expectations.
  6. Several references appear to be to unpublished or in-progress work ([97], work in progress). Where these are load-bearing for the symplectic application (Theorem 1.27), the conditional nature is clear, but the reader should be informed of the status.
  7. In the proof of Theorem 9.1, the obstruction theory computation uses Serre's result that π_i(S)⊗F_p = 0 for 0 < i < 2p−3. The bound N ≥ q/2 + 1 is derived, but the exact relationship between q and the vanishing range could be stated more explicitly.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for a careful and substantive report. The core algebraic results (Theorems 1.5, 1.15) and the construction of the cap product via prismatic subdivision are acknowledged as sound. The three major comments all concern the proof of Theorem 8.4 in Section 8.2, specifically the verification of commutativity of diagram (8.6) and the role of the p-torsion-free hypothesis. We address these below.

read point-by-point responses

  1. Referee: Theorem 8.4 is the load-bearing step connecting the spectral noncommutative Cartier formula (Theorem 6.17) to the algebraic inverse Cartier transform. Its proof constructs diagram (8.8) and identifies its top row with (1⊗F'⊗1)∘F*(κ')⊗1, then invokes Lemma 8.1 to conclude this is the Cartier transform. This identification depends on the commutativity of diagram (8.6), a large diagram involving E∞-ring maps between HH(R/F_p), HH(R_{Z_p}/Z_p), THH(R̃), their Tate constructions, and Ω^1. The author states this commutativity 'follows from Theorem 7.14 and the diagrams (6.13) and (6.14),' but the verification requires checking compatibility of multiple E∞-ring maps and module structures across a diagram with roughly a dozen objects. The text does not trace through this verification in sufficient detail for the reader to confirm it. Since this is the single step that cannot be shortcut by a文献引用

    Authors: The referee is correct that the verification of diagram (8.6) is insufficiently detailed in the current manuscript. The diagram (8.6) is assembled from three pieces: (i) the middle three rows, which are constructed by Theorem 7.14 and the diagrams (6.13)–(6.14); (ii) the top row, obtained by pushout along Z_p → F_p; and (iii) the bottom row, obtained by pushout along Z_p^{tCp} → F_p^{tCp}. The commutativity of the outer rectangle involving the second and fourth columns of (6.7) induces the vertical arrows connecting rows 2–3 and rows 4–5 of (8.6). What is missing is an explicit lemma that isolates the compatibility of the E∞-ring maps and module structures across this assembly. We will add such a lemma (to be labeled Lemma 8.5 in the revision) that states precisely which compatibilities are needed and verifies them. Specifically, the key points are: (a) the map Z_p → THH(F_p) of E∞-rings in cyclotomic spectra (constructed from TC(F_p) or from the cyclotomic trace) is compatible with the collapse maps to F_p and Z_p respectively, as verified in Lemma 6.6; (b) the commutativity of (6.13) and (6.14) as diagrams of E∞-S^1-rings, which follows from the construction of the cyclotomic structure maps on THH of commutative ring spectra as natural E∞-maps; and (c) the compatibility of the lax monoidal structure of the Tate construction with the pushouts, which is a formal consequence of the symmetric monoidal structure on the Tate construction for cofibrant objects in the model structure of Proposition 2.9. The lemma will make explicit that these three ingredients suffice, and that no additional compatibilities are needed. We believe this addresses the referee's concern, but acknowledge that the current text is inadequate without it. revision: yes

  2. Referee: Section 8.2, proof of Theorem 8.4: the identification of the top row of (8.8) with the map (1⊗F'⊗1)∘F*(κ')⊗1 is stated to follow from 'the outer square (consisting of rows 2 and 4 mapping to rows 1 and 5) of the diagram (8.6), together with the factorization claim about F' in Theorem 7.14.' The reader needs to see how the module structures over the various E∞-rings in (8.6) are transported through (8.8), particularly how the T HH(R)-module structure on the domain of ϕ_R interacts with the R_{Z_p}-module structure appearing in the top row of (8.8). A more explicit verification, or at minimum a lemma isolating the compatibility of module structures across (8.6), would strengthen this critical step.

    Authors: This comment is closely related to the first, and we agree that the transport of module structures through (8.8) needs to be made explicit. The key subtlety is that the domain of ϕ_R is THH(A) ∧_{THH(R)} R^{tCp}, where R^{tCp} is an R-module via the Tate-valued Frobenius, while the top row of (8.8) involves HH(A_{Z_p}/R_{Z_p}) ⊗_{R_{Z_p}} R^{tCp}, which is an R_{Z_p}-module. The bridge between these is provided by Proposition 6.12, which identifies THH(A) ∧_{THH(R)} R^{tCp} with HH(A/R) ⊗_R R^{tCp} when R = R̃ ∧ F_p for R̃ a standard cyclotomic base. The identification proceeds through the chain of equivalences in (6.15), which passes through Z_p-coefficients. The module structure compatibility that needs to be checked is that the THH(R)-module structure on the domain of ϕ_R, when transported through the equivalences of (6.15), agrees with the R_{Z_p}-module structure on HH(A_{Z_p}/R_{Z_p}) ⊗_{R_{Z_p}} R^{tCp}. This follows from the fact that all maps in the left column of (8.6) are maps of E∞-S^1-rings, and the module structures are induced by restriction along these maps. We will add an explicit verification of this compatibility as part of the revised Lemma 8.5 mentioned above, and will also expand the discussion surrounding (8.8) to spell out how the module structures are transported at each step. revision: yes

  3. Referee: Theorem 1.23 requires that HH*(C/R) is p-torsion-free, which restricts to p > f(m,n,N,r). The proof of Theorem 1.23 also cannot use Petrov-Vaintrob-Vologodsky [93] directly because 'there is no comparison between Kaledin's noncommutative Cartier map and corresponding spectral constructions available in the literature' (Section 8). This necessitating an independent reproof. While the reproof via Theorem 8.4 is a reasonable strategy, the p-torsion-free hypothesis is essential for the identification in Theorem 8.4 (via Proposition 6.16, which requires dualizability and thus the lift to a standard cyclotomic base). The author should clarify whether the p-torsion-free condition enters only through the applicability of Proposition 6.16, or whether it is also needed for the commutativity of (8.6) itself.

    Authors: We can clarify this point. The p-torsion-free condition enters in two distinct places, and the commutativity of (8.6) itself does not require it. Specifically: (1) The commutativity of diagram (8.6) is a statement about E∞-ring maps and module structures involving THH, HH, and Tate constructions of the base rings R, R_{Z_p}, and F_p. This commutativity holds unconditionally — it is a formal consequence of the functoriality of THH as a symmetric monoidal functor to cyclotomic spectra, the properties of the map Z_p → THH(F_p), and the lax monoidal structure of the Tate construction. No p-torsion-free hypothesis is needed here. (2) The p-torsion-free condition enters through Proposition 6.16, which requires that à be dualizable over R̃ (i.e., smooth and proper) so that ϕ_R is an equivalence. The p-torsion-free condition on HH*(C/R) is used to ensure that the lift à over S[1/N] (or a related spherical lift) exists and is smooth and proper, via Theorem 9.1 and Proposition 9.12. The p-torsion-free condition ensures that the relevant obstruction groups vanish for p large enough, allowing the inductive construction of the lift. (3) Additionally, the p-torsion-free condition is used in the comparison of Theorem 7.16, where two connections on a free module with the same flat sections are identified; this requires the module to be p-torsion-free (see Remark 10.13). We will add a remark after Theorem 8.4 making this threefold role of the p-torsion-free condition explicit. revision: yes

Circularity Check

0 steps flagged

No significant circularity; one minor self-citation dependency in the bridge theorem, but the central algebraic results are independently derived.

full rationale

The paper's main theorems (1.5, 1.15) are derived from explicit point-set constructions of the cap product (Section 3) and the cyclotomic structure map (via the HHR norm [6], an external reference). The key diagram (4.5) commutes 'essentially by construction' — but this is a genuine point-set verification using the prismatic subdivision of McClure-Smith [87], not a definitional tautology. The HHR diagonal is an isomorphism on cofibrant objects by Proposition 2.9(7), which is cited from [20] (Blumberg, a different author). The bridge theorem (Theorem 8.4) connecting spectral to algebraic Cartier transforms does rely on the commutativity of diagram (8.6), whose verification is attributed to 'Theorem 7.14 and the diagrams (6.13) and (6.14).' Theorem 7.14 is proven within the paper (Sections 7.2-7.4, H.1), and diagrams (6.13)-(6.14) are verified from the cyclotomic structure of Z_p and F_p using results of Nikolaus-Scholze [90]. The author explicitly notes the inability to cite Petrov-Vaintrob-Vologodsky [93] due to missing comparisons, necessitating an independent reproof — this is the opposite of circular. The symplectic application (Theorem 1.27) depends on Assumptions A-G, which are clearly flagged as conditional and external. The only minor concern is that Theorem 8.4's proof involves checking a large diagram (8.6) whose commutativity is asserted rather than fully traced, but this is a completeness gap, not circularity: the components are independently established, not defined in terms of the conclusion. No step reduces to its inputs by construction or by a self-citation chain that is itself unverified. The author's prior work [96] is cited for motivation and context but is not load-bearing for any algebraic theorem. Score 2 reflects the minor dependency on unverified diagram commutativity in the bridge step, which is a correctness risk rather than circularity.

Axiom & Free-Parameter Ledger

0 free parameters · 5 axioms · 1 invented entities

The paper introduces no free parameters or fitted constants. The axioms are primarily domain assumptions from equivariant stable homotopy theory ([6, 20, 53, 87, 90]) and arithmetic geometry ([91, 93]). The one ad hoc element is Assumptions A-G on the Fukaya category, which are standard in symplectic topology but unverified in the needed generality. The cyclotomic Deligne conjecture is an invented framework but is not load-bearing for the main theorems.

axioms (5)
  • domain assumption The convenient model structures of [20] (Proposition 2.9) provide symmetric monoidal model structures on orthogonal S1-spectra with the required properties (cofibrant commutative ring spectra forget to cofibrant spectra, geometric fixed points commute with smash products on cofibrant objects, HHR di
    The entire point-set proof strategy depends on these model structures. Proposition 2.9 is quoted from [20, Thm C] and its properties are used throughout Sections 3-5.
  • domain assumption The cyclotomic structure map on THH(A) can be described via the HHR norm as the geometric realization of levelwise diagonal maps A^{∧n} → Φ^{Cp} N^{Cp}_e A^{∧n} (equation 4.2), following [6].
    This description of the cyclotomic structure is the input to the proof of Theorem 1.5 in Section 4. It is attributed to [6] and is a standard result in cyclotomic spectra theory.
  • ad hoc to paper Assumptions A-G on the Fukaya category (Section 12): integrality of curve counts (A), nondegeneracy/OC isomorphism (C), equivariant open-closed comparison (E), p-torsion-free cohomology (F), etc.
    These are the assumptions under which Theorem 1.27 (symplectic p-curvature conjecture) holds. The author states that all except nondegeneracy are expected to hold unconditionally but are included due to the state of Fukaya-categorical foundations.
  • domain assumption The Getzler-Gauss-Manin connection agrees with the u-connection induced by the circle action on THH (Theorem 7.16).
    This comparison is proven in Section 10 for R = k[x], k[x,x^{-1}] (characteristic zero) and extended to k[[x]], k((x)). It is needed to connect the algebraic formula to the GGM connection.
  • standard math The inverse Cartier transform's p-curvature is given by -F*(θ) (Lemma 8.2, from [91, Theorem 2.8]).
    This is a known result in arithmetic geometry, used in Section 8 to connect the Cartier transform to p-curvature.
invented entities (1)
  • Cyclotomic Deligne conjecture (Appendix A) no independent evidence
    purpose: A conjectural framework describing compatibility between the E2-structure on THC, the S1-action on THH, the module structure, and the cyclotomic structure. Would provide an operadic proof of Theorem 1.5.
    Stated as Conjecture A.2. The paper proves a special case (one point in O_∩(p)) but the general conjecture is unproven. No falsifiable prediction beyond the paper's own theorems is made.

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