The reviewed record of science sign in
Pith

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

Reviewed by Pith2026-07-07 14:47 UTCglm-5.2pith:ADDTADFFopen to challenge →

classification math.AT math.AGmath.KTmath.SG
keywords mathbbsymplecticformulaquantumanalogcartiercurvatureproduct
0
0 comments X

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.

pith-pipeline@v1.1.0-glm · 69962 in / 4416 out tokens · 587518 ms · 2026-07-07T14:47:12.427961+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

119 extracted references · 119 canonical work pages · 18 internal anchors

  1. [1]

    On the Floer homology of cotangent bun- dles

    Alberto Abbondandolo and Matthias Schwarz. “On the Floer homology of cotangent bun- dles”. In:Communications on Pure and Applied Mathematics: A Journal Issued by the Courant Institute of Mathematical Sciences59.2 (2006), 254–316

    work page 2006

  2. [2]

    A geometric criterion for generating the Fukaya category

    Mohammed Abouzaid. “A geometric criterion for generating the Fukaya category”. In:Pub- lications Math´ ematiques de l’IH´ES112 (2010), 191–240

    work page 2010

  3. [3]

    J. F. Adams.Stable Homotopy and Generalised Homology. Chicago Lectures in Mathematics. University of Chicago Press, 1974

    work page 1974

  4. [4]

    Prerequisites (on equivariant stable homotopy) for Carlssons’s lecture

    John Frank Adams. “Prerequisites (on equivariant stable homotopy) for Carlssons’s lecture”. In:Algebraic Topology Aarhus 1982: Proceedings of a conference held in Aarhus, Denmark, August 1–7, 1982. Springer. 2006, 483–532

    work page 1982

  5. [5]

    Topological Hochschild homology ofℓandko

    Vigleik Angeltveit, Michael A. Hill, and Tyler Lawson. “Topological Hochschild homology ofℓandko”. In:Amer. J. Math.132.2 (2010), 297–330

    work page 2010

  6. [6]

    Topological cyclic homology via the norm

    Vigleik Angeltveit et al. “Topological cyclic homology via the norm”. In:Documenta math- ematica23 (2018)

    work page 2018

  7. [7]

    On the Blumberg–Mandell K¨ unneth theorem for TP

    Benjamin Antieau, Akhil Mathew, and Thomas Nikolaus. “On the Blumberg–Mandell K¨ unneth theorem for TP”. In:Selecta Mathematica24.5 (2018), 4555–4576

    work page 2018

  8. [8]

    Quantum Adams operations in quasimap K-theory

    Shaoyun Bai and Jae Hee Lee. “Quantum Adams operations in quasimap K-theory”. In: arXiv preprint arXiv:2510.09335(2025)

    work page arXiv 2025

  9. [9]

    Andr´ e–Quillen Cohomology of CommutativeS-Algebras

    Maria Basterra. “Andr´ e–Quillen Cohomology of CommutativeS-Algebras”. In:Journal of Pure and Applied Algebra144.2 (1999), 111–143.doi:10.1016/S0022-4049(98)00151-6

    work page doi:10.1016/s0022-4049(98)00151-6 1999

  10. [10]

    Homology and Cohomology ofE ∞ Ring Spectra

    Maria Basterra and Michael A. Mandell. “Homology and Cohomology ofE ∞ Ring Spectra”. In:Mathematische Zeitschrift249.4 (2005), 903–944.doi:10.1007/s00209-004-0709-7

    work page doi:10.1007/s00209-004-0709-7 2005

  11. [11]

    Koszul duality in higher topoi

    Jonathan Beardsley and Maximilien P´ eroux. “Koszul duality in higher topoi”. In:Homology, Homotopy and Applications25.1 (2023)

    work page 2023

  12. [12]

    Pierre Berthelot and Arthur Ogus.Notes on crystalline cohomology. Vol. 21. Princeton Uni- versity Press, 2015

    work page 2015

  13. [13]

    Fedosov quantization in positive characteris- tic

    Roman Bezrukavnikov and Dmitry Kaledin. “Fedosov quantization in positive characteris- tic”. In:Journal of the American Mathematical Society21.2 (2008), 409–438

    work page 2008

  14. [14]

    Topological Hochschild homology and integralp-adic Hodge theory

    Bhargav Bhatt, Matthew Morrow, and Peter Scholze. “Topological Hochschild homology and integralp-adic Hodge theory”. In:Publications Math´ ematiques de l’IH ´ES129.1 (2019), 199–310.doi:10.1007/s10240-019-00106-9. 76 REFERENCES

    work page doi:10.1007/s10240-019-00106-9 2019

  15. [15]

    Andrew Blumberg and Michael Mandell.The strong K¨ unneth theorem for topological periodic cyclic homology. Vol. 301. 1508. American Mathematical Society, 2024

    work page 2024

  16. [16]

    Open-closed field theories, string topology, and Hochschild homology

    Andrew J Blumberg, Ralph L Cohen, and Constantin Teleman. “Open-closed field theories, string topology, and Hochschild homology”. In:arXiv preprint arXiv:0906.5198(2009)

    work page internal anchor Pith review Pith/arXiv arXiv 2009

  17. [17]

    Localization theorems in topological Hochschild homology and topological cyclic homology

    Andrew J Blumberg and Michael A Mandell. “Localization theorems in topological Hochschild homology and topological cyclic homology”. In:Geometry & Topology16.2 (2012), 1053– 1120

    work page 2012

  18. [18]

    United States multicenter clinical trial of corneal collagen crosslinking for keratoconus treatment

    Andrew J. Blumberg, David Gepner, and Gon¸ calo Tabuada. “Uniqueness of the multiplica- tive cyclotomic trace”. In:Advances in Mathematics260 (2014), 191–232.doi:10.1016/j. aim.2014.02.004

    work page doi:10.1016/j 2014

  19. [19]

    E 2 structures and derived Koszul duality in string topology

    Andrew J. Blumberg and Michael A. Mandell. “E 2 structures and derived Koszul duality in string topology”. In:Algebr. Geom. Topol.19.1 (2019), 239–279

    work page 2019

  20. [20]

    Blumberg and Michael A

    Andrew J. Blumberg and Michael A. Mandell.The homotopy theory of cyclotomic spectra: 10 years later

    work page

  21. [21]

    Blumberg and Michael A

    Andrew J. Blumberg and Michael A. Mandell.The Strong K¨ unneth Theorem for Topological Periodic Cyclic Homology. Vol. 301. Memoirs of the American Mathematical Society 1508. Providence, RI: American Mathematical Society, 2024, v+102.doi:10.1090/memo/1508

    work page doi:10.1090/memo/1508 2024

  22. [22]

    Relative cyclotomic structures and equivariant complex cobordism

    Andrew J. Blumberg, Michael A. Mandell, and Allen Yuan. “Relative cyclotomic structures and equivariant cobordism”. In:arXiv preprint arXiv:2310.02348(2023)

    work page internal anchor Pith review Pith/arXiv arXiv 2023

  23. [23]

    The cyclotomic trace and algebraic K-theory of spaces

    Marcel B¨ okstedt, Wu Chung Hsiang, and Ib Madsen. “The cyclotomic trace and algebraic K-theory of spaces”. In:Inventiones mathematicae111 (1993), 465–539

    work page 1993

  24. [24]

    A. K. Bousfield and D. M. Kan.Homotopy Limits, Completions and Localizations. Vol. 304. Lecture Notes in Mathematics. Berlin–New York: Springer-Verlag, 1972.doi:10.1007/978- 3-540-38117-4

    work page doi:10.1007/978- 1972

  25. [25]

    The cyclic Deligne conjecture and Calabi-Yau structures

    Christopher Brav and Nick Rozenblyum. “The cyclic Deligne conjecture and Calabi-Yau structures”. In:arXiv preprint arXiv:2305.10323(2023)

    work page internal anchor Pith review Pith/arXiv arXiv 2023

  26. [26]

    A note on the Segal conjecture for large objects

    Robert Burklund and Vignesh Subramanian. “A note on the Segal conjecture for large ob- jects”. In:arXiv preprint arXiv:2403.06724(2024)

    work page internal anchor Pith review Pith/arXiv arXiv 2024

  27. [27]

    Philip Candelas, Xenia de la Ossa, and Fernando Rodriguez-Villegas.Calabi-Yau Manifolds Over Finite Fields, I. 2000. eprint:arXiv:hep-th/0012233

    work page internal anchor Pith review Pith/arXiv arXiv 2000

  28. [28]

    Une nouvelle op´ eration sur les formes diff´ erentielles

    Pierre Cartier. “Une nouvelle op´ eration sur les formes diff´ erentielles”. In:Comptes Rendus Hebdomadaires des Seances de le Academie des Sciences244.4 (1957), 426–428

    work page 1957

  29. [29]

    On operadic open-closed maps in characteristic $p$

    Zihong Chen. “On operadic open-closed maps in characteristicp”. In:arXiv preprint arXiv:2402.06183 (2024)

    work page internal anchor Pith review Pith/arXiv arXiv 2024

  30. [30]

    Quantum Steenrod operations and Fukaya categories

    Zihong Chen. “Quantum Steenrod operations and Fukaya categories”. In:arXiv preprint arXiv:2405.05242(2024)

    work page internal anchor Pith review Pith/arXiv arXiv 2024

  31. [31]

    Dustin Clausen.Duality and linearization for p-adic lie groups. 2025. eprint:arXiv:2506. 18174

    work page 2025

  32. [32]

    Topological conformal field theories and Calabi–Yau categories

    Kevin Costello. “Topological conformal field theories and Calabi–Yau categories”. In:Ad- vances in Mathematics210.1 (2007), 165–214

    work page 2007

  33. [33]

    Equivariant maps which are self homotopy equivalences

    Emmanuel Dror, William G Dwyer, and Daniel M Kan. “Equivariant maps which are self homotopy equivalences”. In:Proceedings of the American Mathematical Society80.4 (1980), 670–672

    work page 1980

  34. [34]

    Ordinal subdivision and special pasting in qua- sicategories

    Philip John Ehlers and Timothy Porter. “Ordinal subdivision and special pasting in qua- sicategories”. In:Advances in Mathematics217.2 (2008), 489–518.issn: 0001-8708.doi: https://doi.org/10.1016/j.aim.2007.05.023.url:https://www.sciencedirect. com/science/article/pii/S0001870807002289

    work page doi:10.1016/j.aim.2007.05.023.url:https://www.sciencedirect 2008

  35. [35]

    A. D. Elmendorf et al.Rings, modules, and algebras in stable homotopy theory. Vol. 47. Math- ematical Surveys and Monographs. With an appendix by M. Cole. American Mathematical Society, Providence, RI, 1997, xii+249

    work page 1997

  36. [36]

    The tangent complex and Hochschild cohomology ofE n-rings

    John Francis. “The tangent complex and Hochschild cohomology ofE n-rings”. In:Compositio Mathematica149.3 (2013), 430–480

    work page 2013

  37. [37]

    Derived algebraic geometry overE n-rings

    John John Nathan Kirkpatrick Francis. “Derived algebraic geometry overE n-rings”. PhD thesis. Massachusetts Institute of Technology, 2008. REFERENCES 77

    work page 2008

  38. [38]

    Unobstructed Immersed Lagrangian Correspondence and Filtered A- infty Functor

    Kenji Fukaya et al. “Unobstructed Immersed Lagrangian Correspondence and Filtered A- infty Functor”. In:SIGMA. Symmetry, Integrability and Geometry: Methods and Applica- tions21 (2025), 031

    work page 2025

  39. [39]

    AMS/IP Studies in Advanced Math- ematics

    Kenji Fukaya et al.Lagrangian intersection floer theory. AMS/IP Studies in Advanced Math- ematics. Providence, RI: American Mathematical Society, Jan. 2009

    work page 2009

  40. [40]

    Cyclic homology, S1–equivariant Floer cohomology and Calabi–Yau struc- tures

    Sheel Ganatra. “Cyclic homology, S1–equivariant Floer cohomology and Calabi–Yau struc- tures”. In:Geometry & Topology27.9 (2023), 3461–3584

    work page 2023

  41. [41]

    Symplectic cohomology and duality for the wrapped Fukaya category

    Sheel Ganatra. “Symplectic cohomology and duality for the wrapped Fukaya category”. In: arXiv preprint arXiv:1304.7312(2013)

    work page internal anchor Pith review Pith/arXiv arXiv 2013

  42. [42]

    Mirror symmetry: from categories to curve counts

    Sheel Ganatra, Timothy Perutz, and Nick Sheridan. “Mirror symmetry: from categories to curve counts”. In:arXiv preprint arXiv:1510.03839(2015)

    work page internal anchor Pith review Pith/arXiv arXiv 2015

  43. [43]

    The cyclic open–closed map and variations of Hodge structures

    Sheel Ganatra and Nick Sheridan. “The cyclic open–closed map and variations of Hodge structures”. In:arXiv preprint arXiv:2511.04498(2025)

    work page arXiv 2025

  44. [44]

    Sheel Ganatra et al.Integrality of mirror maps and arithmetic homological mirror symmetry for Greene–Plesser mirrors. 2023. eprint:arXiv:2312.01949

    work page internal anchor Pith review Pith/arXiv arXiv 2023

  45. [45]

    Cartan homotopy formulas and the Gauss-Manin connection in cyclic homol- ogy

    Ezra Getzler. “Cartan homotopy formulas and the Gauss-Manin connection in cyclic homol- ogy”. In:Israel Math. Conf. Proc. Vol. 7. Citeseer. 1993, 65–78

    work page 1993

  46. [46]

    A ∞-algebras and the cyclic bar complex

    Ezra Getzler, John DS Jones, et al. “A ∞-algebras and the cyclic bar complex”. In:Illinois J. Math34.2 (1990), 256–283

    work page 1990

  47. [47]

    Cyclic homology, derivations, and the free loopspace

    Thomas G Goodwillie. “Cyclic homology, derivations, and the free loopspace”. In:Topology 24.2 (1985), 187–215

    work page 1985

  48. [48]

    J. P. C. Greenlees and J. Peter May.Generalized Tate Cohomology. Vol. 113. Memoirs of the American Mathematical Society 543. Providence, RI: American Mathematical Society, 1995, viii+178.isbn: 0821826034

    work page 1995

  49. [49]

    A motivic filtration on the topological cyclic homology of commutative ring spectra

    Jeremy Hahn, Arpon Raksit, and Dylan Wilson. “A motivic filtration on the topological cyclic homology of commutative ring spectra”. In:Annals of Mathematics(2025). to appear. url:https://annals.math.princeton.edu/articles/22003

    work page 2025

  50. [50]

    On the p-typical curves in Quillen’s K-theory

    Lars Hesselholt. “On the p-typical curves in Quillen’s K-theory”. In:Acta Math177 (1996), 1–53

    work page 1996

  51. [51]

    Cyclic polytopes and the K -theory of truncated polynomial algebras

    Lars Hesselholt and Ib Madsen. “Cyclic polytopes and the K -theory of truncated polynomial algebras”. In:Invent. Math.130.1 (Sept. 1997), 73–97

    work page 1997

  52. [52]

    On the K-theory of finite algebras over Witt vectors of perfect fields

    Lars Hesselholt and Ib Madsen. “On the K-theory of finite algebras over Witt vectors of perfect fields”. In:Topology36.1 (1997), 29–101

    work page 1997

  53. [54]

    On the nonexistence of elements of Kervaire invariant one

    Michael A Hill, Michael J Hopkins, and Douglas C Ravenel. “On the nonexistence of elements of Kervaire invariant one”. In:Annals of Mathematics(2016), 1–262

    work page 2016

  54. [55]

    Philip S Hirschhorn.Model categories and their localizations. 99. American Mathematical Soc., 2003

    work page 2003

  55. [56]

    Differential Forms On Regular Affine Algebras

    G Hochschild, Bertram Kostant, and Alex Rosenberg. “Differential Forms On Regular Affine Algebras”. In:Transactions of the American Mathematical Society102.3 (1962), 383–408

    work page 1962

  56. [57]

    Mark Hovey.Model categories. 63. American Mathematical Soc., 2007

    work page 2007

  57. [58]

    The cyclic open–closed map, u-connections and R-matrices

    Kai Hugtenburg. “The cyclic open–closed map, u-connections and R-matrices”. In:Selecta Mathematica30.2 (2024), 29

    work page 2024

  58. [59]

    Cyclic homology with coefficients

    D Kaledin. “Cyclic homology with coefficients”. In:Algebra, Arithmetic, and Geometry: Volume II: In Honor of Yu. I. Manin. Springer, 2010, 23–47

    work page 2010

  59. [60]

    Trace theories, Bokstedt periodicity and Bott periodicity

    D. Kaledin.Trace theories, Bokstedt periodicity and Bott periodicity. 2020. eprint:arXiv: 2004.04279

    work page internal anchor Pith review Pith/arXiv arXiv 2020

  60. [61]

    Cartier isomorphism and Hodge Theory in the non-commutative case

    Dmitry Kaledin. “Cartier isomorphism and Hodge theory in the non-commutative case”. In: arXiv preprint arXiv:0708.1574(2007)

    work page internal anchor Pith review Pith/arXiv arXiv 2007

  61. [62]

    Non-commutative Hodge-to-de Rham degeneration via the method of Deligne-Illusie

    Dmitry Kaledin. “Non-commutative Hodge-to-de Rham degeneration via the method of Deligne-Illusie”. In:Pure and Applied Mathematics Quarterly4.3 (2008), 785–876

    work page 2008

  62. [63]

    Cyclic homology, comodules, and mixed complexes

    Christian Kassel. “Cyclic homology, comodules, and mixed complexes”. In:Journal of Alge- bra107.1 (1987), 195–216

    work page 1987

  63. [64]

    Algebraic solutions of differential equations (p-curvature and the Hodge filtration)

    Nicholas M Katz. “Algebraic solutions of differential equations (p-curvature and the Hodge filtration)”. In:Inventiones mathematicae18.1 (1972), 1–118. 78 REFERENCES

    work page 1972

  64. [65]

    An analytic Hochschild-Kostant- Rosenberg theorem

    Jack Kelly, Kobi Kremnizer, and Devarshi Mukherjee. “An analytic Hochschild-Kostant- Rosenberg theorem”. In:Advances in Mathematics410 (2022), 108694

    work page 2022

  65. [66]

    Integrality of instanton num- bers and p-adic B-model

    Maxim Kontsevich, Albert Schwarz, and Vadim Vologodsky. “Integrality of instanton num- bers and p-adic B-model”. In:Physics Letters B637.1 (2006), 97–101.issn: 0370-2693. doi:https : / / doi . org / 10 . 1016 / j . physletb . 2006 . 04 . 012.url:https : / / www . sciencedirect.com/science/article/pii/S0370269306004515

    work page 2006

  66. [67]

    Notes onA ∞-algebras,A ∞-categories and non- commutative geometry

    Maxim Kontsevich and Yan Soibelman. “Notes onA ∞-algebras,A ∞-categories and non- commutative geometry”. In:Homological Mirror Symmetry: New Developments and Per- spectives. Springer, 2008, 1–67

    work page 2008

  67. [68]

    On the Quantum K-theory of Quiver Varieties at Roots of Unity

    Peter Koroteev and Andrey Smirnov. “On the Quantum K-theory of Quiver Varieties at Roots of Unity”. In:arXiv preprint arXiv:2412.19383(2024)

    work page internal anchor Pith review Pith/arXiv arXiv 2024

  68. [69]

    Springer, 1986

    Ernst Kunz.K¨ ahler differentials. Springer, 1986

    work page 1986

  69. [70]

    Unwinding the relative Tate diagonal

    Tyler Lawson. “Unwinding the relative Tate diagonal”. In:Journal of Topology14.2 (2021), 674–699

    work page 2021

  70. [71]

    Homotopy theory ofA ∞ ring spectra and applications to MU-modules

    Andrey Lazarev. “Homotopy theory ofA ∞ ring spectra and applications to MU-modules”. In:K-theory24.3 (2001), 243–281

    work page 2001

  71. [72]

    Quantum Steenrod operations of symplectic resolutions

    Jae Hee Lee. “Quantum Steenrod operations of symplectic resolutions”. In:arXiv e-prints (2023), arXiv–2312

    work page 2023

  72. [73]

    Jean-Louis Loday.Cyclic homology. Vol. 301. Springer Science & Business Media, 2013

    work page 2013

  73. [74]

    Commuting Homotopy Limits and Smash Products

    Wolfgang L¨ uck, Holger Reich, and Marco Varisco. “Commuting Homotopy Limits and Smash Products”. In:K-Theory30.2 (2003), 137–165

    work page 2003

  74. [75]

    Derived Algebraic Geometry IV: Deformation Theory

    Jacob Lurie. “Derived algebraic geometry IV: Deformation theory”. In:arXiv preprint arXiv:0709.3091 (2007)

    work page internal anchor Pith review Pith/arXiv arXiv 2007

  75. [76]

    Cambridge, MA: Self-Published, 2017

    Jacob Lurie.Higher Algebra. Cambridge, MA: Self-Published, 2017. Version on Lurie’s web- site

    work page 2017

  76. [77]

    Jacob Lurie.Higher topos theory (AM-170). en. Annals of Mathematics Studies. Princeton, NJ: Princeton University Press, July 2009

    work page 2009

  77. [78]

    Spectral algebraic geometry

    Jacob Lurie. “Spectral algebraic geometry”. In:preprint(2018)

    work page 2018

  78. [79]

    String topology and the based loop space

    Eric James Malm. “String topology and the based loop space”. Available athttps : / / ericmalm.net/ac/research/string- topology/string- top- based- loop- space.pdf. PhD thesis. Palo Alto, CA: Stanford University, 2010

    work page 2010

  79. [80]

    Amer- ican Mathematical Soc., 2002

    Michael A Mandell and J Peter May.Equivariant orthogonal spectra andS-modules. Amer- ican Mathematical Soc., 2002

    work page 2002

  80. [81]

    Model categories of diagram spectra

    Michael A Mandell et al. “Model categories of diagram spectra”. In:Proceedings of the London Mathematical Society82.2 (2001), 441–512

    work page 2001

Showing first 80 references.