REVIEW 4 major objections 5 minor 17 references
Fourier--Mukai equivalences for formal groups and elliptic Hochschild homology
T0 review · 4 major / 5 minor · reviewed 2026-08-16 · deepseek-v4-flash
Pith's one-line read A Fourier–Mukai duality for formal groups unifies elliptic Hochschild homology
desk verdict A genuinely new comparison between two definitions of elliptic Hochschild homology, held together by an unpublished Lurie theorem that the authors need to make explicit. read the letter →
The pith
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
The reading
What carries the argument
The load-bearing object is the twisted circle $S^1_G := B G^\vee$, the classifying stack of the Cartier dual of a one-dimensional formal group $G$. The key identity is the symmetric monoidal equivalence $\mathrm{QCoh}(G)^\star \simeq \mathrm{QCoh}(S^1_G)^\otimes$: convolution along the group law on $G$ becomes ordinary tensor product on the circle. The proof's crucial computation is $\mathrm{QCoh}(G) \simeq \mathrm{coMod}_{\mathcal{O}(G)^*}$, identifying quasi-coherent sheaves with comodules over the Hopf algebra of distributions, and then identifying that same comodule category with sheaves on $B G^\vee$; monoidality is established by showing both tensor structures reduce to tensor product of comodules over $R$. The skyscraper sheaf $\mathcal{O}_e$ at the identity, as the unit for convolution, supplies the endomorphism algebra whose cospectrum recovers the circle.
What would settle it
Take a specific elliptic curve over a non-field ring such as the integers, compute the $E_\infty$-algebra of endomorphisms of the skyscraper sheaf at the identity in the convolution category, and compute the $E_\infty$-algebra of global functions on the elliptic curve; if these are not equivalent, then $\operatorname{Aff}(E) \simeq S^1_{\widehat{E}}$ is false and the two Hochschild definitions differ.
Extended reading notes
Core claim
Central claim: for every one-dimensional abelian formal group $G$ over a discrete commutative ring, the category of quasi-coherent sheaves on $G$ with convolution product is symmetric monoidally equivalent to the category of quasi-coherent sheaves on the twisted circle $S^1_G := B G^\vee$ with ordinary tensor product. The proof passes through comodules over the distribution Hopf algebra $\mathcal{O}(G)^*$ and identifies the same comodule category with sheaves on the classifying stack of the Cartier dual $G^\vee$. When $G = \widehat{E}$ is the completion of an elliptic curve, this equivalence composes with the Fourier–Mukai duality for the elliptic curve itself to give $\operatorname{Aff}(E) \simeq S^1_{\widehat{E}}$. Because maps out of an affinization agree with maps out of the original curve for affine targets, the two definitions of elliptic Hochschild homology—functions on maps from the formal-group circle, and functions on maps from the elliptic curve—coincide for affine schemes. The same pattern identifies the affinizations of nodal and cuspidal cubics with the multiplicative and additive formal-group circles, so ordinary Hochschild and Hodge Hochschild homology appear as degenerate cubic cases.
Load-bearing premise
The central comparison rests on an unpublished, quoted Fourier–Mukai equivalence for dual abelian varieties over arbitrary ring spectra; if that equivalence fails in the stated generality, the identification between the affinization of an elliptic curve and its formal-group circle—and therefore the equality of the two Hochschild theories—does not follow.
Editorial extensions
If this is right
- For any one-dimensional formal group, convolution sheaf theory is represented by a circle stack, so Fourier–Mukai-style dualities are not special to elliptic curves.
- The equivalence of the formal-group and mapping-stack definitions of elliptic Hochschild homology for affine schemes means results proven on either side transfer directly.
- Nodal and cuspidal degenerations of the same global theory recover ordinary Hochschild and Hodge Hochschild homology, placing all three theories in one family.
- The sheafified theories over the moduli stacks of elliptic and cubic curves have stalks $\widehat{E}$-Hochschild at elliptic fibers, $\widehat{\mathbb{G}}_m$-Hochschild at nodal fibers, and $\widehat{\mathbb{G}}_a$-Hochschild at cuspidal fibers, so they interpolate between the three theories.
Reading between the lines
- Because the equivalence turns the affinization of an elliptic curve into a classifying stack, computations of elliptic Hochschild homology in mixed characteristic might be reducible to finite group-quotient presentations, where both sides are otherwise hard to compute.
- If the paper's conjectural Fourier–Mukai duality for arbitrary cubic curves holds, the same mechanism would imply that the formal completion of every cubic curve determines its affinization, making the universal tmf theory genuinely universal.
- The formal-group circle suggests a spectral lift: over $E_\infty$-ring spectra, the same comparison should identify elliptic Hochschild homology with functions on a spectral mapping stack, tying the affine statement to topological modular forms.
Signed reviews
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The paper compares two existing definitions of elliptic Hochschild homology: the formal-group-based twisted Hochschild homology of Moulinos–Robalo–Toën [MRT22] and the mapping-stack construction of Sibilla–Tomasini [ST23]. The main tool is a Fourier–Mukai-type equivalence for one-dimensional formal groups, stated as Theorem A, which identifies the convolution category of quasi-coherent sheaves on a formal group with the ordinary tensor category of quasi-coherent sheaves on its Cartier-dual circle. For an elliptic curve E, the completion at the identity is then compared with E itself via Lurie's Fourier–Mukai theory for dual abelian varieties, yielding the central comparison HH_{\hat E}(X) ≃ HH_E(X) (Corollary 2.12 / Theorem B). The paper also treats nodal and cuspidal cubics, proving directly that their affinizations give the multiplicative and additive formal-group circles, and builds global sheaves of Hochschild-type invariants over moduli stacks of elliptic and cubic curves, with conjectural connections to filtered Hochschild homology and TMF.
Significance. If correct, the paper establishes a genuinely useful bridge between two independent constructions in derived algebraic geometry and gives a clean conceptual explanation of why elliptic Hochschild homology can be computed either from the formal completion or from the full elliptic curve. The comparison is not circular: the definitions being identified come from different sources and are not restatements of one another. The paper is also commendably explicit about its limitations, stating open conjectures for non-smooth cubics and for the filtered circle. Its main technical assets are the formal-group Fourier–Mukai equivalence, the direct push-out arguments for nodal and cuspidal curves, and the construction of global TMF-type Hochschild theories with correct stalk behavior. However, the central comparison depends on substantial external inputs—an unpublished theorem of Lurie and a recent preprint of Torii—and on a derived-geometry dictionary that is asserted rather than proved. These dependencies make the main result conditional as written.
major comments (4)
- [§2.2, Theorem 2.8 and Corollary 2.9] Theorem 2.8 is quoted verbatim from Proposition 5.1.3 of the unpublished manuscript [Lurb], and it is the only input that connects the elliptic curve E to its formal completion. Corollary 2.9 and hence Theorem B collapse if this theorem is not available in the required form. The paper gives no proof, no public reference, and no discussion of how Lurie's statement, formulated for spectral abelian varieties over E-infinity rings, specializes to classical elliptic curves over discrete rings. It also does not verify that the quoted result includes the symmetric monoidal enhancement needed to identify endomorphism algebras as E-infinity algebras. The authors should either supply a proof or an exact public reference for the needed statement, or explicitly reformulate Theorem B as conditional on that external theorem.
- [§2.1 and Corollary 2.12] The paper switches from the simplicial-commutative-algebra setting SCR_R used by [MRT22] to the connective-E-infinity-algebra setting CAlg_cn_R with only the remark that the two approaches are 'largely parallel' and that 'in many situations one can freely switch between the two.' This is load-bearing because Theorem B is a comparison with the [MRT22] definition, which is formulated in SCR_R. The manuscript should state precisely which mapping stacks and function algebras are identified under this change of setting, and either prove the identification or cite a theorem that covers the specific sources S^1_{\hat E} and E considered here.
- [§2.2, proof of Proposition 2.5, step (2)] The proof of the equivalence QCoh(G) ≃ coMod_{O(G)^*} relies on base change along the atlas a : Spec R → BG^∨ and cites [BZFN10, Proposition 3.10], which requires a to be perfect. The text says this is immediate because a is affine, since it is the atlas of the classifying stack. That implication is not valid as stated: when G^∨ = G_m, the pullback of a along itself is G_m, whose structure sheaf is not a perfect R-module, so BZFN perfectness is not automatic. The authors need to prove the required base-change statement in this setting or replace it with a theorem covering classifying stacks of non-finite flat affine group schemes.
- [§2.2, monoidality in Proposition 2.5 and Corollary 2.7] The symmetric monoidal enhancement of Proposition 2.5 is established only by invoking [Tor25], a preprint, for the statement that the forgetful functor from comodules is strong symmetric monoidal, and by a sketch that the two tensor products on the intermediate comodule category coincide. Since the monoidal structure is used essentially in Corollary 2.9 to compare E-infinity algebras of endomorphisms, this dependency is load-bearing. The paper should either include a self-contained proof of the monoidality of the composite equivalence, or clearly list the precise statement from [Tor25] that is being assumed.
minor comments (5)
- [§1.1] The sentence 'when G is either ˆGa, ˆGa or ˆE' contains a typo: the second group should presumably be ˆGm.
- [Corollary 2.12] The proof asserts t0(Map_{dSt_R}(E,X)) ≃ t0(X) without justification; a one-line argument or a reference would be helpful.
- [Proposition 2.13] The notation T[-1]T for the shifted tangent bundle is not defined; it should be introduced explicitly.
- [§3] The notation HHTMF_*, HHTmf_*, and HHtmf_* is used both for sheaves over the relevant moduli stack and for their global sections; this is confusing and should be disambiguated.
- [References] The paper relies on [Lurb], [Tor25], and [FPT], all of which are unpublished or in preparation; it would be helpful to indicate in the text which results are needed from each and to mark these dependencies clearly in the introduction.
Circularity Check
No significant circularity: the main comparison is a genuine theorem whose load-bearing input is an external (unpublished) Lurie result, not a restatement of the paper's own conclusion.
full rationale
The paper compares two independently introduced definitions of elliptic Hochschild homology: HH_{\hat E} from Moulinos–Robalo–Toën [MRT22] and HH_E from Sibilla–Tomasini [ST23]. The comparison is not a restatement because the two definitions are distinct constructions coming from different groups. The central bridge is Corollary 2.9, asserting Aff(E) ≃ S^1_{\hat E}; its proof combines Proposition 2.5, a formal-group Fourier–Mukai equivalence proved in the paper from Cartier duality and comodule identifications, with Theorem 2.8, quoted from Lurie's unpublished Proposition 5.1.3. Neither input is the paper's conclusion: Theorem 2.8 supplies the nontrivial symmetric monoidal Fourier–Mukai equivalence for the elliptic curve itself, and Proposition 2.5 is independently proved. The only self-citation, [ST23], appears as the definition being compared, not as evidence for the equivalence, so it is not load-bearing. The global theories in Section 3 are constructed by taking functions on mapping stacks from universal curves; their stalks are computed by base change together with Corollary 2.12, which is a sheafification argument rather than a fitted parameter renamed as a prediction. The nodal and cuspidal comparisons are proved directly from pushout presentations, independent of the elliptic case. The main caveat is external: Theorem 2.8 is taken from an unpublished manuscript [Lurb], so the paper's central claim depends on an unverified external result. That is a correctness risk, not a circularity, and it does not make the derivation equivalent to its own inputs by construction.
Assumptions & free parameters
assumptions (5)
- domain assumption Lurie's Fourier-Mukai equivalence for dual abelian varieties over E-infinity rings: QCoh(E)^star ~ QCoh(E)^tensor for elliptic curves E.
- ad hoc to paper The derived-geometry settings SCR_R (used by MRT22) and CAlg_cn_R (used here) are interchangeable for G-Hochschild homology.
- domain assumption Cartier duality over arbitrary base rings applies to one-dimensional formal groups as developed in [Mou24], with the dual G^v defined on Ab(fSch)_{1-dim}.
- domain assumption The forgetful functor from comodules to modules is symmetric monoidal; cited to [Tor25] in Proposition 2.5.
- standard math Standard facts about mapping stacks, comonadicity, and base change in derived algebraic geometry (e.g., [BZFN10, Proposition 3.10]) hold in the present setting.
Cite this review
Pith. "Pith review of Fourier--Mukai equivalences for formal groups and elliptic Hochschild homology." pith.science (2026). https://pith.science/paper/2F7DGIGA
@misc{pith2026250500172,
author = {Pith},
title = {Pith review of: Fourier--Mukai equivalences for formal groups and elliptic Hochschild homology},
year = {2026},
howpublished = {\url{https://pith.science/paper/2F7DGIGA}},
note = {Machine review of arXiv:2505.00172}
}
abstract
This paper establishes a unifying framework for various forms of twisted Hochschild homology by comparing two definitions of elliptic Hochschild homology: one introduced by Moulinos--Robalo--To\"en and the other by Sibilla--Tomasini. Central to our approach is a new Fourier--Mukai duality for formal groups. We prove that when $\widehat{E}$ is the formal group associated to an elliptic curve $E$, the resulting $\widehat{E}$-Hochschild homology coincides with the mapping stack construction of Sibilla--Tomasini. This identification also recovers ordinary and Hodge Hochschild homology as degenerate limits corresponding to nodal and cuspidal cubics, respectively. Building on this, we introduce global versions of elliptic Hochschild homology over the moduli stacks of elliptic and cubic curves, which interpolate between these theories and suggest a universal form of TMF-Hochschild homology.
Reference graph
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Reviewed August 16, 2026 · model on record in the stance chip above.
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