The relative Riemann-Roch theorem from Hochschild homology

Ajay C. Ramadoss

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The relative Riemann-Roch theorem from Hochschild homology

Ajay C. Ramadoss

Algebraic Geometry · Submitted 6 Mar 2006 · 19 references

Pith's verdict: A careful, honest repair of Markarian's Hochschild-homology proof of relative Riemann-Roch, with a genuinely new explicit pairing computation; the proof relies on an imported Duflo-type theorem that is not proved here, but the central argument is likely sound.

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arXiv IDmath/0603127 (version 5)

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CitationAjay C. Ramadoss. "The relative Riemann-Roch theorem from Hochschild homology." arXiv:math/0603127 (2006). https://pith.science/paper/3FHJGIO6

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Editor's note

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I agree with the reader's conditional verdict. This is an honest, detailed repair of Markarian's approach to relative Riemann-Roch via Hochschild homology. The genuinely new content is Proposition 5, which explicitly computes the pairing on Hodge cohomology to which the Mukai pairing descends under the HKR map twisted by the square root of the Todd genus, and the corrected Theorem 1 that makes the computation work. The relative Riemann-Roch theorem itself was already available through Markarian and classical Riemann-Roch, so the novelty is moderate, but this paper gives a much more transparent route and pinpoints exactly why Todd appears: the HKR map does not preserve Caldararu's Mukai pairing, only a close variant of it. That settles part of Caldararu's conjecture with a precise negative twist.

The paper does several things well. It explicitly flags the nontrivial error in Markarian's original preprint and distinguishes which results come from which source. The Lie-theory dictionary in Sections 2.3, 2.5, and 4.3 is genuinely illuminating, and the local linear algebra in Section 3 is written out in enough detail to check. The paper does not oversell itself; it repeatedly identifies what is imported and what is computed.

The main soft spot is exactly the one the stress-test note identifies. Theorem 2', the Duflo-type description of how the HKR map fails to respect comultiplication, is the engine of the proof, and it is simply recalled from the author's own [1] and Markarian's revised [6], with no proof reproduced here. Given that the original version of this result had a nontrivial error, this is a real gap in self-containedness. It is not evidence of circularity, because [6] is an independent revised source, but a referee should ask the author to either include a proof of Theorem 2' or state it as a black box with a much more explicit disclaimer. The proof of Lemma 3 also uses a non-canonical element-wise computation, which the author acknowledges; it is convincing, but it is the least clean part of the paper.

These are addressable issues, not fatal ones. For anyone working on HKR maps, Mukai pairings, or Caldararu's conjecture, this is a useful and trustworthy reference. I would not desk-reject it. Send it to a knowledgeable referee, with the expectation that the author might be asked to make the dependence on Theorem 2' clearer.

Referee report

Major objections (3)

[Section 2.4, Theorem 2 and Theorem 2'] Theorem 2' is the engine of the paper: it is used to identify Φ_R and Φ_L with the images of α_R and α_L under the HKR map, and it enters Proposition 30, Proposition 31, Proposition 35, Lemma 2, and Lemma 5. Yet no proof of Theorem 2' is given here; its proof consists of the sentence that diagram (17) follows by applying the functor RD to the diagram in Theorem 2, while Theorem 2 is only 'recalled from [1]' with a comment that [6] has a similar result. Since the introduction states that the original version in [2] had a nontrivial error and that this paper is meant to repair that approach, the reader needs either a proof of Theorem 2' with the exact conventions used in the rest of the paper, or a precise statement with matching sign and normalization conventions and a clear reference to the published/correct source. As written, the final formula J(α)∧td(T_X) depends on an external result with a documented history of error, and this dependency is load-bearing.
[Section 4.4, proof of Lemma 3] The proof of Lemma 3 is crucial because Lemma 3 is used in the proof of Lemma 5 and hence in the final computation of Theorem 1. However, the proof is carried out by choosing elements a,b ∈ RHom_X(O_X, S^•(Ω[1])) and then using an explicit elementwise computation. The remark immediately after the proof states that this computation 'can very easily be rewritten in a canonical manner' but that this canonical version is not supplied. Since the argument as written is formally incomplete (it relies on the assertion that the elementwise computation is sufficient), the proof of Lemma 3 needs to be made fully rigorous, whether by supplying the canonical argument or by explaining precisely why the elementwise computation is legitimate in the derived setting.
[Section 5.1, Proposition 36] The proof of Proposition 36 depends on the explicit description of the Serre duality map q from Markarian [6] and on a scalar-factor argument that combines local coordinate checks with a global dimension count for Hom_{D^b(X)}(O_X, Δ_*O_Δ⊗S_X^{-1}). This argument is load-bearing because it identifies the morphism β used in the final computation. The local verification is only sketched on 'good' affine open subschemes, and the passage from local commutativity up to scalar to global commutativity with scalar 1 is compressed, especially in the trace-compatibility check (55). I would like to see a clearer and more complete justification of these steps, since a sign or scalar error here would affect the final Todd-class formula.

Minor objections (4)

[Abstract and Introduction] The abstract contains a duplicated phrase: 'This write up attempts to clarify a preprint by Markarian [2] which proves This paper attempts to clarify a preprint of Markarian [2].' This should be corrected.
[Section 4.2] The phrase 'I cant see how the above proposition follows immediately from Proposition 7' is informal for a journal article; the author should rephrase, for example, as 'The author does not see a direct way to obtain this from Proposition 7 alone, so a concrete proof is given below.'
[Section 2.2, proof of Proposition 7] In the paragraph after equation (16), the text refers to '(2.2)', but no such numbered equation appears; the reference seems to be to the displayed formula immediately above, which should be given an explicit number or the reference should be updated.
[Notation, Sections 3.2 and 4.2] The symbol 1_n is used for several different maps: the map OX → ∧^n T[−n]⊗Ω^n[n], the unit-like section in Section 3.1, and the map appearing in Lemma 4 and in diagram (38). The notation is overloaded and should be made more uniform or explicitly disambiguated.

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The paper

The paper claims

The relative Riemann-Roch theorem follows from an explicit computation of how the HKR map, twisted by the square root of the Todd genus, carries the Mukai pairing on Hochschild homology to a Mukai-like pairing on Hodge cohomology.

The reading

This paper tries to establish that the relative Riemann-Roch theorem can be proved by a direct computation of how the Mukai pairing on Hochschild homology descends to Hodge cohomology. The central claim is a commutative diagram (Theorem 1) relating the Hochschild-Kostant-Rosenberg map, its twisted version by the square root of the Todd genus, and a duality map; from it, the paper computes the pairing explicitly. If the argument is right, the Todd class is not an external input but a Duflo-type correction measuring the HKR map's failure to respect comultiplication, and the relative Riemann-Roch theorem follows from the adjointness of the Mukai pairing. Along the way the paper repairs and elaborates computations from an earlier preprint and settles part of a conjecture about the Mukai pairing on Hodge cohomology.

What carries the argument

The load-bearing mechanism is the completed Hochschild chain complex $\hat{C}^\bullet(X)$, viewed as a Hopf-algebra object in the derived category, together with two connections $\alpha_R$ and $\alpha_L$ on it. The argument is carried by Theorem 2', the Duflo-type formula describing how the HKR map fails to commute with comultiplication, recalled from the author's earlier work and a revised preprint. In the dictionary with Lie groups, this formula is the pullback of a left-invariant 1-form by the exponential map. Through Lemmas 3 and 4, it yields the adjoint formula $\Phi_R^+ = i(f)\circ \Phi_R\circ i(f^{-1})$, and hence the computation of the duality map. The Todd class appears as the determinant $f = \det(At_T/(\exp(At_T)-1))$, the Jacobian-like factor that corrects the pairing.

Extended reading notes

Core claim

The paper's central claim is that for a smooth proper scheme $X$ over a field of characteristic $0$, the duality map $D_\Delta: RHom_X(\mathcal{O}_X, \Delta_*\mathcal{O}_\Delta) \to RHom_X(\Delta_*\mathcal{O}_\Delta, S_X)$ is explicitly determined by the HKR map: after identifying Hochschild homology with Hodge cohomology through $I_{HKR}$, $D_\Delta$ corresponds to multiplication by the Todd class composed with the sign involution $J$. Equivalently, applying the twisted HKR map $\hat{I}_{HKR}$ to $td(X)\wedge J(y)$ reproduces $D_\Delta(I_{HKR}^{-1}(y))$. From this, the paper proves that the Mukai pairing on Hochschild homology becomes, under $I_{HKR}$ twisted by $\sqrt{td}$, a specific Mukai-like pairing $\langle , \rangle_M$ on Hodge cohomology, close to but not always identical to the generalized Mukai pairing. The relative Riemann-Roch theorem is then a formal consequence of the adjointness of this pairing and the Chern character to Hochschild homology.

Load-bearing premise

Everything rests on Theorem 2', the Duflo-type formula for how the HKR map fails to respect comultiplication; the paper takes this theorem as an input from the author's earlier work and a revised preprint, and if it is wrong the computation of the duality map and the Riemann-Roch corollary collapse.

Editorial extensions

If this is right

The Mukai pairing on Hochschild homology, transferred by the HKR map twisted by $\sqrt{td}$, lands on an explicit pairing $\langle , \rangle_M$ on Hodge cohomology, settling the part of the Mukai-pairing conjecture that asks which pairing is preserved.
The relative Riemann-Roch theorem follows: for a proper morphism $f:X\to Y$ and a vector bundle $E$ on $X$, $\int_X f^*(l)\,\mathrm{ch}(E)\,td(TX) = \int_Y l\,\mathrm{ch}(f_*E)\,td(TY)$ for every $l\in H^*(Y)$.
The Todd genus is explained as the Duflo-type correction for the HKR map's failure to respect comultiplication, rather than an independent axiom of the theorem.
The duality map $D_\Delta$ is computed explicitly, giving a direct bridge between Hochschild homology and Serre-duality-based Hom groups.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The Lie-theoretic dictionary developed in the paper suggests the same mechanism should extend to settings with a Duflo isomorphism, such as derived algebraic geometry or families with a fiberwise Atiyah class; checking whether the adjoint formula survives in those settings would test the generality.
Because the proof locates the Todd correction as a determinant of $At_T/(\exp(At_T)-1)$, one could try to extract higher Todd-type corrections by varying the pair of connections or working with higher Hochschild chains; the paper does not do this.
The near-miss with the generalized Mukai pairing on K3 surfaces indicates that the correct pairing on Hodge cohomology depends on the chosen identification with Hochschild homology, which may matter for defining Mukai vectors in settings without a preferred HKR quasi-isomorphism.

Original abstract

This write up attempts to clarify a preprint by Markarian [2] which proves This paper attempts to clarify a preprint of Markarian [2]. The preprint by Markarian [2] proves the relative Riemann-Roch theorem using a result describing how the HKR map fails to respect comultiplication. This paper elaborates on the core computations in [2]. These computations show that the HKR map twisted by the square root of the Todd genus "almost preserves" the Mukai pairing. This settles a part of a conjecture of Caldararu[3]. The relative Riemann-Roch theorem follows from this and a result of Caldararu[4].

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The paper's own sentences, arranged, kept beside the review and separate from it. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.

Compiled by classifier-1.13.0 via TypeSafe API, question bank abstract-p1-20260923, 2026-09-23: 1 request, 4,806 input tokens.

What the abstract itself states, as probabilities: claims first 0.04 · states limitation 0.26 · human participants 0.01 · controlled comparison 0.02 · randomized assignment 0.01 · states quantitative result 0.05 · compares with prior results 0.70 · mentions code or data availability 0.02.

The abstract, sentence by sentence

arXiv listing abstract v5. Results and the main claim are underlined; show every role.

This write up attempts to clarify a preprint by Markarian [2] which proves This paper attempts to clarify a preprint of Markarian [2]. The preprint by Markarian [2] proves the relative Riemann-Roch theorem using a result describing how the HKR map fails to respect comultiplication. This paper elaborates on the core computations in [2]. These computations show that the HKR map twisted by the square root of the Todd genus "almost preserves" the Mukai pairing. This settles a part of a conjecture of Caldararu[3]. The relative Riemann-Roch theorem follows from this and a result of Caldararu[4].

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
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s0	This write up attempts to clarify a preprint by Markarian [2] which proves This paper attempts to clarify a preprint of Markarian [2].	contribution 0.69	0.57	0.05	asserted 0.58
s1	The preprint by Markarian [2] proves the relative Riemann-Roch theorem using a result describing how the HKR map fails to respect comultiplication.	background 0.79	0.05	0.27	asserted 1.00
s2	This paper elaborates on the core computations in [2].	contribution 0.96	0.74	0.05	asserted 0.98
s3	These computations show that the HKR map twisted by the square root of the Todd genus "almost preserves" the Mukai pairing.	result 0.99	0.44	0.32	asserted 0.97
s4	This settles a part of a conjecture of Caldararu[3].	interpretation 0.79	0.45	0.20	asserted 1.00
s5	The relative Riemann-Roch theorem follows from this and a result of Caldararu[4].	result 0.58	0.28	0.37	asserted 1.00

What kind of paper the classifier reports this is: theoretical result 0.64 · evidence from proof or derivation 0.95.

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References

Works cited (19)

[1]

Ramadoss, A., The big Chern classes and the Chern character, preprint, math.AG/0512104

View on Pith arXiv
[6]

Markarian, N., The Atiyah class, Hochschild cohomology and the Riemann-Roch theorem, preprint, math.AG/0610553

View on Pith arXiv
[2]

Markarian, N., Poincare-Birkhoff-Witt isomorphism, Hochschild homology and Riemann-Roch theorem, preprint(2001), MPI 2001-52, available at the Max Planck Institute's website

View on Pith 2001
[3]

Caldararu. A., The Mukai pairing II, preprint, math.AG/0308080

View on Pith arXiv
[4]

Caldararu. A., The Mukai pairing I, preprint, math.AG/0308079

View on Pith arXiv
[5]

Yekutieli, A., The continuous Hochschild cochain complex of a scheme , Canad. J. Math. 54 (2002), no. 6, 1319-1337

View on Pith 2002
[7]

Kapranov, M., Rozansky-Witten invariants via Atiyah classes, Compositio Math. 115 (1999), no. 1, 71-113

View on Pith 1999
[8]

Roberts, J. Willerton, S., On the Rozansky-Witten weight systems,preprint, math.DG/0602653

View on Pith arXiv
[9]

Loday, J. L., Cyclic Homology , Second Edition, Grundlehren Vol. 301, Springer-Verlag, Berlin, (1998)

View on Pith 1998
[10]

Kontsevich, M., Deformation quantization of Poisson manifolds, I, preprint, math.QA/9709040

View on Pith arXiv
[11]

Milnor, J. Moore, J. C., On the structure of Hopf algebras, Ann. of Math. (2) 81 (1965), 211-264

View on Pith 1965
[12]

Roberts, J., Rozansky-Witten theory, preprint, math.QA/0112209

View on Pith arXiv
[13]

Ramadoss, A., On the nonexistence of certain morphisms from Grassmannian to Grassmannian in characteristic 0, Ph.D thesis, preprint, math.AG/0501130

View on Pith arXiv
[14]

Thurston, D., Wheeling: A diagrammatic analogue of the Duflo isomorphism, preprint, math.QA/0006083

View on Pith arXiv
[15]

Reutenauer, C., Free Lie algebras , London Mathematical Society Monographs, new series,no. 7, Oxford University Press 1993

View on Pith 1993
[16]

Ramadoss, A., On the nonexistence of certain morphisms from Grassmannian to Grassmannian in Characteristic 0, preprint

View on Pith
[17]

Getzler, E. Jones, J.D.S., Operads, Homotopy algebras and iterated integrals of double loop spaces, preprint, hep-th/9403055

View on Pith Pith review arXiv
[18]

Ramadoss, A., The relative Riemann-Roch theorem from Hochschild homology, preprint, math.AG/0603127

View on Pith arXiv
[19]

Bahturin, Yu. A., et. al, Infinite dimensional Lie superalgebras , De-Gruyter Expositions in Mathematics 7, De-Gruyter 1992

View on Pith 1992

How the literature uses it

3 arXiv papers cite this in 9 sentences

Each citing sentence is taken from the citing paper's LaTeX source and typed by Pith's classifier. A label appears at probability 0.8 or more (0.95 for support), a contest at 0.7 or more; the quoted words are the citing authors'. How each label was checked · JSON

Papers citing it · 8 sentences without a confident label

Uses its method or tool 1 paper

Firstly, we will show how the recent paper of Ramadoss [ref] can be used to establish Căldăraru's conjecture for varieties with trivial canonical bundle.

Remarks on derived equivalences of Ricci-flat manifolds · The multiplicative structure of Hochschild homology · 0.91

Discussion

Disagree with the review, or know something it missed? Leave a comment, highlight any passage above to comment on it, or challenge this review formally.

[1] [1]

Ramadoss, A., The big Chern classes and the Chern character, preprint, math.AG/0512104

View on Pith arXiv

[2] [6]

Markarian, N., The Atiyah class, Hochschild cohomology and the Riemann-Roch theorem, preprint, math.AG/0610553

View on Pith arXiv

[3] [2]

Markarian, N., Poincare-Birkhoff-Witt isomorphism, Hochschild homology and Riemann-Roch theorem, preprint(2001), MPI 2001-52, available at the Max Planck Institute's website

View on Pith 2001

[4] [3]

Caldararu. A., The Mukai pairing II, preprint, math.AG/0308080

View on Pith arXiv

[5] [4]

Caldararu. A., The Mukai pairing I, preprint, math.AG/0308079

View on Pith arXiv

[6] [5]

Yekutieli, A., The continuous Hochschild cochain complex of a scheme , Canad. J. Math. 54 (2002), no. 6, 1319-1337

View on Pith 2002

[7] [7]

Kapranov, M., Rozansky-Witten invariants via Atiyah classes, Compositio Math. 115 (1999), no. 1, 71-113

View on Pith 1999

[8] [8]

Roberts, J. Willerton, S., On the Rozansky-Witten weight systems,preprint, math.DG/0602653

View on Pith arXiv

[9] [9]

Loday, J. L., Cyclic Homology , Second Edition, Grundlehren Vol. 301, Springer-Verlag, Berlin, (1998)

View on Pith 1998

[10] [10]

Kontsevich, M., Deformation quantization of Poisson manifolds, I, preprint, math.QA/9709040

View on Pith arXiv

[11] [11]

Milnor, J. Moore, J. C., On the structure of Hopf algebras, Ann. of Math. (2) 81 (1965), 211-264

View on Pith 1965

[12] [12]

Roberts, J., Rozansky-Witten theory, preprint, math.QA/0112209

View on Pith arXiv

[13] [13]

Ramadoss, A., On the nonexistence of certain morphisms from Grassmannian to Grassmannian in characteristic 0, Ph.D thesis, preprint, math.AG/0501130

View on Pith arXiv

[14] [14]

Thurston, D., Wheeling: A diagrammatic analogue of the Duflo isomorphism, preprint, math.QA/0006083

View on Pith arXiv

[15] [15]

Reutenauer, C., Free Lie algebras , London Mathematical Society Monographs, new series,no. 7, Oxford University Press 1993

View on Pith 1993

[16] [16]

Ramadoss, A., On the nonexistence of certain morphisms from Grassmannian to Grassmannian in Characteristic 0, preprint

View on Pith

[17] [17]

Getzler, E. Jones, J.D.S., Operads, Homotopy algebras and iterated integrals of double loop spaces, preprint, hep-th/9403055

View on Pith Pith review arXiv

[18] [18]

Ramadoss, A., The relative Riemann-Roch theorem from Hochschild homology, preprint, math.AG/0603127

View on Pith arXiv

[19] [19]

Bahturin, Yu. A., et. al, Infinite dimensional Lie superalgebras , De-Gruyter Expositions in Mathematics 7, De-Gruyter 1992

View on Pith 1992