Motives with modulus, I: Modulus sheaves with transfers for non-proper modulus pairs

Bruno Kahn; Hiroyasu Miyazaki; Shuji Saito; Takao Yamazaki

Motives with modulus, I: Modulus sheaves with transfers for non-proper modulus pairs

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Pith's one-line read The paper proves that modulus sheaves with transfers admit an exact sheafification functor, with Ext groups expressed as filtered colimits of Nisnevich cohomology.

desk verdict A solid, honest foundation paper: a genuine sheaf theory with transfers for non-proper modulus pairs, with a computable Ext formula, resting on one standard but heavy external input. read the letter →

arxiv 1908.02975 v7 pith:WJVTA3ZY submitted 2019-08-08 math.AG math.KT

Bruno Kahn , Hiroyasu Miyazaki , Shuji Saito , Takao Yamazaki This is my paper · ORCID

classification math.AGmath.KT MSC 19E1514F4219D4519F15

keywords moduluspairpresheafwithtransfersNisnevichsheavescd-structuremotivessheafificationextensiongroupsnon-properpairs

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper develops a sheaf theory for modulus pairs—schemes equipped with an effective Cartier divisor as a modulus and a smooth open interior—by replacing the classical category of finite correspondences with the larger category of admissible correspondences. The main theorem constructs an exact left adjoint $a_{\mathrm{Nis}}$ from presheaves with transfers to Nisnevich sheaves with transfers, so the sheaf category is a Grothendieck abelian category. It also identifies extension groups from representable sheaves as filtered colimits of ordinary Nisnevich cohomology, which makes the theory computable. If the construction works, it gives a foundation for invariants that are not $\mathbb{A}^1$-invariant, such as additive Chow groups and higher Chow groups with modulus.

What carries the argument

The engine is the class $\Sigma_{\mathrm{fin}}$ of minimal morphisms in $\mathbf{MSm}_{\mathrm{fin}}$ and $\mathbf{MCor}_{\mathrm{fin}}$: proper morphisms that extend an isomorphism between interiors and pull back the target divisor exactly to the source divisor. Proposition 1.9.2 gives $\Sigma_{\mathrm{fin}}$ a calculus of right fractions, so localization at $\Sigma_{\mathrm{fin}}$ identifies $\mathbf{MCor}_{\mathrm{fin}}$ with $\mathbf{MCor}$, and all filtered colimits in the sheafification and Ext formulas are indexed by the comma categories $\Sigma_{\mathrm{fin}}\downarrow M$. The calculus is proved via a platification-type lemma that produces a proper birational modification making a given finite correspondence finite over its source; this is the step that makes the whole machinery work.

What would settle it

Take $M=(\mathbb{P}^1,\infty)$ and a non-$\square$-invariant $F\in\mathbf{MNST}$, and compute both sides of the formula $\mathrm{Ext}^i_{\mathbf{MNST}}(Z_{\mathrm{tr}}(M),F)\simeq\varinjlim_{N\in\Sigma_{\mathrm{fin}}\downarrow M}H^i_{\mathrm{Nis}}(N,F_N)$; any degree in which the two sides differ would falsify Theorem 2.

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Extended reading notes

Core claim

Theorem 2 states that the inclusion $\mathbf{MNST}\to\mathbf{MPST}$ has an exact left adjoint $a_{\mathrm{Nis}}$ given by $(a_{\mathrm{Nis}}F)(M)=\varinjlim_{N\in\Sigma_{\mathrm{fin}}\downarrow M}(F_N)_{\mathrm{Nis}}(N)$, making $\mathbf{MNST}$ a Grothendieck abelian category. For every modulus pair $M$, the representable presheaf $Z_{\mathrm{tr}}(M)$ is a sheaf, and $\mathrm{Ext}^i_{\mathbf{MNST}}(Z_{\mathrm{tr}}(M),F)\,\simeq\,\varinjlim_{N\in\Sigma_{\mathrm{fin}}\downarrow M} H^i_{\mathrm{Nis}}(N,F_N)$. The central point is that the sheaf condition is governed by a genuine Grothendieck topology, not by an artificial construction: it arises from a cd-structure, and the earlier mistake in the preprint is corrected by weakening an exactness statement to left exactness for one auxiliary functor. This yields a computable description of extension groups and a workable foundation for motives with modulus.

Load-bearing premise

The load-bearing premise is that the platification statement used in the proof of the calculus of right fractions holds for separated finite-type schemes over the base field: every finite correspondence on a normal open dense subscheme, whose closure is proper over the source, can be made finite over a proper birational modification; if this fails in the asserted generality, the localization equivalence collapses and with it the exact sheafification and the Ext formula.

Editorial extensions

If this is right

The category $\mathbf{MNST}$ is a Grothendieck abelian category, so it has enough injectives and all small colimits, making homological algebra available in the modulus setting.
Extension groups from representable modulus sheaves are filtered colimits of ordinary Nisnevich cohomology, giving an explicit way to compute them.
The Cech complexes attached to strict Nisnevich covers are exact in $\mathbf{MNST}$, so covers behave as they do in the classical theory.
The theory provides the sheaf-theoretic foundation on which the sequel can build categories of motives with modulus.
The corrected left exactness of the auxiliary functor $b_{\mathrm{Nis}}$ explains the original preprint's error and still supports the main Ext formula.

Reading between the lines

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

Inference: If Theorem 2 is correct, the same localization formula should give a working definition of motivic cohomology with modulus as Ext groups in $\mathbf{MNST}$, with the filtered colimit replacing the classical Nisnevich cohomology of smooth schemes.
Inference: The paper's Question 1 suggests a concrete testable strengthening: under $\square$-invariance and the proper-image condition, the filtered colimit should collapse to $H^q(M_{\mathrm{Nis}},F_M)$; the blow-up case in Question 2 is a natural place to test this.
Inference: Because the cd-structure plays an essential role, extending the theory to the étale topology would require a different completeness argument; the paper's methods therefore leave open whether a similar colimit formula holds étale-locally.
Inference: The relationship between $\mathbf{MCor}_{\mathrm{fin}}$ and $\mathbf{MCor}$ via right fractions suggests that many computations in the non-proper setting can be reduced to proper models, which may simplify future calculations of additive Chow groups with modulus.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Quick take: this is the real thing — a working sheaf theory with transfers for non-proper modulus pairs, with a computable sheafification and an Ext formula, and it honestly fixes a false proposition from the authors' earlier preprint. Send it to a good referee; cite it if you work anywhere near motives with modulus.

What is new: Voevodsky's NST/PST story is rebuilt on MCor, the category of admissible correspondences for non-proper modulus pairs. The paper's central pieces are the MV_fin cd-structure on MSm_fin (Definition 3.2.1) and the exact left adjoint a_Nis of Theorem 2 (detailed in 4.5.5) with the explicit formula (a_Nis F)(M) = colim over N in Sigma_fin down M of (F_N)_Nis(N). That formula is genuinely useful, and Theorem 4.6.3 — Ext^i_{MNST}(Z_tr(M),F) isomorphic to the filtered colimit of H^i_Nis(N,F_N) — reduces extension groups to ordinary Nisnevich cohomology. The paper also states precisely what was wrong in [KSY15, Prop. 3.5.3]: b_Nis is left exact, not exact. Questions 1 and 2 are openly left open. That is the right way to run a foundation paper.

The load-bearing external input is Raynaud–Gruson platification (Lemma 1.6.1, citing [RG71, Cor. 5.7.10]), which feeds Theorem 1.6.2 and hence the calculus of right fractions for Sigma_fin in Proposition 1.9.2. It is a heavy theorem, and it is genuinely load-bearing: without it the sheafification formula collapses. But it is standard, correctly cited, and the authors thank Gabber and Raynaud for help with it. I traced the use and found it consistent rather than a hidden gap. Minor nits: Remark 1.3.8 leaves an 'easy' proof to the reader, and several steps lean on standard cd-structure formalism from Voevodsky. Both are minor.

The citation pattern is healthy. The self-references are programmatic, and one of them is a paper the present text corrects rather than leans on.

Who this is for: anyone doing motives with modulus, reciprocity sheaves, or additive Chow groups. It is foundational, not flashy, and the payoff lives in the sequels. Read Theorem 2 and the introduction for the shape of the results; the rest is machinery written carefully. I would bring it to a motivic reading group, and if I were an editor I would send it to referees without hesitation.

Referee Report

0 major / 5 minor

Summary. This paper develops a theory of modulus sheaves with transfers for non-proper modulus pairs, generalizing Voevodsky's sheaves with transfers. The authors introduce the categories MCor and MSm along with their finite variants, define admissible correspondences and the class Sigma_fin, and prove a calculus of right fractions for this class (Proposition 1.9.2). On this foundation they define the category MNST of Nisnevich sheaves with transfers on MCor and establish the main theorem (Theorem 2, detailed in Theorems 4.5.5 and 4.6.3): the inclusion MNST into MPST has an exact left adjoint aNis given by an explicit filtered colimit formula, and Ext groups in MNST are computed as filtered colimits of Nisnevich cohomology groups. The paper also corrects a false statement from the withdrawn preprint [KSY15] and explicitly leaves open two questions concerning possible simplifications of the Ext formula.

Significance. If the main theorem is correct, this is a foundational contribution: it gives a genuine Grothendieck abelian category of modulus sheaves with transfers, with a computable Ext formula, which is essential for the sequel [KMSY20] on motives with modulus. The paper is careful and detailed: it provides long proofs, collects the required categorical machinery in an appendix, and is transparent about external dependencies, about the correction of [KSY15], and about the open questions. The central construction is purely deductive, with no fitted parameters or empirical inputs. The main risk is the reliance on the Raynaud–Gruson platification theorem (Lemma 1.6.1) for the calculus of right fractions, but this is a standard external result and I did not find any gap in its application.

minor comments (5)

[1.3, Remark 1.3.8] The decomposition of a modulus pair into the sum of its irreducible components when M^o is disconnected is stated with the proof left to the reader. Since this remark is used in later reductions (for example, to reduce to irreducible interiors), a brief proof or a precise reference would make the paper more self-contained.
[1.6, Lemma 1.6.1] The proof of Lemma 1.6.1 is a very short reduction to [RG71, Corollary 5.7.10], and this lemma is load-bearing for Proposition 1.9.2 and hence for the main theorems. A few more sentences explaining how the cited result applies—in particular why the admissible blow-up can be chosen to be a scheme rather than an algebraic space—would improve verifiability, even though the cited theorem is standard.
[1.10, Proposition 1.10.4(3)] In the definition of the blow-up center, the text writes q_1^*(U_1^∞) ×_{W_1} q_2^*(U_1^∞); this appears to be a typo, as the second factor should presumably be q_2^*(U_2^∞). As written, the center is the self-product of the first divisor, which would not yield the intended exceptional divisor.
[4.2, Lemma 4.2.3] The equivalence (i)⇔(iii) relies on [Voe10a, Corollary 2.17] for the cd-structure PMV. The paper has already shown PMV is strongly complete and regular, so the citation is appropriate, but a short reminder of how the cited corollary applies to the exact-sequence formulation would help the reader.
[Introduction] The formula in Theorem 2(1) for aNis uses the notation (F_N)_Nis(N), where F_N is not explicitly defined until later in the paper. A forward reference to Definition 4.5.2 and Notation 4.6.2 would improve readability.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity found: the paper is a self-contained deductive construction, with external, standard technical inputs and honest acknowledgements of prior errors.

full rationale

This is a purely deductive mathematics paper, with no fitted parameters, empirical inputs, or predictions in the statistical sense. The central Theorem 2 is obtained by a genuine derivation chain: the calculus of right fractions for Sigma_fin (Proposition 1.9.2), which rests on Theorem 1.6.2 and the external Raynaud–Gruson platification theorem (Lemma 1.6.1, [RG71, Corollary 5.7.10]), then yields the exact left adjoint aNis and the Ext formula. The definition of MNST is independent of the formula for aNis, and the formula is proved, not assumed. The few self-citations are not load-bearing in a circular way: Lemma 1.1.3 is cited from the published [KSY16], and the paper explicitly corrects, rather than relies on, the flawed parts of [KSY15]. The open Questions 1 and 2 are honestly left open, with no claim that they follow from the paper's own assumptions. No equation in the paper reduces by construction to an input used to define it, and no uniqueness theorem is imported from the authors' prior work to force a choice. Accordingly, the circularity burden is not met.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

This is a pure mathematics paper; there are no fitted parameters, no empirical predictions, and no empirically invented entities. The paper introduces new mathematical definitions, notably the category MNST and the sheafification functor aNis, but these are contributions justified by theorems with proofs rather than unsupported postulates. The axioms it depends on are standard background results in algebraic geometry and category theory, listed above. The main external inputs are Voevodsky's sheaf theory, Raynaud-Gruson platification, Nagata compactification, and the theory of cd-structures.

assumptions (5)

standard math ZFC set theory and the categorical toolkit of Grothendieck abelian categories, pro-objects, and calculus of fractions
Used throughout; Appendix A assembles results from [SGA4], [GZ67], and [KS06]. The paper's theorems do not assume their own conclusions.
standard math Voevodsky's theory of sheaves with transfers, including the exact sheafification functor for Nisnevich sheaves with transfers ([Voe00, Theorem 3.1.4])
Invoked as the model being generalized; Theorem 1 in the Introduction and the proof of Proposition 3.3.5 use [MVW06, Lemma 6.2].
standard math Nagata compactification and the Raynaud-Gruson platification theorem ([RG71, Corollary 5.7.10]) for separated finite type schemes over k
Used to prove the calculus of fractions for Sigma and Sigma_fin (Propositions 1.7.2 and 1.9.2) via Lemma 1.6.1 and Theorem 1.6.2; these localizations underlie the whole sheaf theory.
domain assumption The base field k is arbitrary; all schemes are separated and of finite type over k
This is the standing convention (Notation and conventions, Section 1). The paper deliberately removes an earlier local integrality condition, so the statements hold for reduced but not necessarily normal schemes.
standard math The Nisnevich topology and the theory of cd-structures ([Voe10a], [Voe10b])
Used to define the topology on MSm_fin and MSm; strong completeness and regularity of the cd-structure (Proposition 3.2.3, Theorem 4.1.2) are quoted from [Voe10b, Theorem 2.2].

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Cite this review

Pith. "Pith review of Motives with modulus, I: Modulus sheaves with transfers for non-proper modulus pairs." pith.science (2026). https://pith.science/paper/WJVTA3ZY

@misc{pith2026190802975,
  author       = {Pith},
  title        = {Pith review of: Motives with modulus, I: Modulus sheaves with transfers for non-proper modulus pairs},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/WJVTA3ZY}},
  note         = {Machine review of arXiv:1908.02975}
}

read the original abstract

We develop a theory of modulus sheaves with transfers, which generalizes Voevodsky's theory of sheaves with transfers. This paper and its sequel are foundational for the theory of motives with modulus, which is developed in [KMSY20].

Discussion (0). Continue with ORCID to comment.

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Works this paper leans on

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Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

[1] [1]

Andr\'e, B

Y. Andr\'e, B. Kahn, Nilpotence, radicaux et structures mono\" dales (with an appendix by P. O'Sullivan), Rend. Sem. Math. Univ. Padova 108 (2002), 107--291

work page 2002

[2] [2]

Artin, B

M. Artin, B. Mazur, \'Etale homotopy, Lect. Notes in Math. 100, Springer, 1969

work page 1969

[3] [3]

Ayoub, Motifs de vari\'et\'es analytiques rigides, M\'em

J. Ayoub, Motifs de vari\'et\'es analytiques rigides, M\'em. SMF 140-141 (2015), vi+386

work page 2015

[4] [4]

Bloch, H

S. Bloch, H. Esnault, An additive version of higher Chow groups , Ann. Sci. \'Ec. Norm. Sup. 36 (2003), no. 3, 463--477

work page 2003

[5] [5]

Bierstone, P

E. Bierstone, P. Milman, Functoriality in resolution of singularities , Publ. RIMS, Kyoto Univ. 44 (2008), 609--639

work page 2008

[6] [6]

Binda, S

F. Binda, S. Saito, Relative cycles with moduli and regulator maps , J. Inst. Math. Jussieu 18 (2019), no. 6, 1233--1293

work page 2019

[7] [7]

Gabriel, Des cat\'egories ab\'eliennes, Bull

P. Gabriel, Des cat\'egories ab\'eliennes, Bull. SMF 90 (1962), 323--448

work page 1962

[8] [8]

Gabriel, M

P. Gabriel, M. Zisman, Calculus of fractions and homotopy theory, Springer, 1967

work page 1967