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arxiv: 2605.22494 · v1 · pith:3CU6SSMKnew · submitted 2026-05-21 · 🧮 math.AG · math.KT

Divisibility phenomena in motivic Bloch--Ogus theory

Jean-Louis Colliot-Th\'el\`ene , Stefan Schreieder This is my paper

Pith reviewed 2026-05-22 02:14 UTC · model grok-4.3

classification 🧮 math.AG math.KT
keywords motivic cohomologyBloch-Ogus filtrationunramified classesMilnor K-theorydivisibilityfunction fieldssmooth projective varietiesalgebraic cycles
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The pith

Unramified classes in the Milnor K-group of a function field over a separably closed field lie in the n-divisible subgroup for n invertible in the field.

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

The authors prove divisibility results for unramified elements in motivic cohomology for smooth projective varieties over suitable fields. When the base field k is separably closed, any unramified class in the Milnor K-group of the function field is n-divisible if n is invertible in k. This result generalizes to unramified motivic cohomology in any bidegree. For finite or separably closed fields, the Bloch-Ogus filtration on motivic cohomology has l-divisible pieces except for the last one, up to torsion, and similar statements hold for quasi-projective schemes.

Core claim

For a smooth projective variety X over a separably closed field k, the subgroup of unramified classes in K_i^M(k(X)) is contained in the n-divisible elements of K_i^M(k(X)) for any integer n invertible in k. This generalizes to unramified motivic cohomology of arbitrary bidegree. Furthermore, if k is finite or separably closed and l is a prime invertible in k, then all but the last step in the Bloch-Ogus filtration of the motivic cohomology of X are l-divisible up to torsion.

What carries the argument

The unramified subgroup of Milnor K-groups and the Bloch-Ogus filtration on motivic cohomology.

Load-bearing premise

The standard definitions from motivic cohomology theory for unramified classes and the Bloch-Ogus filtration are used, with the base field being separably closed or finite as required.

What would settle it

Finding a smooth projective variety X over a separably closed field k, an integer i, and n invertible in k, together with a class in the unramified part of K_i^M(k(X)) that is not annihilated by multiplication by n, would disprove the main claim.

read the original abstract

Let X be a smooth projective variety over a field k. For k separably closed, we prove that the subgroup of unramified classes in the Milnor K-group $K^M_i(k(X))$ of the function field of X is contained in the subgroup of n-divisible elements of $K^M_i(k(X))$ for any integer n invertible in k. This generalizes to a statement for unramified motivic cohomology of arbitrary bidegree. We further show that whenever k is finite or separably closed and l is a prime invertible in k, then all but the last step in the Bloch--Ogus filtration of the motivic cohomology of X are l-divisible up to torsion. Generalizations of this last result to arbitrary quasi-projective k-schemes are also proven.

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

0 major / 2 minor

Summary. The paper proves divisibility results for unramified classes in Milnor K-theory and motivic cohomology. For a smooth projective variety X over a separably closed field k, the unramified subgroup of K^M_i(k(X)) lies in the n-divisible subgroup whenever n is invertible in k; this extends to unramified motivic cohomology in arbitrary bidegree. When k is finite or separably closed and l is a prime invertible in k, all but the final step of the Bloch-Ogus filtration on the motivic cohomology of X is l-divisible up to torsion. The results are generalized to arbitrary quasi-projective k-schemes via localization and dévissage.

Significance. If the claims hold, the work supplies concrete divisibility statements that refine the structure of unramified subgroups and Bloch-Ogus filtrations in motivic cohomology. The arguments rest on standard Gersten resolutions, localization sequences, and coefficient divisibility for separably closed or finite fields, without requiring resolution of singularities beyond the literature. Such results can facilitate explicit computations and comparisons with other filtrations in algebraic K-theory and motivic theory.

minor comments (2)
  1. [§1] §1 (Introduction): the statement of the main theorem for arbitrary bidegree motivic cohomology would benefit from an explicit reference to the precise bidegree notation (e.g., H^{p,q}) used in the body of the paper.
  2. The generalization to quasi-projective schemes is stated in the abstract and conclusion; a short paragraph clarifying how the localization sequence adapts when X is no longer projective would improve readability.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their accurate summary of our results on divisibility in motivic Bloch-Ogus theory and for their positive assessment of the significance of the work. We appreciate the recommendation for minor revision. As no specific major comments appear in the report, we address the overall evaluation below and note our readiness to handle any minor editorial matters.

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper establishes divisibility results for unramified subgroups in Milnor K-theory and motivic cohomology by direct application of standard Gersten resolutions, localization sequences, and coefficient divisibility properties that hold when the base field is separably closed or finite and n is invertible. These steps invoke only the usual definitions of unramified classes and Bloch-Ogus filtrations from the literature, without any reduction of the target statements to quantities defined in terms of the paper's own fitted inputs, self-citations that carry the central load, or ansatzes imported from the authors' prior work. The derivation chain therefore remains self-contained and externally verifiable against existing motivic cohomology machinery.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The results rest on the established framework of motivic cohomology, Milnor K-theory, and the Bloch-Ogus filtration; no new free parameters, ad-hoc axioms, or invented entities are introduced in the statements given.

axioms (1)
  • standard math Standard definitions and functoriality properties of Milnor K-groups, unramified cohomology, and the Bloch-Ogus filtration over fields and schemes.
    The paper invokes these as background from prior literature in the field.

pith-pipeline@v0.9.0 · 5667 in / 1473 out tokens · 56057 ms · 2026-05-22T02:14:08.953052+00:00 · methodology

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Works this paper leans on

56 extracted references · 56 canonical work pages

  1. [1]

    Th.\ Alexandrou, Torsion in Griffiths groups , arXiv:2303.04083, to appear in Algebraic Geometry

    work page arXiv

  2. [2]

    11:e53 1--21

    Th.\ Alexandrou and S.\ Schreieder, On Bloch's map for torsion cycles over non-closed fields , Forum of Mathematics, Sigma (2023), Vol. 11:e53 1--21

    work page 2023

  3. [3]

    Th.\ Alexandrou and S.\ Schreieder, Truncated pushforwards and refined unramified cohomology , Advances in Mathematics 458 (2024) 109979, https://doi.org/10.1016/j.aim.2024.109979

    work page doi:10.1016/j.aim.2024.109979 2024

  4. [4]

    Th.\ Alexandrou and L.\ Zhou, Torsion higher Chow cycles modulo l , Preprint 2025, arXiv:2503.20004

    work page arXiv 2025

  5. [5]

    S.\ Balkan and S.\ Schreieder, Cycle conjectures and birational invariants over finite fields , Sel. Math. New Ser. 32, 37 (2026). https://doi.org/10.1007/s00029-026-01142-0

    work page doi:10.1007/s00029-026-01142-0 2026

  6. [6]

    Soul\'e , November 1, 1982

    A.\ A.\ Beilinson, Letter to C. Soul\'e , November 1, 1982

    work page 1982

  7. [7]

    B.\ Bhatt and P.\ Scholze, The pro-\'etale topology of schemes , Ast\'erisque 369 (2015), 99--201

    work page 2015

  8. [8]

    S.\ Bloch and A.\ Ogus, Gersten's conjecture and the homology of schemes , Ann.\ Sci.\ \'Ec.\ Norm.\ Sup\'er., 7 (1974), 181--201

    work page 1974

  9. [9]

    S.\ Bloch, Algebraic cycles and values of L-functions II , Duke Math.\ J.\ 52 (1985), 379--397

    work page 1985

  10. [10]

    in Math.,\ 61 (1986), 267--304

    S.\ Bloch, Algebraic cycles and higher K-theory , Adv. in Math.,\ 61 (1986), 267--304

    work page 1986

  11. [11]

    S.\ Bloch, The moving lemma for higher Chow groups , J.\ Algebraic Geom.\ 3 (1994), 537--568

    work page 1994

  12. [12]

    S.\ Bloch and H.\ Esnault, The coniveau filtration and non-divisibility for algebraic cycles , Math.\ Ann.\ 304 (1996), 303--314

    work page 1996

  13. [13]

    J.-L.\ Colliot-Th\'el\`ene, Birational invariants, purity and the Gersten conjecture , K-theory and algebraic geometry: connections with quadratic forms and division algebras (Santa Barbara, CA, 1992), 1--64, Proc.\ Sympos.\ Pure Math.\, 58, AMS, Providence, RI, 1995

    work page 1992

  14. [14]

    Soul\'e, Torsion dans le groupe de Chow de codimension deux , Duke Math.\ J.\ 50 (1983), 763--801

    J.-L.\ Colliot-Th\'el\`ene, J.-J.\ Sansuc, and C. Soul\'e, Torsion dans le groupe de Chow de codimension deux , Duke Math.\ J.\ 50 (1983), 763--801

    work page 1983

  15. [15]

    J.-L.\ Colliot-Th\'el\`ene and W.\ Raskind, K_2 -Cohomology and the Second Chow Group , Math.\ Ann.\ 270 (1985), 165--199

    work page 1985

  16. [16]

    J.-L.\ Colliot-Th\'el\`ene, Cycles de torsion et K -th\'eorie alg\'ebrique ,

    work page

  17. [17]

    P.\ Deligne, Th\'eorie de Hodge II , Publ.\ Math.\ I.H.\'E.S., 40 (1971), 5--58

    work page 1971

  18. [18]

    P.\ Deligne, La conjecture de Weil, I , Publ.\ Math.\ I.H.\'E.S., 43 (1974) 273--307

    work page 1974

  19. [19]

    P.\ Deligne, La conjecture de Weil, II , Publ.\ Math.\ I.H.\'E.S., 52 (1980) 137--252

    work page 1980

  20. [20]

    J.\ de Jong, Smoothness, semi-stability and alterations , Publ.\ Math.\ I.H.\'E.S., 83 (1996) 51--93

    work page 1996

  21. [21]

    H.\ Diaz, Nondivisible cycles on products of very general Abelian varieties , J.\ Algebraic Geom.\ 30 (2021), 407--432

    work page 2021

  22. [22]

    T.\ Ekedahl, On the adic formalism , Grothendieck Festschrift, Vol.\ II, Progr.\ Math.\ 87, Birkh\"auser, 1990, 197--218

    work page 1990

  23. [23]

    T.\ Geisser and M.\ Levine, The Bloch--Kato conjecture and a theorem of Suslin–Voevodsky , J.\ reine angew.\ Math.\ 530 (2001), 55--103

    work page 2001

  24. [24]

    Geisser, Duality via cycle complexes , Annals of Mathematics, 172 (2010), 1095--1126

    T. Geisser, Duality via cycle complexes , Annals of Mathematics, 172 (2010), 1095--1126

    work page 2010

  25. [25]

    T.\ Geisser, On the structure of \'etale motivic cohomology , J.\ Pure Appl.\ Algebra 221 (2017), 1614--1628

    work page 2017

  26. [26]

    108 (1997), 107--121

    A.\ Huber, Mixed perverse sheaves for schemes over number fields , Compositio Math. 108 (1997), 107--121

    work page 1997

  27. [27]

    U.\ Jannsen, Continuous \'etale cohomology , Math.\ Ann.\ 280 (1988), 207--245

    work page 1988

  28. [28]

    U.\ Jannsen, Mixed Motives and Algebraic K-Theory , Lecture Notes in Mathematics 1400, Springer, 1990

    work page 1990

  29. [29]

    U.\ Jannsen, Weights in Arithmetic Geometry , Japanese Journal of Mathematics 5 (2010) 73--102

    work page 2010

  30. [30]

    B.\ Kahn, Divisibility properties of motivic cohomology , Preprint 2009, arXiv:1801.06010

    work page arXiv 2009

  31. [31]

    B.\ Kahn, Classes de cycles motiviques \'etales , Algebra & Number Theory 6 (2012), 1369--1407

    work page 2012

  32. [32]

    M.\ Kerz, The Gersten conjecture for Milnor K-theory , Invent.\ math.\ 175 (2009), 1--33

    work page 2009

  33. [33]

    M.\ Kerz and S.\ Saito, Cohomological Hasse principle and motivic cohomology for arithmetic schemes , Publ.\ Math.\ I.H.\'E.S.\ 115 (2012), 123--183

    work page 2012

  34. [34]

    K.\ Kok, On the failure of the integral Hodge/Tate conjecture for products with projective hypersurfaces , Preprint 2023, arXiv:2305.08961

    work page arXiv 2023

  35. [35]

    K.\ Kok and L.\ Zhou, Higher Chow groups with finite coefficients and refined unramified cohomology , Advances in Mathematics 458 (2024) 109972, https://doi.org/10.1016/j.aim.2024.109972

    work page doi:10.1016/j.aim.2024.109972 2024

  36. [36]

    M.\ Levine, K-theory and motivic cohomology of schemes, I , Preprint (2004), https://www.esaga.uni-due.de/f/marc.levine/publ/KthyMotI12.01.pdf

    work page 2004

  37. [37]

    C.\ Mazza, V.\ Voevodsky and C.\ Weibel, Lecture Notes on Motivic Cohomology , American Mathematical Society (AMS); Cambridge, MA: Clay Mathematics Institute, 2006

    work page 2006

  38. [38]

    A. S. Merkurjev, Torsion in the Milnor K-groups of fields , (Russian) Math. USSR-Sb. 59 (1988), no. 1, 95--112; translated from Mat. Sb. (N.S.) 131(173) (1986), no. 1, 94--112, 127

    work page 1988

  39. [39]

    S.\ Merkurjev and A

    A. S.\ Merkurjev and A. A.\ Suslin, K -cohomology of Severi--Brauer varieties and norm residue homomorphism , Izv.\ Akad.\ Nauk SSSR 46 (1982), 1011--1146

    work page 1982

  40. [40]

    S.\ Morel, Mixed -adic complexes for schemes over number fields , Doc.\ Math.\ 30 (2025), 105--181

    work page 2025

  41. [41]

    K. H. Paranjape, Some Spectral Sequences for Filtered Complexes and Applications , J. Algebra 186 (1996), 793--806

    work page 1996

  42. [42]

    Riou, La conjecture de Bloch--Kato (d'apr\`es M

    J. Riou, La conjecture de Bloch--Kato (d'apr\`es M. Rost et V. Voevodsky), S\'eminaire Bourbaki, Volume 2012/2013, SMF, Ast\'erisque 361, 421--463, Exp. No. 1073 (2014)

    work page 2012

  43. [43]

    A.\ Rosenschon and V.\ Srinivas, \'Etale motivic cohomology and algebraic cycles , J.\ Inst.\ Math.\ Jussieu\ 15 (2016), 511--537

    work page 2016

  44. [44]

    Tata Inst.\ Fund.\ Res., Mumbai, 2010

    A.\ Rosenschon and V.\ Srinivas, The Griffiths group of the generic abelian 3-fold , Cycles, motives and Shimura varieties, 449--467. Tata Inst.\ Fund.\ Res., Mumbai, 2010

    work page 2010

  45. [45]

    4, Paper No

    F.\ Scavia, Varieties over with infinite Chow groups modulo almost all primes , J.\ London Math.\ Soc.\ 110 (2024), no. 4, Paper No. e12994, 20 pp

    work page 2024

  46. [46]

    C.\ Schoen, Complex varieties for which the Chow group mod n is not finite , J.\ Alg.\ Geom.\ 11 (2002), 41--100

    work page 2002

  47. [47]

    Farkas et al

    S.\ Schreieder, Unramified cohomology, algebraic cycles and rationality, in: G. Farkas et al. (eds), Rationality of Varieties, Progress in Mathematics 342, Birkhäuser (2021), 345--388

    work page 2021

  48. [48]

    S.\ Schreieder, Refined unramified cohomology of schemes , Compositio Mathematica, 159 (2023), 1466--1530

    work page 2023

  49. [49]

    DOI 10.4171/JEMS/1419

    S.\ Schreieder, Infinite torsion in Griffiths groups , J.\ Eur.\ Math.\ Soc.\ 27 (2025), 2571--2601. DOI 10.4171/JEMS/1419

    work page doi:10.4171/jems/1419 2025

  50. [50]

    Voisin, Article No

    S.\ Schreieder, A moving lemma for cohomology with support , Special volume in honour of C. Voisin, Article No. 20 (2024), 50 pages

    work page 2024

  51. [51]

    Suslin, Higher Chow groups and \'etale cohomology , in: Cycles, Transfer, and Motivic Homology Theories, Annals of Math.\ Studies, Princeton University Press, Princeton, 1999

    A. Suslin, Higher Chow groups and \'etale cohomology , in: Cycles, Transfer, and Motivic Homology Theories, Annals of Math.\ Studies, Princeton University Press, Princeton, 1999

    work page 1999

  52. [52]

    117--189, NATO Sci

    A.\ Suslin and V.\ Voevodsky, Bloch--Kato conjecture and motivic cohomology with finite coefficient s, In The arithmetic and geometry of algebraic cycles (Banff, AB, 1998), pp. 117--189, NATO Sci. Ser. C Math. Phys. Sci. 548, Kluwer Acad. Publ., Dordrecht, 2000

    work page 1998

  53. [53]

    B.\ Totaro, Complex varieties with infinite Chow groups modulo 2 , Ann.\ of Math.\ 183 (2016), 363--375

    work page 2016

  54. [54]

    V.\ Voevodsky, Motivic cohomology groups are isomorphic to higher Chow groups in any characteristic , Int.\ Math.\ Res.\ Not.\ 7 (2002), 351--355

    work page 2002

  55. [55]

    V.\ Voevodsky, Motivic cohomology with /2 -coefficients , Publ.\ Math.\ IHES 98 (2003), 59--104

    work page 2003

  56. [56]

    V.\ Voevodsky, On motivic cohomology with /l -coefficients , Ann.\ of Math.\ 174 (2011), 401--438

    work page 2011