On a relation of a conjecture of Goncharov to the co-Lie algebra of Bloch-Kriz mixed Tate motives

Kenichiro Kimura

arxiv: 2604.17671 · v1 · submitted 2026-04-19 · 🧮 math.AG

On a relation of a conjecture of Goncharov to the co-Lie algebra of Bloch-Kriz mixed Tate motives

Kenichiro Kimura This is my paper

Pith reviewed 2026-05-10 04:58 UTC · model grok-4.3

classification 🧮 math.AG

keywords Goncharov conjecturemixed Tate motivesBloch-Krizmotivic polylogarithmsBeilinson-Soulé conjectureK-groupsco-Lie algebraB_n(F)

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The pith

A linear map may exist from Goncharov's B_n(F) to the co-Lie algebra of Bloch-Kriz mixed Tate motives using motivic polylogarithms.

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

The paper explores the possibility of defining a linear map from the group B_n(F) that Goncharov introduced for any field F and integer n greater than 1 into the co-Lie algebra of the category of mixed Tate motives constructed by Bloch and Kriz. The map would be given explicitly in terms of motivic polylogarithms. Supporting results are supplied under the assumption that a portion of the Beilinson-Soulé conjecture holds, specifically the vanishing of certain K-groups of fields. If such a map can be constructed, it would supply a direct link between Goncharov's groups and the Lie-algebraic structure of these motives.

Core claim

We consider the possibility of defining a linear map from B_n(F) to the co-Lie algebra of the category of mixed Tate motives defined by Bloch and Kriz, in terms of motivic polylogarithms, and give results which support this possibility assuming part of the conjecture by Beilinson and Soulé on vanishing of K-groups of fields.

What carries the argument

The proposed linear map from B_n(F) to the Bloch-Kriz co-Lie algebra, constructed via motivic polylogarithms as the explicit connecting data.

If this is right

The map, if defined, would furnish a concrete relation between Goncharov's groups and the co-Lie algebra structure on mixed Tate motives.
The supporting results become available precisely when the relevant part of the Beilinson-Soulé vanishing conjecture is assumed.
Elements of B_n(F) would acquire an interpretation inside the motivic framework through this construction.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The map could be used to translate questions about Goncharov's conjecture into computations inside the category of mixed Tate motives.
It might suggest similar maps for other groups defined by Goncharov or related authors once the vanishing assumption is removed or weakened.
Testing the construction on explicit low-degree cases such as n=2 or n=3 for number fields would give concrete evidence for or against the possibility.

Load-bearing premise

Part of the Beilinson-Soulé conjecture holds, namely that certain K-groups of fields vanish.

What would settle it

An explicit field F and n > 1 where the motivic polylogarithms fail to define a linear map from B_n(F) into the co-Lie algebra even though the relevant K-groups vanish.

read the original abstract

Goncharov defined for each field $F$ and an integer $n$ greater than 1 a certain group $B_n(F)$. We consider the possibility of defining a linear map from $B_n(F)$ to the co-Lie algebra of the category of mixed Tate motives defined by Bloch and Kriz, in terms of motivic polylogarithms. We give results which support this possibility assuming part of the conjecture by Beilinson and Soul\'{e} on vanishing of $K$-groups of fields.

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.

Desk Editor's Note private letter to a colleague

Kimura sketches a conditional link from Goncharov's B_n groups to Bloch-Kriz motives via polylogarithms, but it all depends on an open conjecture.

read the letter

The main thing here is that Kimura is exploring whether Goncharov's groups B_n(F) can map linearly into the co-Lie algebra of Bloch-Kriz mixed Tate motives, using motivic polylogarithms as the bridge. He provides some supporting results, but only after assuming part of the Beilinson-Soulé vanishing conjecture for K-groups. The new part is the concrete suggestion of how motivic polylogarithms might define that map. The paper does well in spelling out the setup and showing what the assumption buys you. It avoids overclaiming by keeping everything conditional. The main limitation is that dependence on the open conjecture. The supporting results don't seem to include any independent verification or new computations that could stand without it. Since the details are technical, a full read would be needed to check the derivations, but the abstract shows no circularity or hidden fitting. This kind of work is for people already working in the intersection of motives, polylogarithms, and K-theory. Someone chasing Goncharov's conjectures might want to see this perspective. It deserves a serious referee because the idea is cleanly framed and could be useful for future work even if the conjecture remains open for now. A referee can assess whether the supporting arguments hold up under the stated hypothesis. I would send it out for review rather than desk reject.

Referee Report

0 major / 2 minor

Summary. The paper considers the possibility of a linear map from Goncharov's group B_n(F) to the co-Lie algebra of the Bloch-Kriz category of mixed Tate motives, constructed in terms of motivic polylogarithms. It supplies supporting results for this map under the assumption of a portion of the Beilinson-Soulé conjecture on the vanishing of K-groups of fields.

Significance. If the supporting results hold under the stated assumption, the work provides concrete evidence linking Goncharov's B_n(F) to the co-Lie structure of mixed Tate motives. This conditional connection could aid progress on motivic polylogarithms and the structure of the Bloch-Kriz co-Lie algebra, particularly in contexts where the Beilinson-Soulé vanishing is accepted or verified for specific fields.

minor comments (2)

The introduction would benefit from a brief explicit statement of which precise part of the Beilinson-Soulé conjecture (e.g., the vanishing range for K_i(F) with i > 1 or similar) is being assumed, to make the scope of the supporting results immediately clear.
Notation for the co-Lie algebra and the motivic polylogarithms should be defined or referenced to a standard source in the first section where they appear, rather than relying solely on the abstract.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their concise summary of the manuscript and for the positive assessment of its potential significance under the Beilinson-Soulé vanishing assumptions. We note the recommendation for minor revision.

Circularity Check

0 steps flagged

No circularity; central construction conditional on independent external conjecture

full rationale

The paper proposes exploring a linear map from Goncharov's B_n(F) to the Bloch-Kriz co-Lie algebra via motivic polylogarithms and provides supporting results only under the explicit assumption of part of the Beilinson-Soulé vanishing conjecture for K-groups. This assumption is external and independent of the paper's own constructions or citations. No derivation step reduces by definition, by fitting, or by self-citation chain to the target claim; the work is framed as conditional support rather than an unconditional derivation. The approach is self-contained against the stated hypothesis with no load-bearing internal circularity.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The results depend on assuming a standard but unproven conjecture in algebraic K-theory; no free parameters or new entities are introduced in the abstract.

axioms (1)

domain assumption Part of the Beilinson-Soulé conjecture on vanishing of K-groups of fields
Invoked to support the possibility of the linear map as per the abstract.

pith-pipeline@v0.9.0 · 5382 in / 1226 out tokens · 54873 ms · 2026-05-10T04:58:00.999151+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

19 extracted references · 19 canonical work pages · 1 internal anchor

[1]

Algebraic Geom.3, (1994), 537–568

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

work page 1994
[2]

and Kriz, I.,Mixed Tate Motives, Ann

Bloch, S. and Kriz, I.,Mixed Tate Motives, Ann. Math.140(1994), 557-605

work page 1994
[3]

Borel, A.,Stable real cohomology of arithmetic groups, Ann. Sci. ´Ecole Norm. Sup. (4)7, (1974), 235-272

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Scuola Norm

Borel, A.,Cohomologie deSL n et valeurs de fonction zˆ eta aux points entiers, Ann. Scuola Norm. Sup. Pisa Cl. Sci (4)4(4) (1977), 613–636. 20 K Kimura

work page 1977
[5]

Deligne, P., Goncharov, A.,Groupes fondamentaux motiviques de Tate mixte, Ann. Sci. ´Ecole Norm. Sup. (4)38(2005), no. 1, 1–56

work page 2005
[6]

Dupont, C.,Progre` es r´ ecents sur la conjecture de Zagier et le programme de Goncharov (d’ apr` es Goncharov, Rudenko, Gangl,...), S´ eminaire Bourbaki 1176, Ast´ erisque430(2021), 295–343

work page 2021
[7]

A construction of the polylogarithm motive

Dupont, C. and Fres´ an, J.,A construction of the polylogarithm motive, arXiv preprint 2305.00789, 2023

work page internal anchor Pith review Pith/arXiv arXiv 2023
[8]

and Mac Lane, S.,On the groups ofH(Π, n)

Eilenberg, S. and Mac Lane, S.,On the groups ofH(Π, n). I, Ann. of Math. (2)58(1953), 55–106

work page 1953
[9]

Goncharov, A.,Polylogarithm and motivic Galois groups, Proc. Sympos. Pure Math.55Part 2, Amer. Math. Soc., Providence, RI, (1994), 43–96

work page 1994
[10]

Math114 (2) (1995), 197–318

Goncharov, A.,Geometry of configuration, polylogarithms, and motivic cohomology, Adv. Math114 (2) (1995), 197–318

work page 1995
[11]

I, K-Theory1(1987) no

Hain, R.,The de Rham homotopy theory of complex algebraic varieties. I, K-Theory1(1987) no. 3, 271–324

work page 1987
[12]

Hanamura, M.,Mixed motives and algebraic cycles. I, Math. Res. Lett.2(6) (1995), 811–821

work page 1995
[13]

Kriz, I., May, J.P.,Operads, algebras modules and motives, Part IV, Derived categories from the topological point of view, Asterisque233, Soc. Math. de France 1995

work page 1995
[14]

Levine, M.,Tate motives and the vanishing conjectures for algebraicK-theory, AlgebraicK-theory and algebraic topology (Lake Louise, AB), NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci407, Kluwer Acad. Publ., Dordrecht, 167–188

work page
[15]

22610 (1994), 235–320

Levine, M.,Bloch’s higher Chow groups revisited,K-theory (Strasbourg, 1992), Ast´ erisque No. 22610 (1994), 235–320

work page 1992
[16]

Levine, M.,Mixed motives, Mathematical Surveys and Monographs,57, American Mathematical Soci- ety, Providence, RI, 1998

work page 1998
[17]

P.,Simplicial Objects in Algebraic Topology, Van Nostrand Math

May, J. P.,Simplicial Objects in Algebraic Topology, Van Nostrand Math. Studies, No. 11, Van Nos- trand, (1967)

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[18]

Totaro, B.,MilnorK-theory is the Simplest Part of AlgebraicK-Theory,K-Theory6(1992), 177–189

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[19]

Voevodsky, V.,Triangulated categories of motives over a field, Cycles, transfers, and motivic homology theories, Ann. of. Math. Stud.143, Princeton Univ. Press, Princeton, NJ, 188–238. 21

work page

[1] [1]

Algebraic Geom.3, (1994), 537–568

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

work page 1994

[2] [2]

and Kriz, I.,Mixed Tate Motives, Ann

Bloch, S. and Kriz, I.,Mixed Tate Motives, Ann. Math.140(1994), 557-605

work page 1994

[3] [3]

Borel, A.,Stable real cohomology of arithmetic groups, Ann. Sci. ´Ecole Norm. Sup. (4)7, (1974), 235-272

work page 1974

[4] [4]

Scuola Norm

Borel, A.,Cohomologie deSL n et valeurs de fonction zˆ eta aux points entiers, Ann. Scuola Norm. Sup. Pisa Cl. Sci (4)4(4) (1977), 613–636. 20 K Kimura

work page 1977

[5] [5]

Deligne, P., Goncharov, A.,Groupes fondamentaux motiviques de Tate mixte, Ann. Sci. ´Ecole Norm. Sup. (4)38(2005), no. 1, 1–56

work page 2005

[6] [6]

Dupont, C.,Progre` es r´ ecents sur la conjecture de Zagier et le programme de Goncharov (d’ apr` es Goncharov, Rudenko, Gangl,...), S´ eminaire Bourbaki 1176, Ast´ erisque430(2021), 295–343

work page 2021

[7] [7]

A construction of the polylogarithm motive

Dupont, C. and Fres´ an, J.,A construction of the polylogarithm motive, arXiv preprint 2305.00789, 2023

work page internal anchor Pith review Pith/arXiv arXiv 2023

[8] [8]

and Mac Lane, S.,On the groups ofH(Π, n)

Eilenberg, S. and Mac Lane, S.,On the groups ofH(Π, n). I, Ann. of Math. (2)58(1953), 55–106

work page 1953

[9] [9]

Goncharov, A.,Polylogarithm and motivic Galois groups, Proc. Sympos. Pure Math.55Part 2, Amer. Math. Soc., Providence, RI, (1994), 43–96

work page 1994

[10] [10]

Math114 (2) (1995), 197–318

Goncharov, A.,Geometry of configuration, polylogarithms, and motivic cohomology, Adv. Math114 (2) (1995), 197–318

work page 1995

[11] [11]

I, K-Theory1(1987) no

Hain, R.,The de Rham homotopy theory of complex algebraic varieties. I, K-Theory1(1987) no. 3, 271–324

work page 1987

[12] [12]

Hanamura, M.,Mixed motives and algebraic cycles. I, Math. Res. Lett.2(6) (1995), 811–821

work page 1995

[13] [13]

Kriz, I., May, J.P.,Operads, algebras modules and motives, Part IV, Derived categories from the topological point of view, Asterisque233, Soc. Math. de France 1995

work page 1995

[14] [14]

Levine, M.,Tate motives and the vanishing conjectures for algebraicK-theory, AlgebraicK-theory and algebraic topology (Lake Louise, AB), NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci407, Kluwer Acad. Publ., Dordrecht, 167–188

work page

[15] [15]

22610 (1994), 235–320

Levine, M.,Bloch’s higher Chow groups revisited,K-theory (Strasbourg, 1992), Ast´ erisque No. 22610 (1994), 235–320

work page 1992

[16] [16]

Levine, M.,Mixed motives, Mathematical Surveys and Monographs,57, American Mathematical Soci- ety, Providence, RI, 1998

work page 1998

[17] [17]

P.,Simplicial Objects in Algebraic Topology, Van Nostrand Math

May, J. P.,Simplicial Objects in Algebraic Topology, Van Nostrand Math. Studies, No. 11, Van Nos- trand, (1967)

work page 1967

[18] [18]

Totaro, B.,MilnorK-theory is the Simplest Part of AlgebraicK-Theory,K-Theory6(1992), 177–189

work page 1992

[19] [19]

Voevodsky, V.,Triangulated categories of motives over a field, Cycles, transfers, and motivic homology theories, Ann. of. Math. Stud.143, Princeton Univ. Press, Princeton, NJ, 188–238. 21

work page