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arxiv: 2606.23863 · v1 · pith:KTK4AFCV · submitted 2026-06-22 · math.KT · math.AT· math.NT

The Goncharov Lie coalgebra of a field

Alexander Kupers , Daniil Rudenko , Ismael Sierra This is my paper

pith:KTK4AFCVreviewed 2026-06-26 05:55 UTCmodel grok-4.3open to challenge →

classification math.KT math.ATmath.NT
keywords Goncharov Lie coalgebraalgebraic K-theorySteinberg modulespolylogarithmic complexRognes rank spectral sequencemotivic realizationHodge realization
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The pith

The Goncharov Lie coalgebra, built from general linear group homology, expresses the rational algebraic K-theory of fields in weight 3 via polylogarithms.

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

The paper defines the Goncharov Lie coalgebra using the E-infinity homology of general linear groups over a field. It derives a presentation from Steinberg modules, computes the Lie cobracket, and constructs motivic and Hodge realizations. These structures are then combined with the Rognes rank spectral sequence to produce explicit symbolic descriptions of the rationalized K-theory groups. The descriptions cover K four of weight three and the indecomposable part of K five of weight three, written in terms of Goncharov's polylogarithmic complex.

Core claim

We introduce the Goncharov Lie coalgebra defined in terms of the E∞-homology of general linear groups. Using Steinberg modules, we find a presentation, compute its Lie cobracket, and construct motivic and Hodge realisations. Combining these results with the Rognes rank spectral sequence, we give symbolic descriptions of the rationalisation of the algebraic K-theory of fields beyond the cases studied by Matsumoto-Milnor and Bloch-Suslin: we express K4(3)(F) and the indecomposable part of K5(3)(F) in terms of Goncharov's polylogarithmic complex of weight 3.

What carries the argument

The Goncharov Lie coalgebra, defined via the E∞-homology of general linear groups and presented using Steinberg modules, which supports the Lie cobracket computation and the motivic and Hodge realizations.

If this is right

  • The rational algebraic K-theory groups K4(3)(F) and the indecomposable summand of K5(3)(F) admit explicit descriptions in terms of the weight-3 polylogarithmic complex.
  • The method extends the earlier symbolic results known for weight 2 and weight 3 in lower degrees.
  • Motivic and Hodge realizations are available for the Goncharov Lie coalgebra itself.
  • The Rognes rank spectral sequence interacts with the new coalgebra structure to produce these identifications.

Where Pith is reading between the lines

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

  • If analogous presentations exist in higher weights, the same method could supply descriptions for additional K-theory groups.
  • The explicit link to the polylogarithmic complex may allow numerical checks for particular fields whose K-theory is already known by other means.
  • The coalgebra and its realizations could be compared with other spectral sequences that compute rational K-theory.

Load-bearing premise

The presentation of the Goncharov Lie coalgebra from Steinberg modules, together with its Lie cobracket and realizations, is compatible with the Rognes rank spectral sequence in the manner required to produce the claimed symbolic descriptions of the K-theory groups.

What would settle it

A direct calculation of the group K4(3)(F) for a specific field F whose structure or dimension disagrees with the prediction coming from the weight-3 polylogarithmic complex.

Figures

Figures reproduced from arXiv: 2606.23863 by Alexander Kupers, Daniil Rudenko, Ismael Sierra.

Figure 1
Figure 1. Figure 1: in particular, they vanish for d ď 2n ´ 2 with the exception of pn, dq “ p1, 0q. The Goncharov Lie coalgebra is defined as those entries on the critical line above where this vanishing result applies: GpFq “ à ně1 H E8 n,2n´1 pBGLpFqQq. Its name is justified by Koszul duality between the nonunital commutative operad and the suspended Lie cooperad, which implies that E8-indecomposables admit the structure o… view at source ↗
Figure 2
Figure 2. Figure 2: A generator of St3pFq (on the left) and a generator of St2 3 pFq (on the right). To explain further structure, we organise all Steinberg modules into a single object. Let VectF denote the groupoid of finite-dimensional vector spaces over F and linear isomorphisms, then we can think of the Steinberg modules as a functor (with StpV q in degree dimpV q) St: VectF ÝÑ GrModQ V ÞÝÑ StpV q. The category VectF adm… view at source ↗
Figure 3
Figure 3. Figure 3: A generator of St8 3 pFq. We believe that the infinite Steinberg module is a remarkable object worth studying inde￾pendently. It can also be obtained from the common basis complex [Rog92] or the partial decomposition poset [HHS25, BPW24]. By Koszul duality, St 8 admits the structure of a Lie coalgebra with respect to ‘. Charlton, Radchenko, and Rudenko gave a presentation of the infinite Steinberg modules … view at source ↗
Figure 4
Figure 4. Figure 4: A pictorial interpretation of duals to cyclic words, compatible with our discussion of infinite Steinberg modules, is as polygons. 3.2.6. Dual Lie coalgebras. Linearly dualising the discussions in Sections 3.2.1 and 3.2.3, degreewise by putting the generators in degree 1, we obtain from Section 3.2.4 a pair of surjective maps of Lie coalgebras as_ S ÝÑ lie_ S and sder_ S ÝÑ sder_ S which are compatible wit… view at source ↗
Figure 5
Figure 5. Figure 5: The E1-homology of BGLpFqQ is, up to a shift by ´1, the free graded-commutative algebra on the E8-homology, and thus vanishes for d ă 2n ´ 2 as indicated by the dashed line. We use the abbreviations Λ k :“ Λ kF ˆ Q and σ :“ Qtσu. 4.4. Recollections from proof of Theorem 4.3. We will need some ingredients of the proof of Theorem 4.3 from [GKRW25b]. We recall those here. 4.4.1. Steinberg modules and building… view at source ↗
Figure 6
Figure 6. Figure 6: The E1 -page E1 p,q of the Rognes rank spectral sequence, converging to KqpFq. The colours and dashed lines denote the `1-eigenspaces and -1-eigenspaces . The d 1 -differential has bidegree p´1, ´1q and necessarily sends ˘1-eigenspaces to ˘1-eigenspaces. 10.2. Computing the d 1 differentials in terms of the cobracket. We shall explain how to obtain most of the d 1 -differentials. Recall that the d 1 -diffe… view at source ↗
Figure 7
Figure 7. Figure 7: The E2 -page E2 p,q of the rank spectral sequence, converging to KqpFq. As before, the colours denote the `1-eigenspaces and ´1-eigenspaces . The d r -differential has bidegree p´r, ´1q and necessarily sends ˘1-eigenspaces to ˘1-eigenspaces, so there can be only a few nonzero differentials in this range. Proposition 10.5. There are isomorphisms (i) F ˆ –ÝÑ K1pFq. (ii) KM 2 pFq –ÝÑ K2pFq. (iii) K p2q 3 pFq … view at source ↗
read the original abstract

This paper relates algebraic $K$-theory of fields to polylogarithms via general linear groups. We introduce the Goncharov Lie coalgebra, defined in terms of the $E_\infty$-homology of general linear groups. Using Steinberg modules, we find a presentation, compute its Lie cobracket, and construct motivic and Hodge realisations. Combining these results with the Rognes rank spectral sequence, we give symbolic descriptions of the rationalisation of the algebraic $K$-theory of fields beyond the cases studied by Matsumoto-Milnor and Bloch-Suslin: we express $K^{(3)}_4(F)$ and the indecomposable part of $K^{(3)}_5(F)$ in terms of Goncharov's polylogarithmic complex of weight 3.

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

1 major / 2 minor

Summary. The paper defines the Goncharov Lie coalgebra via E_∞-homology of general linear groups. Using Steinberg modules it obtains a presentation, computes the Lie cobracket, and constructs motivic and Hodge realizations. Combining these with the Rognes rank spectral sequence yields explicit rational descriptions of K_4^{(3)}(F) and the indecomposable summand of K_5^{(3)}(F) in terms of the weight-3 polylogarithmic complex, extending the Matsumoto–Milnor and Bloch–Suslin cases.

Significance. If the compatibility with the Rognes spectral sequence holds, the work supplies new symbolic expressions for rational algebraic K-theory of fields in weight 3. The Steinberg-module presentation, explicit cobracket computation, and realization maps are concrete strengths that could be reusable beyond this application.

major comments (1)
  1. [Combining these results with the Rognes rank spectral sequence] The central claim (abstract and the section combining results with the Rognes rank spectral sequence) rests on the Goncharov Lie coalgebra operations and realizations mapping onto the relevant pages and differentials of the Rognes spectral sequence without hidden extensions or missing filtration terms in weight 3. The manuscript provides no explicit verification or diagram chase confirming this compatibility for the stated K_4^{(3)} and K_5^{(3)} descriptions; this step is load-bearing and requires additional detail or a reference to a prior result establishing the necessary convergence and exactness.
minor comments (2)
  1. Clarify the precise grading and filtration conventions used when passing from the E_∞-homology definition to the Steinberg presentation.
  2. Add a short table or diagram summarizing the motivic and Hodge realization maps on generators.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their detailed and constructive report. The single major comment concerns the need for explicit verification of compatibility between our constructions and the Rognes rank spectral sequence. We address this point directly below and will revise the manuscript to strengthen this aspect of the argument.

read point-by-point responses

  1. Referee: [Combining these results with the Rognes rank spectral sequence] The central claim (abstract and the section combining results with the Rognes rank spectral sequence) rests on the Goncharov Lie coalgebra operations and realizations mapping onto the relevant pages and differentials of the Rognes spectral sequence without hidden extensions or missing filtration terms in weight 3. The manuscript provides no explicit verification or diagram chase confirming this compatibility for the stated K_4^{(3)} and K_5^{(3)} descriptions; this step is load-bearing and requires additional detail or a reference to a prior result establishing the necessary convergence and exactness.

    Authors: We agree that the compatibility of the Goncharov Lie coalgebra operations, cobracket, and realization maps with the pages and differentials of the Rognes rank spectral sequence is central to the claimed descriptions of K_4^{(3)}(F) and the indecomposable summand of K_5^{(3)}(F), and that an explicit verification is required to rule out hidden extensions or missing filtration terms in weight 3. The constructions in the paper are intended to be compatible by design (via the E_∞-homology definition and the Steinberg-module presentation), but we acknowledge that the current text does not contain a dedicated diagram chase or convergence argument for this step. In the revised manuscript we will add a new subsection (or appendix) providing the required diagram chase in weight 3, together with a precise statement of the induced maps on the E_2-page and the absence of obstructions in this range; if the compatibility follows from a prior reference on the Rognes spectral sequence we will cite it explicitly. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation chain is self-contained

full rationale

The paper defines the Goncharov Lie coalgebra from E_∞-homology of general linear groups, obtains a presentation via Steinberg modules, computes the Lie cobracket, and constructs motivic/Hodge realizations as independent steps. These are then combined with the external Rognes rank spectral sequence to produce new symbolic descriptions of K_4^{(3)}(F)_Q and the indecomposable part of K_5^{(3)}(F)_Q. No quoted step reduces a claimed result to a fitted input, self-definition, or self-citation chain by construction. The abstract and described claims treat the constructions as additive and independent, with the final combination presented as yielding results beyond prior cases (Matsumoto-Milnor, Bloch-Suslin). This is the normal non-circular case for a paper whose central output is a synthesis of separately derived objects.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review supplies no information on free parameters, axioms, or invented entities; ledger left empty.

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