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On Goncharov's conjecture in next to Milnor degree

Vasily Bolbachan

TL;DR

The paper tackles Goncharov's conjecture in next to Milnor degree by proving a canonical isomorphism between the motivic cohomology $H^{m-1,m}(\mathrm{Spec}\mathbb{K},\mathbb{Q})$ and the cohomology $H^{m-1}(\Gamma(\mathbb{K},m))$ of the polylogarithmic complex for a field $\mathbb{K}$ of characteristic zero and $m\ge 2$. The authors build an auxiliary complex $\Lambda(\mathbb{K},m)$ and relate it to higher Chow groups via a map $\mathcal{W}$, while a Totaro-type map $\mathcal{T}$ connects the polylogarithmic side to the higher Chow side; a crucial step is proving that $\widetilde{\mathcal{W}}$ and $\mathcal{T}'$ are (quasi-)isomorphisms. A strong Suslin reciprocity law is developed to control reciprocity phenomena across valuations and field extensions, including a norm-map machinery to reduce to the algebraically closed case (Galois descent). The result provides a partial extension of Suslin's description of indecomposable $K_3$ and builds a robust framework to approach Goncharov's conjecture in characteristic zero, linking motivic cohomology with polylogarithmic and higher Chow constructions. Overall, the work advances the understanding of how motivic invariants align with explicit polylogarithmic complexes, enabling concrete computations and potential characteristic-generalizations.

Abstract

Let $\mathbb K$ be a field of characteristic zero. We prove that its motivic cohomology in degree $m-1$ and weight $m$ is rationally isomorphic to the cohomology of the polylogarithmic complex. This gives a partial extension of A. Suslin theorem describing the indecomposable $K_3$ of a field.

On Goncharov's conjecture in next to Milnor degree

TL;DR

The paper tackles Goncharov's conjecture in next to Milnor degree by proving a canonical isomorphism between the motivic cohomology and the cohomology of the polylogarithmic complex for a field of characteristic zero and . The authors build an auxiliary complex and relate it to higher Chow groups via a map , while a Totaro-type map connects the polylogarithmic side to the higher Chow side; a crucial step is proving that and are (quasi-)isomorphisms. A strong Suslin reciprocity law is developed to control reciprocity phenomena across valuations and field extensions, including a norm-map machinery to reduce to the algebraically closed case (Galois descent). The result provides a partial extension of Suslin's description of indecomposable and builds a robust framework to approach Goncharov's conjecture in characteristic zero, linking motivic cohomology with polylogarithmic and higher Chow constructions. Overall, the work advances the understanding of how motivic invariants align with explicit polylogarithmic complexes, enabling concrete computations and potential characteristic-generalizations.

Abstract

Let be a field of characteristic zero. We prove that its motivic cohomology in degree and weight is rationally isomorphic to the cohomology of the polylogarithmic complex. This gives a partial extension of A. Suslin theorem describing the indecomposable of a field.
Paper Structure (33 sections, 63 theorems, 151 equations)

This paper contains 33 sections, 63 theorems, 151 equations.

Table of Contents

  1. Introduction
  2. The idea of the proof
  3. The structure of the paper
  4. Acknowledgements
  5. List of notations
  6. Preliminary definitions and lemmas
  7. Strictly regular elements
  8. Higher Chow groups
  9. The complex $\Lambda(\mathbb K,m)$
  10. Definition of the complex $\Lambda(\mathbb K, m)$
  11. Some vanishing statements
  12. The differential is well-defined
  13. The complex $\Lambda(\mathbb K,m)$ and the Bloch's higher Chow groups
  14. The generators of $\Lambda(\mathbb K,m)$
  15. The map $\widetilde{\mathcal{W}}$ is an isomorphism
  16. ...and 18 more sections

Key Result

Theorem 1.2

Let $\mathbb K$ be a field of characteristic zero and $m\in\mathbb Z, m\geq 2$. We have the canonical isomorphism $H^{m-1,m}(\mathop{\mathrm{Spec}}\nolimits\mathbb K,\mathbb Q)\cong H^{m-1}(\Gamma(\mathbb K,m))$.

Theorems & Definitions (120)

  • Conjecture 1.1: Goncharov's conjecture
  • Theorem 1.2
  • Theorem 1.3
  • Definition 1.4
  • Theorem 1.5
  • Proposition 1.6
  • Example
  • Theorem 1.7
  • Proposition 1.8
  • Proposition 3.1
  • ...and 110 more