Goncharov conjecture

In mathematics, the Goncharov conjecture is a conjecture introduced by Goncharov (1995) suggesting that the cohomology of certain motivic complexes coincides with pieces of K-groups. It extends a conjecture due to Zagier (1991).

Statement

Let F be a field. Goncharov defined the following complex called $\Gamma (F,n)$ placed in degrees $[1,n]$:

$\Gamma _{F}(n)\colon {\mathcal {B}}_{n}(F)\to {\mathcal {B}}_{n-1}(F)\otimes F_{\mathbb {Q} }^{\times }\to \dots \to \Lambda ^{n}F_{\mathbb {Q} }^{\times }.$

He conjectured that i-th cohomology of this complex is isomorphic to the motivic cohomology group $H_{mot}^{i}(F,\mathbb {Q} (n))$.

References

• Goncharov, A. B. (1995), "Geometry of configurations, polylogarithms, and motivic cohomology", Advances in Mathematics, 114 (2): 197–318, doi:10.1006/aima.1995.1045, ISSN 0001-8708, MR 1348706
• Zagier, Don (1991), "Polylogarithms, Dedekind zeta functions and the algebraic K-theory of fields", Arithmetic algebraic geometry (Texel, 1989), Progr. Math., vol. 89, Boston, MA: Birkhäuser Boston, pp. 391–430, ISBN 978-0-8176-3513-8, MR 1085270