Goncharov conjecture

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

Statement

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

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

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. MR 1348706.

• Zagier, D. (1991). "Polylogarithms, Dedekind zeta functions and the algebraic K-theory of fields". In van der Geer, G.; Oort, F.; Steenbrink, J. (eds.). Arithmetic Algebraic Geometry. Progress in Mathematics. Vol. 89. Boston: Birkhäuser. pp. 391–430. ISBN 978-0-8176-3513-8. MR 1085270.