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 $\mathrm{\Gamma }(F,n)$ placed in degrees $[1,n]$:

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

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

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. 

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