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 [math]\displaystyle{ \Gamma(F,n) }[/math] placed in degrees [math]\displaystyle{ [1,n] }[/math]:

[math]\displaystyle{ \Gamma_F(n)\colon \mathcal B_n(F)\to \mathcal B_{n-1}(F)\otimes F^\times_\mathbb Q\to\dots\to \Lambda^n F^\times_\mathbb Q. }[/math]

He conjectured that i-th cohomology of this complex is isomorphic to the motivic cohomology group [math]\displaystyle{ H^i_{mot}(F,\mathbb Q(n)) }[/math].

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 
• Zagier, Don (1991), "Polylogarithms, Dedekind zeta functions and the algebraic K-theory of fields", Arithmetic algebraic geometry (Texel, 1989), Progr. Math., 89, Boston, MA: Birkhäuser Boston, pp. 391–430, ISBN 978-0-8176-3513-8