Goncharov conjecture

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

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

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