Goncharov conjecture

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

Statement

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

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

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

References

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