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 {\displaystyle F} be a field. Goncharov defined the following complex called Γ ( F , n ) {\displaystyle \Gamma (F,n)} placed in degrees [ 1 , n ] {\displaystyle [1,n]} :
Γ F ( n ) : B n ( F ) → B n − 1 ( F ) ⊗ F Q × → ⋯ → Λ n F Q × . {\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 i {\displaystyle i} -th cohomology of this complex is isomorphic to the motivic cohomology group H m o t i ( F , Q ( n ) ) {\displaystyle H_{mot}^{i}(F,\mathbb {Q} (n))} .
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