Goncharov conjecture

Today, Goncharov conjecture is a topic that generates great interest and debate in society. For a long time, Goncharov conjecture has been the subject of study and analysis, but over time it has acquired even greater relevance. This topic has captured the attention of experts and professionals from various areas, who have dedicated themselves to researching and delving into its different aspects. Whether due to its impact on daily life, politics, culture or technology, Goncharov conjecture has become an unavoidable reference point today. In this article, we will thoroughly explore the different facets of Goncharov conjecture and its influence on our society.

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

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

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, ISSN 0001-8708, MR 1348706
Zagier, Don (1991), "Polylogarithms, Dedekind zeta functions and the algebraic K-theory of fields", Arithmetic algebraic geometry (Texel, 1989), Progr. Math., vol. 89, Boston, MA: Birkhäuser Boston, pp. 391–430, ISBN 978-0-8176-3513-8, MR 1085270

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