Goncharov conjecture

In mathematics, the Goncharov conjecture is a conjecture introduced by {{harvs|txt|last=Goncharov|authorlink=Alexander Goncharov|year=1995}} suggesting that the cohomology of certain motivic complexes coincides with pieces of K-groups. It extends a conjecture due to {{harvs|txt|last=Zagier|authorlink=Don Zagier|year=1991}}.

Statement

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

: $\Gamma_F(n)\colon \mathcal B_n(F)\to \mathcal B_{n-1}(F)\otimes F^\times_\mathbb Q\to\dots\to \Lambda^n F^\times_\mathbb Q.$

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

References

{{Citation | last1=Goncharov | first1=A. B. | title=Geometry of configurations, polylogarithms, and motivic cohomology | doi=10.1006/aima.1995.1045 | doi-access=free | mr=1348706 | year=1995 | journal=Advances in Mathematics | issn=0001-8708 | volume=114 | issue=2 | pages=197–318}}
{{Citation | last1=Zagier | first1=Don | title=Arithmetic algebraic geometry (Texel, 1989) | publisher=Birkhäuser Boston | location=Boston, MA | series=Progr. Math. | isbn=978-0-8176-3513-8 |mr=1085270 | year=1991 | volume=89 | chapter=Polylogarithms, Dedekind zeta functions and the algebraic K-theory of fields | pages=391–430}}

Category:Conjectures

Category:K-theory

Category:Cohomology theories