In mathematics, the Goncharov conjecture is a conjecture introduced by suggesting that the

cohomology In mathematics, specifically in homology theory and algebraic topology, cohomology is a general term for a sequence of abelian groups, usually one associated with a topological space, often defined from a cochain complex. Cohomology can be viewe ...

of certain

motivic complex In music, a motif IPA: ( /moʊˈtiːf/) (also motive) is a short musical phrase, a salient recurring figure, musical fragment or succession of notes that has some special importance in or is characteristic of a composition: "The moti ...

es coincides with pieces of K-groups. It extends a conjecture due to .

Statement

Let ''F'' be a field. Goncharov defined the following complex called

\Gamma(F,n)

placed in degrees

,n /math>:
: \Gamma_F(n)\colon \mathcal B_n(F)\to \mathcal B_(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_(F,\mathbb Q(n)) .

References

* * Conjectures K-theory Cohomology theories {{abstract-algebra-stub