mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern 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 complexes 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