Beilinson–Soulé vanishing for the γ-filtration in algebraic K-theory

Prove that for every field F, every integer n ≥ 1, and every integer i ≤ 0, the nth graded piece of the γ-filtration on the rational K-group vanishes; equivalently, establish gr^n_γ K_{2n−i}(F)_Q = 0 for all such (F, n, i).

Background

The triangulated category DMT(F) of mixed Tate motives over a field F is generated by pure Tate objects Q(−n). Understanding morphisms between these objects is controlled by graded pieces of the γ-filtration on K-theory via Hom-groups in DMT(F).

The Beilinson–Soulé vanishing is the key input that allows Levine to define the heart MT(F) of a t-structure on DMT(F), making it an abelian tannakian category. The conjecture is known for number fields (by Borel’s results) but remains wide open in general.