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.

References

The following vanishing conjecture, stated independently by Beilinson and Soul\e , plays a central role in the study of mixed Tate motives.

— An introduction to mixed Tate motives  (2404.03770 - Dupont, 2024) in Section 6.1 (The abelian category of mixed Tate motives: definition)

The negative range $2i-j\le-2d$ in \autoref{prop:l-div} is related to the Beilinson--Soul\'e vanishing conjecture (Conjecture~5). If $X$ is regular, the Beilinson--Soul\'e vanishing conjecture predicts $CH{d+i}(X,j)=0 \qquad\text{if $2i-j<-2d$}$. Therefore, the unique $l$-divisibility proved above may be viewed as an unconditional prime-to-$\ch(F)$ substitute for this expected integral vanishing.

— Divisibility and torsion in higher Chow groups over arithmetic fields  (2609.11178 - Hiranouchi et al., 10 Sep 2026) in Remark 2.2, Section 2 (the remark following Proposition 2.1)