Proof of the Parshin's Conjecture
We prove the Parshin's conjecture on the rational triviality of the higher algebraic $K$-theory of smooth projective varieties over finite fields. This is known to imply the Beilinson-Soulé conjecture for the fields of positive characteristic. Especially it implies that for a field $F$ of char $p$ and $n>0$, the only rationally non-trivial weight appearing in $K_n(F)\otimes \mathbb{Q}$ can be $n$, thus $K_n(F)\otimes \mathbb{Q}=K_n^M(F)\otimes \mathbb{Q}$ where $K_n^M(F)$ is the Milnor $K$-theory.
math.KT↗