Proof of the Parshin's Conjecture

TL;DR

This paper proves Parshin's conjecture, establishing the rational triviality of higher algebraic K-theory for smooth projective varieties over finite fields, and confirming related conjectures in positive characteristic.

Contribution

It provides a proof of Parshin's conjecture, linking algebraic K-theory and motivic cohomology over finite fields, and confirms the Beilinson-Soulé conjecture in positive characteristic.

Findings

Higher algebraic K-theory is rationally trivial for smooth projective varieties over finite fields.

The only non-trivial rational weight in K-theory for fields of positive characteristic is the degree itself.

K_n(F)⊗Q equals Milnor K-theory tensor Q for n>0.

Abstract

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-Soul\'e 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.