Parshin's conjecture states that higher algebraic k-theory of smooth projective varieties over finite fields are rationally trivial. This has been shown for curves. Quillen showed that the K-groups in the case of curves are finitely generated. Harder used this to show the K-groups are finite so in particular they are rationally trivial. My questions are:

1) If we know for some smooth projective variety over a finite field, the K-groups are rationally finitely generated (They are finite dimensional vector spaces), does this imply the parshin conjecture for this variety?

2) What exactly is known about the rational algebraic k-theory of these varieties? are they finite dimensional vector spaces?



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