ScalingStacks

Proof. First, note that the higher $K$-theory of finite fields $K_{i}(\operatorname{Spec}(\mathbb{F}_{q}))$ is torsion for $i>0$ by [Qui72, Theorem 8]. The same is true for the algebraic closure, since $K$-theory commutes with filtered colimits. Hence, $K_{i}(k)_{\mathbb{Q}}=0$ for $i>0.$ By the Riemann–Roch theorem for rational higher Chow groups, see [Blo86, Theorem 9.1], $\operatorname{CH}^{m}(k,2m-n)_{\mathbb{Q}}$ is a direct summand of $K_{2m-n}(k)_{\mathbb{Q}}.$ The statement follows from (3.2). ∎