Lemma 3.1 (0MXR) · ScalingStacks

Lemma 3.1. Let $k=\mathbb{F}_{q}$ or $k=\overline{\mathbb{F}}_{p}$ , then

\displaystyle\operatorname{Hom}_{\operatorname{DM}(k,\mathbb{Q})}(\mathbb{Q},\mathbb{Q}(m)[n])=\left\{\begin{array}[]{cl}\mathbb{Q}&\text{ for }n=m=0\text{ and}\\ 0&\text{otherwise.}\\ \end{array}\right.

0MXS

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). ∎