# why Borel's computation implies Beilinson-Soulé?

Let $k$ be a field of characteristic zero and $DM(k)_{\mathrm{Q}}$ Voevodsky's category of motives over $k$ with rational coefficients. The Beilinson-Soulé conjecture says $$\mathrm{Hom}_{DM(k)_{\mathrm{Q}}}(M(X), \mathbb{Q}(p)[q])=0 \quad \quad (\ast)$$ whenever $p>0$ and $q\leq 0$. Here $M$ denotes the functor from smooth variteties over $k$ to $DM(k)_{\mathrm{Q}}$.

For $X=Spec(k)$, where $k$ is a number field, this follows from Borel's computation of the rank of K-theory. Why?

I know that the rough answer is: Hom's in Voevodsky's category are given by higher Chow groups, which are isomorphic to (graded pieces) of $K$-theory, but can anybody give a precise explanation (in particular, with the indices...). Is is true, for instance, that there is only one graded piece for the Adams operations on $K_i(k) \otimes_{\mathrm{Z}} \mathrm{Q}$?

The Beilinson-Soulé conjecture is not necessarily something about motives. Before Voevodsky's definition of the derived category of motives, the Beilinson-Soulé conjecture was formulated as a question about Adams eigenspaces of K-theory. Of course, people were expecting/anticipating the existence of a (derived) category of motives and were using the graded pieces of the $\gamma$-filtration as replacements for motivic cohomology groups.
$$\operatorname{Hom}_{DM(k)_{\mathbb{Q}}}(\mathbb{Q},\mathbb{Q}(p)[q])\cong H^p_{\operatorname{Nis}}(\operatorname{Spec}k,\mathbb{Q}(q))\cong CH^q(\operatorname{Spec}k,2q-p)_{\mathbb{Q}}\cong K_{2q-p}(k)^{(q)}_{\mathbb{Q}}.$$