why Borel's computation implies Beilinson-Soulé?

Question

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 $K_i(k) \otimes_{\mathrm{Z}} \mathrm{Q}$?

Answer 1

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.

The translation between the various forms the Beilinson-Soulé conjecture can take is given by the identifications
$$ \operatorname{Hom}_{DM(k)_{\mathbb{Q}}}(\mathbb{Q},\mathbb{Q}(p)[q])\cong H^p_{\operatorname{Nis}}(\operatorname{Spec}k,\mathbb{Q}(q))\cong CH^q(\operatorname{Spec}k,2q-p)_{\mathbb{Q}}\cong K_{2q-p}(k)^{(q)}_{\mathbb{Q}}. $$

The first identification is close to the definition of motivic cohomology. The second identification is Voevodsky's theorem on identification of motivic cohomology with Bloch's higher Chow groups (paper link here), and the last identification is due to Bloch (S. Bloch. Algebraic cycles and algebraic K-theory. Adv. Math. 61, 1986). Some more details can also be found in the lecture notes on motivic cohomology by Mazza, Voevodsky and Weibel. Eventually, I think the reason for these identifications being true is that motivic cohomology is defined as the cohomology of the sheaf of Bloch's cycle complexes, and these have a suitable fibrancy property. That all of this works out is the mystery hidden in the proof details.

Anyway, the K-theoretic formulation of the Beilinson-Soulé conjecture is a consequence of Borel's computation of the ranks of the rational K-groups plus of course the identification of the weights. The identification of weights can be done using the Beilinson-regulator to Deligne cohomology. A list of the right weights and some explanation can be found in this MO-question. Some more details on how to get the weights can be found in this MO-question.