Finite generation of motivic cohomology of number fields

Question

Let $F$ be a number field ($F=\mathbb Q$ is fine for my purposes) and let $n\geq2$ be an integer. Is it known whether the first motivic cohomology groups $$\mathrm H^1(\mathrm{Spec}(F),\mathbb Z(n))$$ are finitely generated? We know that they are finite-dimensional after tensoring with $\mathbb Q$, since they become identified with the rational $K$-theory $K_{2n-1}(F)_{\mathbb Q}$ whose dimension was computed by Borel. But are they finitely generated integrally? A reference, if it exists, would be appreciated.

Answer 1

Indeed, I believe it is known that these are finitely generated. First, the Gysin sequence shows that the map $$H^1(\mathcal{O}_F;\mathbb{Z}(n))\rightarrow H^1(F;\mathbb{Z}(n))$$ is injective with cokernel given by $$\oplus_\nu H^0(k_\nu;\mathbb{Z}(n-1))$$ where $\nu$ runs over all maximal ideals of $\mathcal{O}_F$ with residue field $k_\nu$. As we assumed $n\geq 2$, this direct sum vanishes, so the question is equivalent to showing that $H^1(\mathcal{O}_F;\mathbb{Z}(n))$ is finitely generated. But now that we've replaced $F$ by $\mathcal{O}_F$, this finite generation actually holds in any degree and weight.

Indeed, the K-groups of $\mathcal{O}_F$ are finitely generated as shown by Quillen (a "simple" argument is to use homological stability to reduce to showing that the homology groups of general linear groups over $\mathcal{O}_F$ are finitely generated, which follows from Borel-Serre). Now, the spectral sequence from motivic cohomology to K-theory degenerates rationally by the Adams operations, but in fact more is true, as noted by Kahn: it degenerates "up to isogeny". So K-theory and motivic cohomology can only differ by bounded torsion. Thus, to deduce finite generation of motivic cohomology from that of K-theory, it suffices to see that (mod $p$) motivic cohomology of $\mathcal{O}_K$ is finitely generated for any prime $p$, in any degree and weight. By another application of Gysin and the fact that mod $p$ motivic cohomology of a finite field of characteristic $p$ is only nontrivial when degree = weight = zero, this reduces to the same claim for $\mathcal{O}_K[1/p]$. Now we are in the Bloch-Kato regime where we can compare to etale cohomology, but we should take a bit of care because $\mathcal{O}_K$ is not itself a field. But if you compare Gysin sequences for motivic cohomology and etale cohomology and use Bloch-Kato for the quotient field and residue fields, you do indeed see that the claim reduces to the finiteness of etale cohomology of $\mathcal{O}_K[1/p]$ with $\mathbb{F}_p(n)$-coefficients.

Answer 2

By the Beilinson–Lichtenbaum conjecture (which is now a theorem), this group is just isomorphic to the corresponding etale (which is also Galois) cohomology one if $n\ge 1$ and zero for $n=0$ (as noted earlier by Denis Nardin).

To my surprise, this appears to imply that for $n\ge 1$ this group is infinite; see Infiniteness of the Galois cohomology over a number field with coefficients in a finite Galois module