Bloch–Kato conjecture on integral motivic cohomology

Let $U$ be a smooth variety over a number field or a local field $F$, and let $\HM^i(U,n)$ be its motivic cohomology. Let $\HMf^i(U,n)$ be the subgroup whose $\ell$-adic Abel–Jacobi images lie in the Bloch–Kato finite subspaces $H^1_f(F_v,V_\ell)$ at every place $v$ and prime $\ell$, and let $\HMO^i(U,n)$ denote the subgroup defined by the integral condition. Bloch–Kato's integrality conjecture.

$$\HMf^i(U,n)=\HMO^i(U,n).$$

For fixed $v$, the subspace

$$\ker\left[\HM^i(U,n)\to H^1(F_v,V_\ell)/H^1_f(F_v,V_\ell)\right]$$

is independent of $\ell$. This is presented as part of the generalisation of the Beilinson conjectures; the source gives no resolution status.