Zero-cycles on varieties over a $\mathfrak{B}_s$-field

A field $F$ is a $\mathfrak{B}_s$-field if, for every finite extension $E'/E$ of $F$, the norm map $K_s^M(E')\to K_s^M(E)$ of the Milnor $K$-groups is surjective.

In particular, finite fields ($s=1$), local fields, and certain global fields (with $s=2$) satisfy this condition.

For such a field $F$ and a $d$-dimensional variety $X$ over $F$, we prove that $CH^{d+n}(X,n)$ is divisible for $n \geq s+1$.

Under a suitable condition on the index of $X$, $CH^{d+s}(X,s)$ is isomorphic to the direct sum of the Milnor $K$-group $K_{s}^M(F)$ and a divisible group.

As an application, we study the Kato homology groups $KH_0^{(n)}(X,\mathbb{Z}/l^r\mathbb{Z})$ for any prime $l$ different from the characteristic of $F$.