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

Abstract

A field $F$ is a ${\mathfrak{B}}_s$-field if, for every finite extension $E^{\prime }/E$ of $F$, the norm map $K_s^M(E^{\prime })\to K_s^M(E)$ 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 $C{H}^{d+n}(X,n)$ is divisible for $n\ge s+1$. Under a suitable condition on the index of $X$, $C{H}^{d+s}(X,s)$ 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 $K{H}_0^{(n)}(X,\mathbb{Z}/l^r\mathbb{Z})$ for any prime $l$ different from the characteristic of $F$.