Zero-cycles on varieties over a Bₛ-field

A field F is a Bₛ-field if, for every finite extension $E'/E$ of F, the norm map KₛM(E')→ KₛM(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 CHᵈ⁺ⁿ(X,n) is divisible for n ≥ s+1, and CHᵈ⁺ˢ(X,s) is isomorphic to the direct sum of the Milnor K-group KₛM(F) and a divisible group. As an application, we study the Kato homology groups KH₀⁽ⁿ⁾(X,Z/lʳZ) for any prime l different from the characteristic of F.