Zero-cycles on varieties over a \mathfrakB_s-field

2025/09/19 by Toshiro Hiranouchi, Rin Sugiyama, Hiranouchi, Toshiro +1
Mathematics · #Algebraic Geometry and Number Theory #Advanced Algebra and Geometry #Tensor decomposition and applications

paper · pdf · doi:10.48550/arxiv.2509.15617

Abstract

A field F is a \mathfrakB_s-field if, for every finite extension E'/E of F, the norm map K_s^M(E')→ 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 ≥ 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 KH₀⁽ⁿ⁾(X,ℤ/l^rℤ) for any prime l different from the characteristic of F.

Related

• Milnor K₂ and field homomorphisms
• On standard norm varieties
• Geometric presentations of Milnor K-groups of certain Artin algebras and Bass-Tate-Kato norms
• Kato-Kuzumaki's properties for function fields over higher local fields
• Some results about zero-cycles on abelian and semi-abelian varieties