Zero-cycles on varieties over a $\mathfrak{B}_s$-field
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866918394636271616 |
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| author | Hiranouchi, Toshiro Sugiyama, Rin |
| author_facet | Hiranouchi, Toshiro Sugiyama, Rin |
| contents | 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 $n \geq s+1$, and $CH^{d+s}(X,s)$ 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 $F$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_15617 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Zero-cycles on varieties over a $\mathfrak{B}_s$-field Hiranouchi, Toshiro Sugiyama, Rin Number Theory 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 $n \geq s+1$, and $CH^{d+s}(X,s)$ 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 $F$. |
| title | Zero-cycles on varieties over a $\mathfrak{B}_s$-field |
| topic | Number Theory |
| url | https://arxiv.org/abs/2509.15617 |