Geometric presentations of Milnor $K$-groups of certain Artin algebras and Bass-Tate-Kato norms
Fuente:
arXiv
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866915642486030336 |
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| author | Park, Jinhyun |
| author_facet | Park, Jinhyun |
| contents | For an arbitrary field $k$, and an arbitrary regular henselian local $k$-scheme $X$ of dimension $1$ with the residue field $k$, we introduce two subcomplexes of the higher Chow complexes of $X$ using certain extended face intersection conditions. We define suitable equivalence relations on them, and prove that their Milnor range cycle class groups offer geometric presentations of the improved (Gabber-Kerz) Milnor $K$-groups of Artin local $k$-algebras of the embedding dimension $1$, and their relative groups, generalizing the theorem of Nesterenko-Suslin and Totaro.
Using these, we prove the existence of the norm and trace maps for the Milnor $K$-groups of the Artin local algebras associated to arbitrary finite extensions of fields, generalizing the Bass-Tate and Kato norms on the Milnor $K$-theory of fields. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2511_22489 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Geometric presentations of Milnor $K$-groups of certain Artin algebras and Bass-Tate-Kato norms Park, Jinhyun Algebraic Geometry K-Theory and Homology Primary 14C25, Secondary 19D45, 14F42 For an arbitrary field $k$, and an arbitrary regular henselian local $k$-scheme $X$ of dimension $1$ with the residue field $k$, we introduce two subcomplexes of the higher Chow complexes of $X$ using certain extended face intersection conditions. We define suitable equivalence relations on them, and prove that their Milnor range cycle class groups offer geometric presentations of the improved (Gabber-Kerz) Milnor $K$-groups of Artin local $k$-algebras of the embedding dimension $1$, and their relative groups, generalizing the theorem of Nesterenko-Suslin and Totaro. Using these, we prove the existence of the norm and trace maps for the Milnor $K$-groups of the Artin local algebras associated to arbitrary finite extensions of fields, generalizing the Bass-Tate and Kato norms on the Milnor $K$-theory of fields. |
| title | Geometric presentations of Milnor $K$-groups of certain Artin algebras and Bass-Tate-Kato norms |
| topic | Algebraic Geometry K-Theory and Homology Primary 14C25, Secondary 19D45, 14F42 |
| url | https://arxiv.org/abs/2511.22489 |