The big de Rham-Witt forms over fields and motives of non-reduced schemes
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866912740395712512 |
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| author | Park, Jinhyun |
| author_facet | Park, Jinhyun |
| contents | Using algebraic cycles as a medium, we prove that the groups of the big (Hesselholt-Madsen) de Rham-Witt forms over arbitrary fields are isomorphic to the relative improved (Gabber-Kerz) Milnor $K$-groups of Artin local algebras of embedding dimension $1$. This answers an old problem on the relative Milnor $K$-groups studied since 1970s, especially in ${\rm char} (k) = p>0$.
Applications include an interpretation of the big de Rham-Witt forms precisely as the vanishing cycles of the Elmanto-Morrow motivic cohomology of non-reduced schemes, as well as a construction of an extended logarithmic derivative map $d\log$ on the Milnor $K$-theory of some Artin rings to the de Rham-Witt forms. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_01254 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The big de Rham-Witt forms over fields and motives of non-reduced schemes Park, Jinhyun Algebraic Geometry K-Theory and Homology Number Theory Primary 14C25, Secondary 19D45, 13F35, 14F42 Using algebraic cycles as a medium, we prove that the groups of the big (Hesselholt-Madsen) de Rham-Witt forms over arbitrary fields are isomorphic to the relative improved (Gabber-Kerz) Milnor $K$-groups of Artin local algebras of embedding dimension $1$. This answers an old problem on the relative Milnor $K$-groups studied since 1970s, especially in ${\rm char} (k) = p>0$. Applications include an interpretation of the big de Rham-Witt forms precisely as the vanishing cycles of the Elmanto-Morrow motivic cohomology of non-reduced schemes, as well as a construction of an extended logarithmic derivative map $d\log$ on the Milnor $K$-theory of some Artin rings to the de Rham-Witt forms. |
| title | The big de Rham-Witt forms over fields and motives of non-reduced schemes |
| topic | Algebraic Geometry K-Theory and Homology Number Theory Primary 14C25, Secondary 19D45, 13F35, 14F42 |
| url | https://arxiv.org/abs/2512.01254 |