Higher Chow groups and not necessarily admissible cycles
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arXiv
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| Format: | Preprint |
| Published: |
2023
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| _version_ | 1866929238057156608 |
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| author | Bolbachan, Vasily |
| author_facet | Bolbachan, Vasily |
| contents | We construct some analog of cubical Bloch's higher Chow groups. Instead of considering cycles in $X\times\mathbb A^n$ we consider varieties $Y$ over $X$ together with a distinguished element in the $n$-th exterior power of the multiplicative group of the field of fraction on $Y$. This definition allows us to make sense of a cycle in $X\times\mathbb A^n$ intersecting faces improperly as an element in this complex.
We prove that this complex is well-defined and study its basic properties: flat pullback, the localization sequence etc. As an application we prove that the cohomology of this complex in degrees $m-1, m$ and weight $m$ isomorphic to the cohomology of polylogarithmic complex. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2311_07567 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Higher Chow groups and not necessarily admissible cycles Bolbachan, Vasily Algebraic Geometry K-Theory and Homology We construct some analog of cubical Bloch's higher Chow groups. Instead of considering cycles in $X\times\mathbb A^n$ we consider varieties $Y$ over $X$ together with a distinguished element in the $n$-th exterior power of the multiplicative group of the field of fraction on $Y$. This definition allows us to make sense of a cycle in $X\times\mathbb A^n$ intersecting faces improperly as an element in this complex. We prove that this complex is well-defined and study its basic properties: flat pullback, the localization sequence etc. As an application we prove that the cohomology of this complex in degrees $m-1, m$ and weight $m$ isomorphic to the cohomology of polylogarithmic complex. |
| title | Higher Chow groups and not necessarily admissible cycles |
| topic | Algebraic Geometry K-Theory and Homology |
| url | https://arxiv.org/abs/2311.07567 |