On the weight zero motivic cohomology
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866909406428397568 |
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| author | Molokov, Semen Vologodsky, Vadim |
| author_facet | Molokov, Semen Vologodsky, Vadim |
| contents | We prove that singular cohomology of the underlying space of Berkovich's analytification of a scheme $X$ locally of finite type over a trivially-valued field $k$ of characteristic $0$ is isomorphic to cdh-cohomology with integer coefficients which is also isomorphic to the weight zero motivic cohomology $H^*(X, \mathbb{Z})$. Using this isomorphism, we demonstrate the vanishing of $RHom_{Sh_{Nis}(cor_k)}(\underline{G},\mathbb{Z})$, where $\underline{G}$ denotes the Nisnevich sheaf with transfers associated with a commutative algebraic group $G$ over $k$. For abelian $k$-varieties $A$ and $B$, we prove that $RHom_{\mathcal{PS}h_{tr}}(\underline{A},\underline{B})$ is isomorphic to $Hom_{\mathbf{Ab_k}}(A,B)$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_18274 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On the weight zero motivic cohomology Molokov, Semen Vologodsky, Vadim Algebraic Geometry We prove that singular cohomology of the underlying space of Berkovich's analytification of a scheme $X$ locally of finite type over a trivially-valued field $k$ of characteristic $0$ is isomorphic to cdh-cohomology with integer coefficients which is also isomorphic to the weight zero motivic cohomology $H^*(X, \mathbb{Z})$. Using this isomorphism, we demonstrate the vanishing of $RHom_{Sh_{Nis}(cor_k)}(\underline{G},\mathbb{Z})$, where $\underline{G}$ denotes the Nisnevich sheaf with transfers associated with a commutative algebraic group $G$ over $k$. For abelian $k$-varieties $A$ and $B$, we prove that $RHom_{\mathcal{PS}h_{tr}}(\underline{A},\underline{B})$ is isomorphic to $Hom_{\mathbf{Ab_k}}(A,B)$. |
| title | On the weight zero motivic cohomology |
| topic | Algebraic Geometry |
| url | https://arxiv.org/abs/2411.18274 |