Weil-étale cohomology and duality for arithmetic schemes in negative weights
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arXiv
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| Formato: | Preprint |
| Publicado: |
2020
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| _version_ | 1866912760394153984 |
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| author | Beshenov, Alexey |
| author_facet | Beshenov, Alexey |
| contents | Flach and Morin constructed in (Doc. Math. 23 (2018), 1425--1560) Weil-étale cohomology $H^i_\text{W,c} (X, \mathbb{Z} (n))$ for a proper, regular arithmetic scheme $X$ (i.e. separated and of finite type over $\operatorname{Spec} \mathbb{Z}$) and $n \in \mathbb{Z}$. In the case when $n < 0$, we generalize their construction to an arbitrary arithmetic scheme $X$, thus removing the proper and regular assumption. The construction uses étale motivic cohomology groups $H^i(X_\text{ét}, \mathbb{Z}^c(n))$, as studied by Geisser (Ann. of Math. (2) 172 (2010), 1095--1126), and assumes their finite generation for $n < 0$. We give a class of X for which finite generation is known, and hence $H^i_\text{W,c} (X, \mathbb{Z} (n))$ is defined unconditionally. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2012_11034 |
| institution | arXiv |
| publishDate | 2020 |
| record_format | arxiv |
| spellingShingle | Weil-étale cohomology and duality for arithmetic schemes in negative weights Beshenov, Alexey Algebraic Geometry Number Theory 14F20, 14F42 Flach and Morin constructed in (Doc. Math. 23 (2018), 1425--1560) Weil-étale cohomology $H^i_\text{W,c} (X, \mathbb{Z} (n))$ for a proper, regular arithmetic scheme $X$ (i.e. separated and of finite type over $\operatorname{Spec} \mathbb{Z}$) and $n \in \mathbb{Z}$. In the case when $n < 0$, we generalize their construction to an arbitrary arithmetic scheme $X$, thus removing the proper and regular assumption. The construction uses étale motivic cohomology groups $H^i(X_\text{ét}, \mathbb{Z}^c(n))$, as studied by Geisser (Ann. of Math. (2) 172 (2010), 1095--1126), and assumes their finite generation for $n < 0$. We give a class of X for which finite generation is known, and hence $H^i_\text{W,c} (X, \mathbb{Z} (n))$ is defined unconditionally. |
| title | Weil-étale cohomology and duality for arithmetic schemes in negative weights |
| topic | Algebraic Geometry Number Theory 14F20, 14F42 |
| url | https://arxiv.org/abs/2012.11034 |