{"about":{"site":"https://codewithpapers.app","non_affiliation":"Code with Papers and Syntology are not affiliated with, endorsed by, or sponsored by Papers with Code, Meta, or the pwc-archive mirror.","licence":"CC BY-SA 4.0","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","attribution":"https://codewithpapers.app/attribution","modified":"archive material modified by Syntology; see the attribution page"},"url":"/paper/weil-etale-cohomology-for-arbitrary-1","title":"Weil-étale cohomology and duality for arithmetic schemes in negative weights","arxiv_id":"2012.11034","date":"2020-12-20","proceeding":null,"authors":["Alexey Beshenov"],"abstract":"Flach and Morin constructed in (Doc. Math. 23 (2018), 1425--1560) Weil-\\'etale 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 \\'etale motivic cohomology groups $H^i(X_\\text{\\'et}, \\mathbb{Z}^c(n))$, 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.","url_abs":"https://arxiv.org/abs/2012.11034v3","url_pdf":"https://arxiv.org/pdf/2012.11034v3.pdf","source":{"archive":"pwc-archive (Hugging Face), CC BY-SA 4.0","snapshot":"2025-07-28","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","row_kind":"links_only","authors_date_abstract":"arXiv metadata, CC0 1.0 (https://info.arxiv.org/help/license), from the Kaggle arXiv metadata snapshot of 2026-09-12"},"code_links":[{"paper_slug":"weil-etale-cohomology-for-arbitrary-1","repo_url":"https://github.com/alexey-beshenov/weil-etale","is_official":0,"mentioned_in_paper":0,"mentioned_in_github":1,"framework":"none","reach":null}],"tasks":[],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"syntology_url":null,"atlas_url":null,"mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}