{"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","title":"Weil-étale cohomology and zeta-values of arithmetic schemes at negative integers","arxiv_id":"2102.12114","date":"2021-02-24","proceeding":null,"authors":["Alexey Beshenov"],"abstract":"Following the ideas of Flach and Morin (Doc. Math. 23 (2018), 1425--1560), we state a conjecture in terms of Weil-\\'etale cohomology for the vanishing order and special value of the zeta function $\\zeta (X, s)$ at $s = n < 0$, where $X$ is a separated scheme of finite type over $\\operatorname{Spec} \\mathbb{Z}$. We prove that the conjecture is compatible with closed-open decompositions of schemes and with affine bundles, and consequently, that it holds for cellular schemes over certain one-dimensional bases. This is a continuation of arXiv:2012.11034, which gives a construction of Weil-\\'etale cohomology for $n < 0$ under the mentioned assumptions on $X$.","url_abs":"https://arxiv.org/abs/2102.12114v2","url_pdf":"https://arxiv.org/pdf/2102.12114v2.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","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}