{"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/fi-and-oi-modules-with-varying-coefficients","title":"FI- and OI-modules with varying coefficients","arxiv_id":"1710.09247","date":"2017-10-25","proceeding":null,"authors":["Uwe Nagel","Tim Römer"],"abstract":"We introduce FI-algebras over a commutative ring $K$ and the category of FI-modules over an FI-algebra. Such a module may be considered as a family of invariant modules over compatible varying $K$-algebras. FI-modules over $K$ correspond to the well studied constant coefficient case where every algebra equals $K$. We show that a finitely generated FI-module over a noetherian polynomial FI-algebra is a noetherian module. This is established by introducing OI-modules. We prove that every submodule of a finitely generated free OI-module over a noetherian polynomial OI-algebra has a finite Gr\\\"obner basis. Applying our noetherianity results to a family of free resolutions, finite generation translates into stabilization of syzygies in any fixed homological degree. In particular, in the graded case this gives uniformity results on degrees of minimal syzygies.","url_abs":"https://arxiv.org/abs/1710.09247v2","url_pdf":"https://arxiv.org/pdf/1710.09247v2.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":"fi-and-oi-modules-with-varying-coefficients","repo_url":"https://github.com/morrowmh/oigroebnerbases","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}