FI- and OI-modules with varying coefficients

Uwe Nagel, Tim Römer

Research output: Contribution to journalArticlepeer-review

14 Scopus citations


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 Gröbner 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.

Original languageEnglish
Pages (from-to)286-322
Number of pages37
JournalJournal of Algebra
StatePublished - Oct 1 2019

Bibliographical note

Publisher Copyright:
© 2019 Elsevier Inc.


  • Category
  • Functor
  • Gröbner basis
  • Noetherian
  • Syzygy

ASJC Scopus subject areas

  • Algebra and Number Theory


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