FI- and OI-modules with varying coefficients

TL;DR

This paper develops a framework for FI- and OI-modules with varying coefficients over rings, establishing their noetherian properties and implications for syzygy stabilization and minimal degree bounds.

Contribution

It introduces FI- and OI-modules over rings, proves their noetherianity over polynomial algebras, and applies these results to syzygy stabilization and minimal degree bounds.

Findings

Finitely generated FI-modules over noetherian polynomial FI-algebras are noetherian.

Submodules of finitely generated free OI-modules have finite Gr"obner bases.

Results imply stabilization of syzygies and bounds on degrees of minimal syzygies.

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.