FI- and OI-modules with varying coefficients

Uwe Nagel; Tim Römer

doi:10.1016/j.jalgebra.2019.06.029

FI- and OI-modules with varying coefficients

Uwe Nagel
, Tim Römer

Mathematics

Research output: Contribution to journal › Article › peer-review

34 Scopus citations

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 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 language	English
Pages (from-to)	286-322
Number of pages	37
Journal	Journal of Algebra
Volume	535
DOIs	https://doi.org/10.1016/j.jalgebra.2019.06.029
State	Published - Oct 1 2019

Bibliographical note

Publisher Copyright:
© 2019 Elsevier Inc.

Funding

The first author was partially supported by Simons Foundation grant #317096. He also is grateful to Daniel Erman and Greg Smith for organizing a very inspiring April 2016 Banff workshop on Free Resolutions, Representations, and Asymptotic Algebra. Both authors thank the referee for helpful comments.

Funders	Funder number
Simons Foundation	317096

Keywords

Category
Functor
Gröbner basis
Noetherian
Syzygy

ASJC Scopus subject areas

Algebra and Number Theory

Access to Document

10.1016/j.jalgebra.2019.06.029

Cite this

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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{\"o}bner 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.",

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AU - Römer, Tim

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AB - 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.

KW - Category

KW - Functor

KW - Gröbner basis

KW - Noetherian

KW - Syzygy

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DO - 10.1016/j.jalgebra.2019.06.029

M3 - Article

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VL - 535

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JO - Journal of Algebra

JF - Journal of Algebra

ER -

FI- and OI-modules with varying coefficients

Abstract

Bibliographical note

Funding

Keywords

ASJC Scopus subject areas

Access to Document

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