Macaulay2 » Documentation
Packages » OIGroebnerBases :: OIGroebnerBases
next | previous | forward | backward | up | index | toc

OIGroebnerBases -- OI-modules over Noetherian polynomial OI-algebras


OIGroebnerBases is a package for Gröbner bases, syzygies and free resolutions for submodules of free OI-modules over Noetherian polynomial OI-algebras. For an introduction to the theory of OI-modules, see [3].

Given a Noetherian polynomial OI-algebra $\mathbf{P} := (\mathbf{X}^{\text{OI},1})^{\otimes c}$ for some integer $c > 0$, one can consider free OI-modules $\mathbf{F} := \bigoplus_{i=1}^s\mathbf{F}^{\text{OI}, d_i}$ over $\mathbf{P}$ for integers $d_i\geq 0$.

Gröbner bases for submodules of $\mathbf{F}$ were introduced in [3]. Free resolutions and homological aspects of submodules have been studied in [2,3]. Using the methods of [1], Gröbner bases, syzygy modules, and free resolutions for submodules can be computed with oiGB, oiSyz and oiRes respectively.


[1] M. Morrow and U. Nagel, Computing Gröbner Bases and Free Resolutions of OI-Modules, Preprint, arXiv:2303.06725, 2023.

[2] N. Fieldsteel and U. Nagel, Minimal and cellular free resolutions over polynomial OI-algebras, Preprint, arXiv:2105.08603, 2021.

[3] U. Nagel and T. Römer, FI- and OI-modules with varying coefficients, J. Algebra 535 (2019), 286-322.



This documentation describes version 1.0.0 of OIGroebnerBases.

Source code

The source code from which this documentation is derived is in the file OIGroebnerBases.m2.


For the programmer

The object OIGroebnerBases is a package.


Polynomial OI-algebras

Free OI-modules

OI-Gröbner bases