Introduction

OSCAR provides two implementations of finitely presented modules, with different mathematical scopes and algorithmic approaches.

Commutative algebra modules are discussed in this section. They work over multivariate polynomial rings, quotients of such rings and their localizations, and employ Gröbner basis methods.
Linear algebra modules are documented in the Linear Algebra modules section. They originate from AbstractAlgebra.jl, work over Euclidean domains and fields, and employ algorithms from linear algebra, such as the Smith Normal Form.

Scope and functionality of commutative algebra modules

The implementation described in this section is limited to finitely presented modules over the following rings:

multivariate polynomial rings (OSCAR type MPolyRing),
quotients of multivariate polynomial rings (OSCAR type MPolyQuoRing), and
localizations of the above rings (OSCAR types MPolyLocRing, MPolyQuoLocRing).

The OSCAR type FreeMod provides a sparse implementation of free modules. More generally, modules are represented as subquotients, that is, as submodules of quotients of free modules. This representation naturally includes both submodules and quotients of free modules.

Note

Most functions in this section rely on Gröbner (standard) basis techniques and should therefore only be applied to modules over the rings listed above.

Note

For simplicity of the presentation, many examples below use modules over multivariate polynomial rings. The same functionality is generally available for modules over the other supported ring types.