Introduction

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

• Linear algebra modules are discussed in this section. They originate from AbstractAlgebra.jl, work over Euclidean domains and fields, and employ algorithms from linear algebra, such as the Smith Normal Form.

• Commutative algebra modules over multivariate polynomial rings, quotients of such rings and their localizations are discussed in the Commutative Algebra chapter. They employ Gröbner basis methods and are designed for multivariate polynomial rings, as well as quotients and localizations thereof.

Scope and functionality of linear algebra modules

The implementation described in this section is limited to finitely presented modules over fields and Euclidean domains.

Free modules and vector spaces are available over fields and Euclidean domains, respectively. Submodule, quotient module and direct sum constructions can then be applied recursively to these.

Invariant decompositions can be computed using the Smith Normal Form. The system also provides module homomorphisms and isomorphisms, building on top of the map interface.