This chapter describes modules over multivariate polynomial rings and
related rings. The fundamental tool for computing with such modules
is the construction of Gröbner bases for modules, since these rings are not
principal ideal rings in general (so standard matrix echelonization
algorithms are not applicable).
In this chapter, unless otherwise indicated,
a ring R will refer to one of the following:
- (a)
- Multivariate Polynomial Ring
(Chapters MULTIVARIATE POLYNOMIAL RINGS and GRÖBNER BASES).
Currently the coefficient ring of such a ring may be a field or Euclidean ring
(even operations such as syzygy modules or free resolutions work over
modules whose coefficient rings are Euclidean but not fields).
- (b)
- Local Polynomial Ring (Localization of a Multivariate Polynomial
Ring: Chapter LOCAL POLYNOMIAL RINGS; new in V2.15). Currently the
coefficient ring of such a ring must be a field.
- (c)
- Affine Algebra (Chapter AFFINE ALGEBRAS).
Currently the coefficient ring of such a ring must be a field.
- (d)
- Exterior Algebra (Chapter FINITELY PRESENTED ALGEBRAS; new in V2.15).
Currently the coefficient ring of such a ring must be a field. Strictly
speaking, this is a skew-commutative ring, so is not a commutative
ring, and the associated modules are left R-modules, but
the operations on R-modules in this chapter are practically all
applicable if R is such an algebra also, so the term `a ring R'
will include such an algebra in this chapter.
In this chapter, the term "
module" will always
refer to an R-module, where R is one of the above types of ring,
and such a module will have type
ModMPol (or may have type
ModMPolGrd if graded; see below). So we assume that the reader is
generally familiar with such base rings and their ideals in Magma; see the
relevant chapters for background. Many of the concepts and tools
of Gröbner basis theory carry over from these types of rings.
V2.28, 13 July 2023