Graded Rings The package &GradedRingForHomalg; defines the classes of graded rings, ring elements and matrices over such rings. These three objects can be used as data structures defined in &MatricesForHomalg; on which the &homalg; project can rely to do homological computations over graded rings.

The graded rings most prominently can be used with methods known from general &homalg; rings. The methods for doing the computations are presented in the appendix (), since they are not for external use. The new attributes and operations are documented here.

Since the objects inplemented here are representations from objects elsewhere in the &homalg; project (i.e. &MatricesForHomalg;), we want to stress that there are many other operations in &MatricesForHomalg;, which can be used in connection with the ones presented here. A few of them can be found in the examples and the appendix of this documentation.

Operations within &MatricesForHomalg; that take matrices as input and produce a matrix as an output produce homogeneous output for homogeneous input in the following cases: the graded ring in question is either a polynomial ring or the exterior algebra residing in &Singular;, and the called operation is one of the following listed below: SyzygiesGeneratorsOfRows SyzygiesGeneratorsOfColumns ReducedSyzygiesGeneratorsOfRows ReducedSyzygiesGeneratorsOfColumns BasisOfRowModule BasisOfColumnModule ReducedBasisOfRowModule ReducedBasisOfColumnModule DecideZeroRows DecideZeroColumns LeftDivide RightDivide These operation trigger Gröbner bases computations in &Singular;, which are always forced to be performed with a tail reduction by &homalg;. In particular, the resulting elements of the Gröbner bases have to be homogeneous.
Graded Rings: Category and Representations <#Include Label="IsHomalgGradedRingRep"> <#Include Label="IsHomalgGradedRingElementRep">
Graded Rings: Constructors <#Include Label="HomalgGradedRingElement">
Graded Rings: Attributes and Properties <#Include Label="DegreeGroup"> <#Include Label="CommonNonTrivialWeightOfIndeterminates"> <#Include Label="WeightsOfIndeterminates"> <#Include Label="IsHomogeneousRingElement">
Graded Rings: Operations and Functions <#Include Label="UnderlyingNonGradedRing:ring"> <#Include Label="UnderlyingNonGradedRing:element"> <#Include Label="Name"> <#Include Label="HomogeneousPartOfRingElement">