GradedRing.xml GradedRingForHomalg package documentation Mohamed Barakat Markus Lange-Hegermann
Copyright (C) 2010, Mohamed Barakat, University of Kaiserslautern Markus Lange-Hegermann, RWTH-Aachen University
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<Chapter Label="GradedRing">
<Heading>Graded Rings</Heading>
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.
<P/>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 (<Ref Appendix="FileOverview" />),
since they are not for external use. The new attributes and
operations are documented here.
<P/>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.
<P/>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:
<List>
<Item><C>SyzygiesGeneratorsOfRows</C></Item>
<Item><C>SyzygiesGeneratorsOfColumns</C></Item>
<Item><C>ReducedSyzygiesGeneratorsOfRows</C></Item>
<Item><C>ReducedSyzygiesGeneratorsOfColumns</C></Item>
<Item><C>BasisOfRowModule</C></Item>
<Item><C>BasisOfColumnModule</C></Item>
<Item><C>ReducedBasisOfRowModule</C></Item>
<Item><C>ReducedBasisOfColumnModule</C></Item>
<Item><C>DecideZeroRows</C></Item>
<Item><C>DecideZeroColumns</C></Item>
<Item><C>LeftDivide</C></Item>
<Item><C>RightDivide</C></Item>
</List>
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.
<Section Label="GradedRings:Category">
<Heading>Graded Rings: Category and Representations</Heading>
<#Include Label="IsHomalgGradedRingRep">
<#Include Label="IsHomalgGradedRingElementRep">
</Section>
<Section Label="GradedRings:Constructors">
<Heading>Graded Rings: Constructors</Heading>
<#Include Label="HomalgGradedRingElement">
</Section>
<Section Label="GradedRings:Attributes and Properties">
<Heading>Graded Rings: Attributes and Properties</Heading>
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