Quelle MatrixOverGradedRing.xml Sprache: XML

<?xml version="1.0" encoding="UTF-8"?>



<Chapter Label="HomogeneousMatrices">

  <Heading>Homogeneous Matrices</Heading>

  The package &GradedRingForHomalg; defines the classes of graded
  rings, ring elements and homogeneous 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="HomogeneousMatrices:Category">

    <Heading>Homogeneous Matrices: Category and Representations</Heading>

    <#Include Label="IsHomalgMatrixOverGradedRingRep">

  </Section>

  <Section Label="Matrices:Constructors">

    <Heading>Homogeneous Matrices: Constructors</Heading>

    <#Include Label="MatrixOverGradedRing">

  </Section>

  <Section Label="HomogeneousMatrices:Attributes">
    <Heading>Homogeneous Matrices: Attributes</Heading>

    <#Include Label="DegreesOfEntries">
    <#Include Label="NonTrivialDegreePerRow">
    <#Include Label="NonTrivialDegreePerColumn">
    <#Include Label="HomogeneousPartOfMatrix">
    <#Include Label="IsMatrixOverGradedRingWithHomogeneousEntries">

  </Section>

  <Section Label="HomogeneousMatrices:Operations">

    <Heading>Homogeneous Matrices: Operations and Functions</Heading>

    <#Include Label="UnderlyingNonGradedRing:matrix">

    <#Include Label="SetMatElm">

    <#Include Label="AddToMatElm">

    <#Include Label="MatElmAsString">

    <#Include Label="MatElm">

  </Section>

</Chapter>

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