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



<Appendix Label="FileOverview">

<Heading>Overview of the &LocalizeRingForHomalg; Package Source Code</Heading>

  This appendix is included in the documentation to shine some light
  on the mathematical backgrounds of this Package. Neither is it
  needed to work with this package nor should the methods presented
  here be called directly. The functions documented here are entries
  of the so called ring table and not to be called directly. There are
  higher level methods in declared and installed in
  &MatricesForHomalg;, which call this functions
  (&see; <C>?MatricesForHomalg:The Basic Matrix Operations</C>).

  <P/>We only present the simpler procedures, where no transformation
  matrices are computed, since the computation of transformation
  matrices carries no further mathematical ideas.

  <Section Label="HomalgTable:Generic">

     <Heading>The generic Methods</Heading>

     There are some methods in localized rings, where homalg is able
     to fall back on procedures of the corresponding global
     ring. Furthermore these methods work quite good together with
     Mora's algorithm as implemented in Singular, since we can treat
     it like a global ring.

     We will present some methods as an example, to show the idea:

     <#Include Label="BasisOfRowModule">

     <#Include Label="DecideZeroRows:generic">

     <#Include Label="SyzygiesGeneratorsOfRows">


  </Section>

  <Section Label="HomalgTable:DecideZero">

    <Heading>The Local Decide Zero trick</Heading>

    <#Include Label="DecideZeroRows">


  </Section>

  <Section Label="LocalizeRing:Tools">

    <Heading>Tools</Heading>

    The package &LocalizeRingForHomalg; also implements tool
    functions. These are referred to from &MatricesForHomalg;
    automatically. We list the implemented methods here are and refer
    to the &MatricesForHomalg; documentation
    (&see; <C>?MatricesForHomalg: The Matrix Tool Operations</C>
    and <C>?MatricesForHomalg:RingElement</C>) for details. All tools
    functions from &MatricesForHomalg; not listed here are also
    supported by fallback tools.

    <List>
        <Item>IsZero</Item>
        <Item>IsOne</Item>
        <Item>Minus</Item>
        <Item>DivideByUnit</Item>
        <Item>IsUnit</Item>
        <Item>Sum</Item>
        <Item>Product</Item>
        <Item>ShallowCopy</Item>
        <Item>ZeroMatrix</Item>
        <Item>IdentityMatrix</Item>
        <Item>AreEqualMatrices</Item>
        <Item>Involution</Item>
        <Item>CertainRows</Item>
        <Item>CertainColumns</Item>
        <Item>UnionOfRows</Item>
        <Item>UnionOfColumns</Item>
        <Item>DiagMat</Item>
        <Item>KroneckerMat</Item>
        <Item>DualKroneckerMat</Item>
        <Item>MulMat</Item>
        <Item>AddMat</Item>
        <Item>SubMat</Item>
        <Item>Compose</Item>
        <Item>NumberRows</Item>
        <Item>NumberColumns</Item>
        <Item>IsZeroMatrix</Item>
        <Item>IsDiagonalMatrix</Item>
        <Item>ZeroRows</Item>
        <Item>ZeroColumns</Item>
    </List>

  </Section>

</Appendix>

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