LocalizeRing.xml

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  LocalizeRing.xml     LocalizeRingForHomalg package documentation             Mohamed Barakat
                                                                        Markus Lange-Hegermann

         Copyright (C) 2009, Mohamed Barakat, Universität des Saarlandes
                      Markus Lange-Hegermann, RWTH-Aachen University
-->

<Chapter Label="LocalizeRing">
  
  <Heading>Localize Rings</Heading>
  
  The package &LocalizeRingForHomalg; defines the classes of local(ized) rings, local ring elements and local matrices. These three objects can be used as data structures defined in &MatricesForHomalg; on which the &homalg; project can rely to do homological computations over localized rings.
  
  <P/>A &homalg; local ring element contains two &homalg; ring elements, a numerator (&see; <Ref Oper="Numerator" Label="for homalg local ring elements"/>) and a denominator (&see; <Ref Oper="Denominator" Label="for homalg local ring elements"/>). A &homalg; local matrix contains a global &homalg; matrix as a numerator (&see; <Ref Oper="Numerator" Label="for homalg local matrices"/>) and a ring element as a denominator (&see; <Ref Oper="Denominator" Label="for homalg local matrices"/>). New constructors for ring elements and matrices are <Ref Func="HomalgLocalRingElement" Label="constructor for local ring elements using numerator and denominator"/> and <Ref Func="HomalgLocalMatrix" Label="constructor for local matrices using numerator and denominator"/> in addition to the standard contructors introduced in other packages of the &homalg; project.
  
  <P/>The local 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 &homalg;, 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.
  
  <Section Label="Rings:Category">

    <Heading>Category and Representations</Heading>
    
    <#Include Label="IsHomalgLocalRingRep">
    
    <#Include Label="IsHomalgLocalRingElementRep">
    
    <#Include Label="IsHomalgLocalMatrixRep">
  
  </Section>
  
  <Section Label="Rings:Attributes">
    <Heading>Rings: Attributes</Heading>
    
    <#Include Label="GeneratorsOfMaximalLeftIdeal">
    <#Include Label="GeneratorsOfMaximalRightIdeal">
  
  </Section>
  
  <Section Label="LocalizeRing:Operations">
  
    <Heading>Operations and Functions</Heading>
      
    <#Include Label="AssociatedGlobalRing:ring">
    
    <#Include Label="AssociatedGlobalRing:element">
    
    <#Include Label="AssociatedGlobalRing:matrix">
    
    <#Include Label="Numerator:element">
    
    <#Include Label="Numerator:matrix">
    
    <#Include Label="Denominator:element">
    
    <#Include Label="Denominator:matrix">
    
    <#Include Label="Name">
    
    <#Include Label="SetMatElm">
    
    <#Include Label="AddToMatElm">
    
    <#Include Label="MatElmAsString">
    
    <#Include Label="MatElm">
    
    <#Include Label="Cancel">
      
    <#Include Label="LocalizeAt">
      
    <#Include Label="LocalizeAtZero">
      
    <#Include Label="LocalizePolynomialRingAtZeroWithMora">
      
    <#Include Label="HomalgLocalRingElement">
      
    <#Include Label="HomalgLocalMatrix">
  
  </Section>
    
</Chapter>