chap0.txt

  
  
                             LocalizeRingForHomalg
  
  
              Localize Commutative homalg Rings at Maximal Ideals
  
  
                               Version 2017.09.02
  
  
                                 September 2015
  
  
                                Mohamed Barakat
  
                             Markus Lange-Hegermann
  
                                   Vinay Wagh
  
  
  
            This  manual  is  best  viewed  as an HTML document. The
            latest version is available online at:
            http://homalg.math.rwth-aachen.de/~markus/LocalizeRingForHomalg/chap0.html
            An   offline   version   should   be   included  in  the
            documentation subfolder of the package.
  
  
  
  Mohamed Barakat
      Email:    mailto:[email protected]
      Homepage: http://www.mathematik.uni-kl.de/~barakat/
      Address:  Department of Mathematics,
                University of Kaiserslautern,
                67653 Kaiserslautern,
                Germany
  
  
  Markus Lange-Hegermann
      Email:    mailto:[email protected]
      Homepage: http://wwwb.math.rwth-aachen.de/~markus
      Address:  Lehrstuhl  B  für  Mathematik,  RWTH Aachen, Templergraben 64,
                52056 Aachen, Germany
  
  
  Vinay Wagh
      Email:    mailto:[email protected]
      Homepage: http://www.iitg.ernet.in/vinay.wagh/
      Address:  E-102, Department of Mathematics,
                Indian Institute of Technology Guwahati,
                Guwahati, Assam, India.
                PIN: 781 039.
  
  
  
  -------------------------------------------------------
  Copyright
  © 2009-2015 by Mohamed Barakat and Markus Lange-Hegermann
  
  This  package  may  be distributed under the terms and conditions of the GNU
  Public License Version 2.
  
  
  -------------------------------------------------------
  Acknowledgements
  We  would  like  to  thank the Aachen forest for being such a nice place for
  jogging.
  
  
  -------------------------------------------------------
  
  
  Contents (LocalizeRingForHomalg)
  
  1 Introduction
    1.1 What is the Role of the LocalizeRingForHomalg Package in the homalg
    Project?
    1.2 Functionality
    1.3 The Math Behind This Package
    1.4 Which Ring to Use?
  2 Installation of the LocalizeRingForHomalg Package
  3 Quick Start
    3.1 Localization of ℤ
  4 Localize Rings
    4.1 Category and Representations
      4.1-1 IsHomalgLocalRingRep
      4.1-2 IsHomalgLocalRingElementRep
      4.1-3 IsHomalgLocalMatrixRep
    4.2 Rings: Attributes
      4.2-1 GeneratorsOfMaximalLeftIdeal
      4.2-2 GeneratorsOfMaximalRightIdeal
    4.3 Operations and Functions
      4.3-1 AssociatedGlobalRing
      4.3-2 AssociatedGlobalRing
      4.3-3 AssociatedGlobalRing
      4.3-4 Numerator
      4.3-5 Numerator
      4.3-6 Denominator
      4.3-7 Denominator
      4.3-8 Name
      4.3-9 SetMatElm
      4.3-10 AddToMatElm
      4.3-11 MatElmAsString
      4.3-12 MatElm
      4.3-13 Cancel
      4.3-14 LocalizeAt
      4.3-15 LocalizeAtZero
      4.3-16 LocalizePolynomialRingAtZeroWithMora
      4.3-17 HomalgLocalRingElement
      4.3-18 HomalgLocalMatrix
  5 Examples
    5.1 An Easy Polynomial Example
    5.2 Hom(Hom(-,Z128),Z16)
    5.3 ResidueClass
    5.4 Testing the Intersection Formula
  A Overview of the LocalizeRingForHomalg Package Source Code
    A.1 The generic Methods
      A.1-1 BasisOfRowModule
      A.1-2 DecideZeroRows
      A.1-3 SyzygiesGeneratorsOfRows
    A.2 The Local Decide Zero trick
      A.2-1 DecideZeroRows
    A.3 Tools