chap0.txt

  
  
                              GradedRingForHomalg
  
  
                Endow Commutative Rings with an Abelian Grading
  
  
                               Version 2017.09.02
  
  
                                 September 2015
  
  
                                Mohamed Barakat
  
                               Sebastian Gutsche
  
                             Markus Lange-Hegermann
  
  
  
            This  manual  is  best  viewed  as an HTML document. The
            latest version is available online at:
            http://homalg.math.rwth-aachen.de/~markus/GradedRingForHomalg/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
  
  
  Sebastian Gutsche
      Email:    mailto:[email protected]
      Homepage: http://wwwb.math.rwth-aachen.de/~gutsche/
      Address:  Lehrstuhl  B  für  Mathematik,  RWTH Aachen, Templergraben 64,
                52056 Aachen, 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
  
  
  
  -------------------------------------------------------
  Copyright
  ©   2008-2015   by   Mohamed   Barakat,   Sebastian   Gutsche,   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 (GradedRingForHomalg)
  
  1 Introduction
    1.1 What is the Role of the GradedRingForHomalg Package in the homalg
    Project?
    1.2 Functionality
    1.3 The Math Behind This Package
  2 Installation of the GradedRingForHomalg Package
  3 Quick Start
  4 Graded Rings
    4.1 Graded Rings: Category and Representations
      4.1-1 IsHomalgGradedRingRep
      4.1-2 IsHomalgGradedRingElementRep
    4.2 Graded Rings: Constructors
      4.2-1 HomalgGradedRingElement
    4.3 Graded Rings: Attributes
      4.3-1 DegreeGroup
      4.3-2 CommonNonTrivialWeightOfIndeterminates
      4.3-3 WeightsOfIndeterminates
      4.3-4 MatrixOfWeightsOfIndeterminates
    4.4 Graded Rings: Operations and Functions
      4.4-1 UnderlyingNonGradedRing
      4.4-2 UnderlyingNonGradedRing
      4.4-3 Name
  5 Homogeneous Matrices
    5.1 Homogeneous Matrices: Category and Representations
      5.1-1 IsHomalgMatrixOverGradedRingRep
    5.2 Homogeneous Matrices: Constructors
      5.2-1 MatrixOverGradedRing
    5.3 Homogeneous Matrices: Attributes
      5.3-1 DegreesOfEntries
      5.3-2 NonTrivialDegreePerRow
      5.3-3 NonTrivialDegreePerColumn
    5.4 Homogeneous Matrices: Operations and Functions
      5.4-1 UnderlyingNonGradedRing
      5.4-2 SetMatElm
      5.4-3 AddToMatElm
      5.4-4 MatElmAsString
      5.4-5 MatElm
  6 Examples
  A The Matrix Tool Operations
    A.1 The Tool Operations without a Fallback Method
    A.2 The Tool Operations with a Fallback Method
      A.2-1 MonomialMatrix
      A.2-2 RandomMatrixBetweenGradedFreeLeftModules
      A.2-3 RandomMatrixBetweenGradedFreeRightModules
      A.2-4 Diff
  B Overview of the GradedRingForHomalg Package Source Code
    B.1 The generic Methods
      B.1-1 BasisOfRowModule
      B.1-2 DecideZeroRows
      B.1-3 SyzygiesGeneratorsOfRows
    B.2 Tools