chap0.txt

  
  
                                    Modules
  
  
     A homalg based Package for the Abelian Category of Finitely Presented
                         Modules over Computable Rings
  
  
                               Version 2018.01.07
  
  
                                 September 2015
  
  
                                Mohamed Barakat
  
                             Markus Lange-Hegermann
  
  
  
            (this manual is still under construction)
            This  manual  is  best  viewed  as an HTML document. The
            latest version is available online at:
            http://homalg.math.rwth-aachen.de/~barakat/homalg-project/Modules/chap0.html
            An   offline   version   should   be   included  in  the
            documentation  subfolder of the package. This package is
            part of the homalg-project:
            http://homalg.math.rwth-aachen.de/index.php/core-packages/modules-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
  
  
  
  -------------------------------------------------------
  Copyright
  © 2007-2015 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 are very much indebted to Alban Quadrat.
  
  
  -------------------------------------------------------
  
  
  Contents (Modules)
  
  1 Introduction
    1.1 What is the role of the Modules package in the homalg project?
      1.1-1 Modules provides ...
      1.1-2 Rings supported in a sufficient way
      1.1-3 Principal limitation
      1.1-4 Ring dictionaries (technical)
      1.1-5 The advantages of the outsourcing concept
      1.1-6 Does this mean that homalg has only algorithms for the generic
      case?
      1.1-7 The principle of least communication (technical)
      1.1-8 Frequently asked questions
    1.2 This manual
  2 Installation of the Modules Package
  3 Quick Start
    3.1 Why are all examples in this manual over ℤ or ℤ/mℤ?
    3.2 gap> ExamplesForHomalg();
    3.3 A typical example
      3.3-1 HomHom
  4 Ring Maps
    4.1 Ring Maps: Attributes
      4.1-1 KernelSubobject
      4.1-2 KernelEmb
    4.2 Ring Maps: Operations and Functions
      4.2-1 Kernel
  5 Relations
    5.1 Relations: Categories and Representations
      5.1-1 IsHomalgRelations
      5.1-2 IsHomalgRelationsOfLeftModule
      5.1-3 IsHomalgRelationsOfRightModule
      5.1-4 IsRelationsOfFinitelyPresentedModuleRep
    5.2 Relations: Constructors
    5.3 Relations: Properties
      5.3-1 CanBeUsedToDecideZeroEffectively
      5.3-2 IsInjectivePresentation
    5.4 Relations: Attributes
    5.5 Relations: Operations and Functions
  6 Generators
    6.1 Generators: Categories and Representations
      6.1-1 IsHomalgGenerators
      6.1-2 IsHomalgGeneratorsOfLeftModule
      6.1-3 IsHomalgGeneratorsOfRightModule
      6.1-4 IsGeneratorsOfModuleRep
      6.1-5 IsGeneratorsOfFinitelyGeneratedModuleRep
    6.2 Generators: Constructors
    6.3 Generators: Properties
      6.3-1 IsReduced
    6.4 Generators: Attributes
      6.4-1 ProcedureToReadjustGenerators
    6.5 Generators: Operations and Functions
  7 Modules
    7.1 Modules: Category and Representations
      7.1-1 IsHomalgModule
      7.1-2 IsFinitelyPresentedModuleOrSubmoduleRep
      7.1-3 IsFinitelyPresentedModuleRep
      7.1-4 IsFinitelyPresentedSubmoduleRep
    7.2 Modules: Constructors
      7.2-1 LeftPresentation
      7.2-2 RightPresentation
      7.2-3 HomalgFreeLeftModule
      7.2-4 HomalgFreeRightModule
      7.2-5 HomalgZeroLeftModule
      7.2-6 HomalgZeroRightModule
      7.2-7 \*
      7.2-8 Subobject
      7.2-9 Subobject
      7.2-10 LeftSubmodule
      7.2-11 RightSubmodule
    7.3 Modules: Properties
      7.3-1 IsCyclic
      7.3-2 IsHolonomic
      7.3-3 IsReduced
      7.3-4 IsPrimeIdeal
      7.3-5 IsPrimeModule
    7.4 Modules: Attributes
      7.4-1 ResidueClassRing
      7.4-2 PrimaryDecomposition
      7.4-3 RadicalDecomposition
      7.4-4 ModuleOfKaehlerDifferentials
      7.4-5 RadicalSubobject
      7.4-6 SymmetricAlgebra
      7.4-7 ExteriorAlgebra
      7.4-8 ElementaryDivisors
      7.4-9 FittingIdeal
      7.4-10 NonFlatLocus
      7.4-11 LargestMinimalNumberOfLocalGenerators
      7.4-12 CoefficientsOfUnreducedNumeratorOfHilbertPoincareSeries
      7.4-13 CoefficientsOfNumeratorOfHilbertPoincareSeries
      7.4-14 UnreducedNumeratorOfHilbertPoincareSeries
      7.4-15 NumeratorOfHilbertPoincareSeries
      7.4-16 HilbertPoincareSeries
      7.4-17 AffineDegree
      7.4-18 DataOfHilbertFunction
      7.4-19 HilbertFunction
      7.4-20 IndexOfRegularity
    7.5 Modules: Operations and Functions
      7.5-1 HomalgRing
      7.5-2 ByASmallerPresentation
      7.5-3 \*
      7.5-4 SubobjectQuotient
  8 Maps
    8.1 Maps: Categories and Representations
      8.1-1 IsHomalgMap
      8.1-2 IsHomalgSelfMap
      8.1-3 IsMapOfFinitelyGeneratedModulesRep
    8.2 Maps: Constructors
      8.2-1 HomalgMap
      8.2-2 HomalgZeroMap
      8.2-3 HomalgIdentityMap
    8.3 Maps: Properties
    8.4 Maps: Attributes
    8.5 Maps: Operations and Functions
      8.5-1 HomalgRing
      8.5-2 PreInverse
  9 Module Elements
    9.1 Module Elements: Category and Representations
      9.1-1 IsHomalgElement
      9.1-2 IsElementOfAModuleGivenByAMorphismRep
    9.2 Module Elements: Constructors
    9.3 Module Elements: Properties
      9.3-1 IsElementOfIntegers
    9.4 Module Elements: Attributes
    9.5 Module Elements: Operations and Functions
      9.5-1 HomalgRing
  10 Functors
    10.1 Functors: Category and Representations
    10.2 Functors: Constructors
    10.3 Functors: Attributes
    10.4 Basic Functors
      10.4-1 functor_Cokernel
      10.4-2 Cokernel
      10.4-3 functor_ImageObject
      10.4-4 ImageObject
      10.4-5 Kernel
      10.4-6 DefectOfExactness
      10.4-7 Functor_Hom
      10.4-8 Hom
      10.4-9 Functor_TensorProduct
      10.4-10 TensorProduct
      10.4-11 Functor_Ext
      10.4-12 Ext
      10.4-13 Functor_Tor
      10.4-14 Tor
      10.4-15 Functor_RHom
      10.4-16 RHom
      10.4-17 Functor_LTensorProduct
      10.4-18 LTensorProduct
      10.4-19 Functor_HomHom
      10.4-20 Functor_LHomHom
    10.5 Tool Functors
    10.6 Other Functors
    10.7 Functors: Operations and Functions
  11 Symmetric Algebra and Koszul Complex
    11.1 Symmetric Algebra: Constructor
      11.1-1 SymmetricPower
    11.2 Symmetric Algebra: Properties and Attributes
      11.2-1 IsSymmetricPower
      11.2-2 SymmetricPowerExponent
      11.2-3 SymmetricPowerBaseModule
  12 Exterior Algebra and Koszul Complex
    12.1 Exterior Algebra: Constructor
      12.1-1 ExteriorPower
    12.2 Exterior Algebra: Properties and Attributes
      12.2-1 IsExteriorPower
      12.2-2 ExteriorPowerExponent
      12.2-3 ExteriorPowerBaseModule
    12.3 Exterior Algebra: Element Properties
      12.3-1 IsExteriorPowerElement
    12.4 Exterior Algebra: Element Operations
      12.4-1 Wedge
      12.4-2 ExteriorPowerElementDual
      12.4-3 SingleValueOfExteriorPowerElement
    12.5 Koszul complex and Cayley determinant
      12.5-1 KoszulCocomplex
      12.5-2 CayleyDeterminant
      12.5-3 Gcd_UsingCayleyDeterminant
  13 Examples
    13.1 ExtExt
    13.2 Purity
    13.3 TorExt-Grothendieck
    13.4 TorExt
  A The Mathematical Idea behind Modules
  B Logic Subpackages
    B.1 LIMOD: Logical Implications for Modules
    B.2 LIHOM: Logical Implications for Homomorphisms of Modules
  C Overview of the Modules Package Source Code
    C.1 Relations and Generators
    C.2 The Basic Objects
    C.3 The High Level Homological Algorithms
    C.4 Logical Implications for homalg Objects