  
  
                                   [1X Modules [101X
  
  
     [1X A homalg based package for the Abelian category of finitely presented
                         modules over computable rings [101X
  
  
                                   2022.11-01
  
  
                                23 November 2022
  
  
                                 Thomas Bächler
  
                                Mohamed Barakat
  
                                Florian Diebold
  
                               Sebastian Gutsche
  
                             Markus Lange-Hegermann
  
                                   Vinay Wagh
  
  
  
  Thomas Bächler
      Email:    [7Xmailto:thomas@momo.math.rwth-aachen.de[107X
      Homepage: [7Xhttp://wwwb.math.rwth-aachen.de/~thomas/[107X
      Address:  [33X[0;14YThomas Bächler[133X
                [33X[0;14YLehrstuhl B für Mathematik[133X
                [33X[0;14YRWTH Aachen[133X
                [33X[0;14YTemplergraben 64[133X
                [33X[0;14Y52062 Aachen[133X
                [33X[0;14YGermany[133X
  
  
  Mohamed Barakat
      Email:    [7Xmailto:mohamed.barakat@uni-siegen.de[107X
      Homepage: [7Xhttps://mohamed-barakat.github.io[107X
      Address:  [33X[0;14YWalter-Flex-Str. 3[133X
                [33X[0;14Y57072 Siegen[133X
                [33X[0;14YGermany[133X
  
  
  Florian Diebold
      Email:    [7Xmailto:diebold@mathematik.uni-kl.de[107X
      Address:  [33X[0;14YDepartment of Mathematics[133X
                [33X[0;14YUniversity of Kaiserslautern[133X
                [33X[0;14Y67653 Kaiserslautern[133X
                [33X[0;14YGermany[133X
  
  
  Sebastian Gutsche
      Email:    [7Xmailto:gutsche@mathematik.uni-siegen.de[107X
      Homepage: [7Xhttps://sebasguts.github.io[107X
      Address:  [33X[0;14YDepartment Mathematik[133X
                [33X[0;14YUniversität Siegen[133X
                [33X[0;14YWalter-Flex-Straße 3[133X
                [33X[0;14Y57072 Siegen[133X
                [33X[0;14YGermany[133X
  
  
  Markus Lange-Hegermann
      Email:    [7Xmailto:markus.lange-hegermann@hs-owl.de[107X
      Homepage: [7Xhttps://www.th-owl.de/eecs/fachbereich/team/markus-lange-hegermann/[107X
      Address:  [33X[0;14YMarkus Lange-Hegermann[133X
                [33X[0;14YHochschule Ostwestfalen-Lippe[133X
                [33X[0;14YLiebigstraße 87[133X
                [33X[0;14Y32657 Lemgo[133X
                [33X[0;14YGermany[133X
  
  
  Vinay Wagh
      Email:    [7Xmailto:waghoba@gmail.com[107X
      Homepage: [7Xhttp://www.iitg.ernet.in/vinay.wagh/[107X
      Address:  [33X[0;14YE-102, Department of Mathematics,[133X
                [33X[0;14YIndian Institute of Technology Guwahati,[133X
                [33X[0;14YGuwahati, Assam, India.[133X
                [33X[0;14YPIN: 781 039.[133X
                [33X[0;14YIndia[133X
  
  
  
  -------------------------------------------------------
  [1XCopyright[101X
  [33X[0;0Y©  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 or (at your option) any later version.[133X
  
  
  -------------------------------------------------------
  [1XAcknowledgements[101X
  [33X[0;0YWe are very much indebted to Alban Quadrat.[133X
  
  
  -------------------------------------------------------
  
  
  [1XContents (Modules)[101X
  
  1 [33X[0;0YIntroduction[133X
    1.1 [33X[0;0YWhat is the role of the [5XModules[105X package in the [5Xhomalg[105X project?[133X
      1.1-1 [33X[0;0Y[5XModules[105X provides ...[133X
      1.1-2 [33X[0;0YRings supported in a sufficient way[133X
      1.1-3 [33X[0;0YPrincipal limitation[133X
      1.1-4 [33X[0;0YRing dictionaries (technical)[133X
      1.1-5 [33X[0;0YThe advantages of the outsourcing concept[133X
      1.1-6 [33X[0;0YDoes this mean that [5Xhomalg[105X has only algorithms for the generic
      case?[133X
      1.1-7 [33X[0;0YThe principle of least communication (technical)[133X
      1.1-8 [33X[0;0YFrequently asked questions[133X
    1.2 [33X[0;0YThis manual[133X
  2 [33X[0;0YInstallation of the [5XModules[105X Package[133X
  3 [33X[0;0YQuick Start[133X
    3.1 [33X[0;0YWhy are all examples in this manual over ℤ or [22Xℤ/mℤ[122X?[133X
    3.2 [33X[0;0Y[10Xgap> ExamplesForHomalg();[110X[133X
    3.3 [33X[0;0YA typical example[133X
      3.3-1 [33X[0;0YHomHom[133X
  4 [33X[0;0YRing Maps[133X
    4.1 [33X[0;0YRing Maps: Attributes[133X
      4.1-1 KernelSubobject
      4.1-2 KernelEmb
    4.2 [33X[0;0YRing Maps: Operations and Functions[133X
      4.2-1 Kernel
  5 [33X[0;0YRelations[133X
    5.1 [33X[0;0YRelations: Categories and Representations[133X
      5.1-1 IsHomalgRelations
      5.1-2 IsHomalgRelationsOfLeftModule
      5.1-3 IsHomalgRelationsOfRightModule
      5.1-4 IsRelationsOfFinitelyPresentedModuleRep
    5.2 [33X[0;0YRelations: Constructors[133X
    5.3 [33X[0;0YRelations: Properties[133X
      5.3-1 CanBeUsedToDecideZeroEffectively
      5.3-2 IsInjectivePresentation
    5.4 [33X[0;0YRelations: Attributes[133X
    5.5 [33X[0;0YRelations: Operations and Functions[133X
  6 [33X[0;0YGenerators[133X
    6.1 [33X[0;0YGenerators: Categories and Representations[133X
      6.1-1 IsHomalgGenerators
      6.1-2 IsHomalgGeneratorsOfLeftModule
      6.1-3 IsHomalgGeneratorsOfRightModule
      6.1-4 IsGeneratorsOfModuleRep
      6.1-5 IsGeneratorsOfFinitelyGeneratedModuleRep
    6.2 [33X[0;0YGenerators: Constructors[133X
    6.3 [33X[0;0YGenerators: Properties[133X
      6.3-1 IsReduced
    6.4 [33X[0;0YGenerators: Attributes[133X
      6.4-1 ProcedureToReadjustGenerators
    6.5 [33X[0;0YGenerators: Operations and Functions[133X
  7 [33X[0;0YModules[133X
    7.1 [33X[0;0YModules: Category and Representations[133X
      7.1-1 IsHomalgModule
      7.1-2 IsFinitelyPresentedModuleOrSubmoduleRep
      7.1-3 IsFinitelyPresentedModuleRep
      7.1-4 IsFinitelyPresentedSubmoduleRep
    7.2 [33X[0;0YModules: Constructors[133X
      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 [33X[0;0YModules: Properties[133X
      7.3-1 IsCyclic
      7.3-2 IsHolonomic
      7.3-3 IsReduced
      7.3-4 IsPrimeIdeal
      7.3-5 IsPrimeModule
    7.4 [33X[0;0YModules: Attributes[133X
      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 [33X[0;0YModules: Operations and Functions[133X
      7.5-1 HomalgRing
      7.5-2 ByASmallerPresentation
      7.5-3 \*
      7.5-4 SubobjectQuotient
  8 [33X[0;0YMaps[133X
    8.1 [33X[0;0YMaps: Categories and Representations[133X
      8.1-1 IsHomalgMap
      8.1-2 IsHomalgSelfMap
      8.1-3 IsMapOfFinitelyGeneratedModulesRep
    8.2 [33X[0;0YMaps: Constructors[133X
      8.2-1 HomalgMap
      8.2-2 HomalgZeroMap
      8.2-3 HomalgIdentityMap
    8.3 [33X[0;0YMaps: Properties[133X
    8.4 [33X[0;0YMaps: Attributes[133X
    8.5 [33X[0;0YMaps: Operations and Functions[133X
      8.5-1 HomalgRing
      8.5-2 PreInverse
  9 [33X[0;0YModule Elements[133X
    9.1 [33X[0;0YModule Elements: Category and Representations[133X
      9.1-1 IsHomalgElement
      9.1-2 IsElementOfAModuleGivenByAMorphismRep
    9.2 [33X[0;0YModule Elements: Constructors[133X
    9.3 [33X[0;0YModule Elements: Properties[133X
      9.3-1 IsElementOfIntegers
    9.4 [33X[0;0YModule Elements: Attributes[133X
    9.5 [33X[0;0YModule Elements: Operations and Functions[133X
      9.5-1 HomalgRing
  10 [33X[0;0YFunctors[133X
    10.1 [33X[0;0YFunctors: Category and Representations[133X
    10.2 [33X[0;0YFunctors: Constructors[133X
    10.3 [33X[0;0YFunctors: Attributes[133X
    10.4 [33X[0;0YBasic Functors[133X
      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 [33X[0;0YTool Functors[133X
    10.6 [33X[0;0YOther Functors[133X
    10.7 [33X[0;0YFunctors: Operations and Functions[133X
  11 [33X[0;0YSymmetric Algebra and Koszul Complex[133X
    11.1 [33X[0;0YSymmetric Algebra: Constructor[133X
      11.1-1 SymmetricPower
    11.2 [33X[0;0YSymmetric Algebra: Properties and Attributes[133X
      11.2-1 IsSymmetricPower
      11.2-2 SymmetricPowerExponent
      11.2-3 SymmetricPowerBaseModule
  12 [33X[0;0YExterior Algebra and Koszul Complex[133X
    12.1 [33X[0;0YExterior Algebra: Constructor[133X
      12.1-1 ExteriorPower
    12.2 [33X[0;0YExterior Algebra: Properties and Attributes[133X
      12.2-1 IsExteriorPower
      12.2-2 ExteriorPowerExponent
      12.2-3 ExteriorPowerBaseModule
    12.3 [33X[0;0YExterior Algebra: Element Properties[133X
      12.3-1 IsExteriorPowerElement
    12.4 [33X[0;0YExterior Algebra: Element Operations[133X
      12.4-1 Wedge
      12.4-2 ExteriorPowerElementDual
      12.4-3 SingleValueOfExteriorPowerElement
    12.5 [33X[0;0YKoszul complex and Cayley determinant[133X
      12.5-1 KoszulCocomplex
      12.5-2 CayleyDeterminant
      12.5-3 Gcd_UsingCayleyDeterminant
  13 [33X[0;0YExamples[133X
    13.1 [33X[0;0YExtExt[133X
    13.2 [33X[0;0YPurity[133X
    13.3 [33X[0;0YTorExt-Grothendieck[133X
    13.4 [33X[0;0YTorExt[133X
  A [33X[0;0YThe Mathematical Idea behind [5XModules[105X[133X
  B [33X[0;0YLogic Subpackages[133X
    B.1 [33X[0;0Y[5XLIMOD[105X: Logical Implications for Modules[133X
    B.2 [33X[0;0Y[5XLIHOM[105X: Logical Implications for Homomorphisms of Modules[133X
  C [33X[0;0YOverview of the [5XModules[105X Package Source Code[133X
    C.1 [33X[0;0YRelations and Generators[133X
    C.2 [33X[0;0YThe Basic Objects[133X
    C.3 [33X[0;0YThe High Level Homological Algorithms[133X
    C.4 [33X[0;0YLogical Implications for [5Xhomalg[105X Objects[133X
  
  
  [32X
