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Modules

A homalg based Package for the Abelian Category of Finitely Presented Modules over Computable Rings

Version 2018.01.07

September 2015

(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: [email protected]
Homepage: http://www.mathematik.uni-kl.de/~barakat/
Address:
Department of Mathematics,
University of Kaiserslautern,
67653 Kaiserslautern,
Germany

Markus Lange-Hegermann
Email: [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

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

3.1 Why are all examples in this manual over ℤ or ℤ/mℤ?

3.2 gap> ExamplesForHomalg();

3.3 A typical example

4.1 Ring Maps: Attributes

4.1-1 KernelSubobject
4.1-2 KernelEmb

4.2 Ring Maps: Operations and Functions

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

6.1 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

7.1 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

7.5 Modules: Operations and Functions

  7.5-1 HomalgRing
  7.5-2 ByASmallerPresentation
  7.5-3 \*
  7.5-4 SubobjectQuotient

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

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

13.3 TorExt-Grothendieck

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

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