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# GradedRingForHomalg

## Endow Commutative Rings with an Abelian Grading

Version 2015.12.04

September 2015

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: barakat@mathematik.uni-kl.de
Homepage: http://www.mathematik.uni-kl.de/~barakat/
Address:
Department of Mathematics,
University of Kaiserslautern,
67653 Kaiserslautern,
Germany

Sebastian Gutsche
Email: sebastian.gutsche@rwth-aachen.de
Homepage: http://wwwb.math.rwth-aachen.de/~gutsche/
Address:
Lehrstuhl B für Mathematik, RWTH Aachen, Templergraben 64, 52056 Aachen, Germany

Markus Lange-Hegermann
Email: markus.lange.hegermann@rwth-aachen.de
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

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
References
Index
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