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# GradedModules

## A homalg based package for the Abelian category of finitely presented graded modules over a computable graded ring

Version 2018.02.04

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/~markus/GradedModules/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/unreleased/gradedmodules

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

Oleksandr Motsak
Email: motsak@mathematik.uni-kl.de
Homepage: http://www.mathematik.uni-kl.de/~motsak
Address:
Department of Mathematics,
University of Kaiserslautern,
67653 Kaiserslautern,
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

### Contents

1 Introduction
2 Installation of the GradedModules Package
3 Quick Start
4 Ring Maps
4.1 Ring Maps: Attributes
4.1-1 KernelSubobject
4.2 Ring Maps: Operations and Functions
4.2-1 SegreMap
4.2-2 PlueckerMap
4.2-3 VeroneseMap
5 GradedModules
5.1 GradedModules: Category and Representations
5.2 GradedModules: Constructors
5.3 GradedModules: Properties
5.4 GradedModules: Attributes
5.4-1 BettiTable
5.4-2 CastelnuovoMumfordRegularity
5.4-3 CastelnuovoMumfordRegularityOfSheafification
5.5 LISHV: Logical Implications for GradedModules
5.6 GradedModules: Operations and Functions
5.6-1 MonomialMap
5.6-2 RandomMatrix
5.6-3 GeneratorsOfHomogeneousPart
5.6-4 SubmoduleGeneratedByHomogeneousPart
5.6-5 RepresentationMapOfRingElement
5.6-6 RepresentationMatrixOfKoszulId
5.6-7 RepresentationMapOfKoszulId
5.6-8 KoszulRightAdjoint
5.6-9 HomogeneousPartOverCoefficientsRing
6 The Tate Resolution
6.1 The Tate Resolution: Operations and Functions
6.1-1 TateResolution
7 Examples
7.1 Betti Diagrams
7.1-1 DE-2.2
7.1-2 DE-Code
7.1-3 Schenck-3.2
7.1-4 Schenck-8.3
7.1-5 Schenck-8.3.3
7.2 Commutative Algebra
7.2-1 Saturate
7.3 Global Section Modules of the Induced Sheaves
7.3-1 Examples of the ModuleOfGlobalSections Functor and Purity Filtrations
7.3-2 Horrocks Mumford bundle
A Overview of the GradedModules Package Source Code
References
Index
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