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# GradedModules

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

2022.09-02

28 September 2022

Mohamed Barakat
Email: mohamed.barakat@uni-siegen.de
Homepage: https://mohamed-barakat.github.io
Address:
Walter-Flex-Str. 3
57072 Siegen
Germany

Sebastian Gutsche
Email: gutsche@mathematik.uni-siegen.de
Homepage: https://sebasguts.github.io
Address:
Department Mathematik
Universität Siegen
Walter-Flex-Straße 3
57072 Siegen
Germany

Sebastian Jambor
Email: sebastian.jambor@rwth-aachen.de
Homepage: http://wwwb.math.rwth-aachen.de/~sebastian/
Address:
Sebastian Jambor
Lehrstuhl B fuer Mathematik, RWTH Aachen
Templergraben 64
52062 Aachen
Germany

Markus Lange-Hegermann
Email: markus.lange-hegermann@hs-owl.de
Homepage: https://www.th-owl.de/eecs/fachbereich/team/markus-lange-hegermann/
Address:
Markus Lange-Hegermann
Hochschule Ostwestfalen-Lippe
Liebigstraße 87
32657 Lemgo
Germany

Arne Lorenz
Email: arne.lorenz@rwth-aachen.de
Homepage: http://wwwb.math.rwth-aachen.de/~arne/
Address:
Arne Lorenz
Lehrstuhl B fuer Mathematik, RWTH Aachen
Templergraben 64
52062 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 or (at your option) any later version.

### Contents

1 Installation of the GradedModules Package
2 Ring Maps
2.1 Ring Maps: Attributes
2.1-1 KernelSubobject
2.2 Ring Maps: Operations and Functions
2.2-1 SegreMap
2.2-2 PlueckerMap
2.2-3 VeroneseMap
3 GradedModules
3.1 GradedModules: Category and Representations
3.2 GradedModules: Constructors
3.3 GradedModules: Properties
3.4 GradedModules: Attributes
3.4-1 BettiTable
3.4-2 CastelnuovoMumfordRegularity
3.4-3 CastelnuovoMumfordRegularityOfSheafification
3.5 LISHV: Logical Implications for GradedModules
3.6 GradedModules: Operations and Functions
3.6-1 MonomialMap
3.6-2 RandomMatrix
3.6-3 GeneratorsOfHomogeneousPart
3.6-4 SubmoduleGeneratedByHomogeneousPart
3.6-5 RepresentationMapOfRingElement
3.6-6 RepresentationMatrixOfKoszulId
3.6-7 RepresentationMapOfKoszulId
3.6-8 KoszulRightAdjoint
3.6-9 HomogeneousPartOverCoefficientsRing
4 The Tate Resolution
4.1 The Tate Resolution: Operations and Functions
4.1-1 TateResolution
5 Examples
5.1 Betti Diagrams
5.1-1 DE-2.2
5.1-2 DE-Code
5.1-3 Schenck-3.2
5.1-4 Schenck-8.3
5.1-5 Schenck-8.3.3
5.2 Commutative Algebra
5.2-1 Saturate
5.3 Global Section Modules of the Induced Sheaves
5.3-1 Examples of the ModuleOfGlobalSections Functor and Purity Filtrations
5.3-2 Horrocks Mumford bundle
References
Index
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