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# LocalizeRingForHomalg

## A Package for Localization of Polynomial Rings

2022.11-01

23 November 2022

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

Vinay Wagh
Email: waghoba@gmail.com
Homepage: http://www.iitg.ernet.in/vinay.wagh/
Address:
E-102, Department of Mathematics,
Indian Institute of Technology Guwahati,
Guwahati, Assam, India.
PIN: 781 039.
India

### Copyright

© 2009-2015 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.

### Contents

1 Introduction
1.1 What is the Role of the LocalizeRingForHomalg Package in the homalg Project?
1.2 Functionality
1.3 The Math Behind This Package
1.4 Which Ring to Use?
2 Installation of the LocalizeRingForHomalg Package
3 Quick Start
3.1 Localization of ℤ
4 Localize Rings
4.1 Category and Representations
4.1-1 IsHomalgLocalRingRep
4.1-2 IsHomalgLocalRingElementRep
4.1-3 IsHomalgLocalMatrixRep
4.2 Rings: Attributes
4.2-1 GeneratorsOfMaximalLeftIdeal
4.2-2 GeneratorsOfMaximalRightIdeal
4.3 Operations and Functions
4.3-1 AssociatedGlobalRing
4.3-2 AssociatedGlobalRing
4.3-3 AssociatedGlobalRing
4.3-4 Numerator
4.3-5 Numerator
4.3-6 Denominator
4.3-7 Denominator
4.3-8 Name
4.3-9 SetMatElm
4.3-10 AddToMatElm
4.3-11 MatElmAsString
4.3-12 MatElm
4.3-13 Cancel
4.3-14 LocalizeAt
4.3-15 LocalizeAtZero
4.3-16 LocalizePolynomialRingAtZeroWithMora
4.3-17 HomalgLocalRingElement
4.3-18 HomalgLocalMatrix
5 Examples
5.1 An Easy Polynomial Example
5.2 Hom(Hom(-,Z128),Z16)
5.3 ResidueClass
5.4 Testing the Intersection Formula
A Overview of the LocalizeRingForHomalg Package Source Code
A.1 The generic Methods
A.1-1 BasisOfRowModule
A.1-2 DecideZeroRows
A.1-3 SyzygiesGeneratorsOfRows
A.2 The Local Decide Zero trick
A.2-1 DecideZeroRows
A.3 Tools
References
Index
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