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<h1>LocalizeRingForHomalg</h1>


<h2>A Package for Localization of Polynomial Rings</h2>

<p>
    2023.10-01</p>

<p>
    5 October 2023
  </p>

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<p><b>
    Mohamed Barakat




  </b>
<br />Email: <span class="URL"><a href="mailto:mohamed.barakat@uni-siegen.de">mohamed.barakat@uni-siegen.de</a></span>
<br />Homepage: <span class="URL"><a href="https://mohamed-barakat.github.io">https://mohamed-barakat.github.io</a></span>
<br />Address: <br />Walter-Flex-Str. 3<br /> 57072 Siegen<br /> Germany<br />
</p><p><b>
    Markus Lange-Hegermann




  </b>
<br />Email: <span class="URL"><a href="mailto:markus.lange-hegermann@hs-owl.de">markus.lange-hegermann@hs-owl.de</a></span>
<br />Homepage: <span class="URL"><a href="https://www.th-owl.de/eecs/fachbereich/team/markus-lange-hegermann/">https://www.th-owl.de/eecs/fachbereich/team/markus-lange-hegermann/</a></span>
<br />Address: <br />Markus Lange-Hegermann<br /> Hochschule Ostwestfalen-Lippe<br /> Liebigstraße 87<br /> 32657 Lemgo<br /> Germany<br />
</p><p><b>
    Vinay Wagh




  </b>
<br />Email: <span class="URL"><a href="mailto:waghoba@gmail.com">waghoba@gmail.com</a></span>
<br />Homepage: <span class="URL"><a href="http://www.iitg.ernet.in/vinay.wagh/">http://www.iitg.ernet.in/vinay.wagh/</a></span>
<br />Address: <br />E-102, Department of Mathematics,<br /> Indian Institute of Technology Guwahati,<br /> Guwahati, Assam, India.<br /> PIN: 781 039.<br /> India<br />
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<h3>Copyright</h3>
<p>© 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.</p>

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<h3>Contents<a id="contents" name="contents"></a></h3>

<div class="ContChap"><a href="chap1.html#X7DFB63A97E67C0A1">1 <span class="Heading">Introduction</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap1.html#X7AC2A9FA80501972">1.1 <span class="Heading">What is the Role of the <strong class="pkg">LocalizeRingForHomalg</strong> Package in the <strong class="pkg">homalg</strong> Project?</span></a>
</span>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap1.html#X87F1120883F5B4D0">1.2 <span class="Heading">Functionality</span></a>
</span>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap1.html#X7845DA5685210CC3">1.3 <span class="Heading">The Math Behind This Package</span></a>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap1.html#X795ABF907CEB327F">1.4 <span class="Heading">Which Ring to Use?</span></a>
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<div class="ContChap"><a href="chap2.html#X7E6144A47929DD91">2 <span class="Heading">Installation of the <strong class="pkg">LocalizeRingForHomalg</strong> Package</span></a>
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<div class="ContChap"><a href="chap3.html#X7EB860EC84DFC71E">3 <span class="Heading">Quick Start</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap3.html#X799572BF810091BC">3.1 <span class="Heading">Localization of ℤ</span></a>
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<div class="ContChap"><a href="chap4.html#X86CBDDB082295E5F">4 <span class="Heading">Localize Rings</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap4.html#X826B3EF37FB188A8">4.1 <span class="Heading">Category and Representations</span></a>
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<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X7AEE9D7F85CD25DD">4.1-1 IsHomalgLocalRingRep</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X7E2AD9FF82990735">4.1-2 IsHomalgLocalRingElementRep</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X80AB357F7883ACF3">4.1-3 IsHomalgLocalMatrixRep</a></span>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap4.html#X867290E7847A5101">4.2 <span class="Heading">Rings: Attributes</span></a>
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<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X7EAE00247B717A0F">4.2-1 GeneratorsOfMaximalLeftIdeal</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X85BA4850780569D7">4.2-2 GeneratorsOfMaximalRightIdeal</a></span>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap4.html#X81CFB50085A4E12F">4.3 <span class="Heading">Operations and Functions</span></a>
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<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X7D8C5E8B8533EE2D">4.3-1 AssociatedGlobalRing</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X8612432F81364A90">4.3-2 AssociatedGlobalRing</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X78AE3B5187FC5B24">4.3-3 AssociatedGlobalRing</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X86B86F85831145D0">4.3-4 Numerator</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X8613D7A27D3D770D">4.3-5 Numerator</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X83E87130831148A4">4.3-6 Denominator</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X863DB91A7DEDEA22">4.3-7 Denominator</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X7FA0CC577E97B6E8">4.3-8 Name</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X80F936B786454DD0">4.3-9 SetMatElm</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X87689354873E0B14">4.3-10 AddToMatElm</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X8789D7998407AD36">4.3-11 MatElmAsString</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X7EFD740F87ADB6E0">4.3-12 MatElm</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X7F0196827D195F88">4.3-13 Cancel</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X7E4E70FE82978F9C">4.3-14 LocalizeAt</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X7D910AA785CEED34">4.3-15 LocalizeAtZero</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X7F790F3B852C1A5C">4.3-16 LocalizePolynomialRingAtZeroWithMora</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X8741B0AE787624CB">4.3-17 HomalgLocalRingElement</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X808317A88773E967">4.3-18 HomalgLocalMatrix</a></span>
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<div class="ContChap"><a href="chap5.html#X7A489A5D79DA9E5C">5 <span class="Heading">Examples</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap5.html#X8426A658837B4911">5.1 <span class="Heading">An Easy Polynomial Example</span></a>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap5.html#X7820475F7C884EA5">5.2 <span class="Heading">Hom(Hom(-,Z128),Z16)</span></a>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap5.html#X7CC8EA507E7AABA4">5.3 <span class="Heading">ResidueClass</span></a>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap5.html#X7958E7417BB312F0">5.4 <span class="Heading">Testing the Intersection Formula</span></a>
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<div class="ContChap"><a href="chapA.html#X7C7697867CBF79C9">A <span class="Heading">Overview of the <strong class="pkg">LocalizeRingForHomalg</strong> Package Source Code</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chapA.html#X87807E467C364A00">A.1 <span class="Heading">The generic Methods</span></a>
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<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chapA.html#X7B04DAB47BAB9165">A.1-1 BasisOfRowModule</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chapA.html#X8196F11E7FD3201C">A.1-2 DecideZeroRows</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chapA.html#X7B1B3662863726EE">A.1-3 SyzygiesGeneratorsOfRows</a></span>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chapA.html#X85822F1B7E006A87">A.2 <span class="Heading">The Local Decide Zero trick</span></a>
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<span class="ContSS"><br /><span class="nocss">  </span><a href="chapA.html#X790256E88417752F">A.2-1 DecideZeroRows</a></span>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chapA.html#X8508AEF8845565A1">A.3 <span class="Heading">Tools</span></a>
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<div class="ContChap"><a href="chapBib.html"><span class="Heading">References</span></a></div>
<div class="ContChap"><a href="chapInd.html"><span class="Heading">Index</span></a></div>
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