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<h1>GradedModules</h1>


<h2>A homalg based package for the Abelian category of finitely presented graded modules over computable graded rings</h2>

<p>
    2024.12-01</p>

<p>
    22 December 2024
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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>
    Sebastian Gutsche




  </b>
<br />Email: <span class="URL"><a href="mailto:gutsche@mathematik.uni-siegen.de">gutsche@mathematik.uni-siegen.de</a></span>
<br />Homepage: <span class="URL"><a href="https://sebasguts.github.io">https://sebasguts.github.io</a></span>
<br />Address: <br />Department Mathematik<br /> Universität Siegen<br /> Walter-Flex-Straße 3<br /> 57072 Siegen<br /> Germany<br />
</p><p><b>
    Sebastian Jambor




  </b>
<br />Email: <span class="URL"><a href="mailto:sebastian.jambor@rwth-aachen.de">sebastian.jambor@rwth-aachen.de</a></span>
<br />Homepage: <span class="URL"><a href="http://wwwb.math.rwth-aachen.de/~sebastian/">http://wwwb.math.rwth-aachen.de/~sebastian/</a></span>
<br />Address: <br />Sebastian Jambor<br /> Lehrstuhl B fuer Mathematik, RWTH Aachen<br /> Templergraben 64<br /> 52062 Aachen<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>
    Arne Lorenz




  </b>
<br />Email: <span class="URL"><a href="mailto:arne.lorenz@rwth-aachen.de">arne.lorenz@rwth-aachen.de</a></span>
<br />Homepage: <span class="URL"><a href="http://wwwb.math.rwth-aachen.de/~arne/">http://wwwb.math.rwth-aachen.de/~arne/</a></span>
<br />Address: <br />Arne Lorenz<br /> Lehrstuhl B fuer Mathematik, RWTH Aachen<br /> Templergraben 64<br /> 52062 Aachen<br /> Germany<br />
</p><p><b>
    Oleksandr Motsak




  </b>
<br />Email: <span class="URL"><a href="mailto:motsak@mathematik.uni-kl.de">motsak@mathematik.uni-kl.de</a></span>
<br />Homepage: <span class="URL"><a href="http://www.mathematik.uni-kl.de/~motsak/">http://www.mathematik.uni-kl.de/~motsak/</a></span>
<br />Address: <br />Department of Mathematics<br /> University of Kaiserslautern<br /> 67653 Kaiserslautern<br /> Germany<br />
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<h3>Copyright</h3>
<p>© 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.</p>

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

<div class="ContChap"><a href="chap1.html#X80C2F0867ADFF1C8">1 <span class="Heading">Installation of the <strong class="pkg">GradedModules</strong> Package</span></a>
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<div class="ContChap"><a href="chap2.html#X7B222197819984A6">2 <span class="Heading">Ring Maps</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap2.html#X7EBF1DD67BD0758F">2.1 <span class="Heading">Ring Maps: Attributes</span></a>
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<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap2.html#X87C00FFB79FA93A8">2.1-1 KernelSubobject</a></span>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap2.html#X7C7401BA7E2221CB">2.2 <span class="Heading">Ring Maps: Operations and Functions</span></a>
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<span class="ContSS"><br /><span class="nocss">  </span><a href="chap2.html#X7B7DDDA17837AEF5">2.2-1 SegreMap</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap2.html#X78E0B36179C5646C">2.2-2 PlueckerMap</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap2.html#X816B9AB287EEF9A5">2.2-3 VeroneseMap</a></span>
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<div class="ContChap"><a href="chap3.html#X78B70E1D86624AC1">3 <span class="Heading">GradedModules</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap3.html#X84BE86BD7CAFCA5F">3.1 <span class="Heading">GradedModules: Category and Representations</span></a>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap3.html#X7B3AF789845366C0">3.2 <span class="Heading">GradedModules: Constructors</span></a>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap3.html#X858BEC417BE013FE">3.3 <span class="Heading">GradedModules: Properties</span></a>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap3.html#X7EEE66BA7E3A4CB8">3.4 <span class="Heading">GradedModules: Attributes</span></a>
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<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X78E2B4AD7F671293">3.4-1 BettiTable</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X854A879B8705130F">3.4-2 CastelnuovoMumfordRegularity</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X7CB0AA408287A8E2">3.4-3 CastelnuovoMumfordRegularityOfSheafification</a></span>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap3.html#X795320C4829D4F67">3.5 <span class="Heading"><strong class="pkg">LISHV</strong>: Logical Implications for GradedModules</span></a>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap3.html#X877CA99B7CB05AD2">3.6 <span class="Heading">GradedModules: Operations and Functions</span></a>
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<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X7E7BA9887C435CD4">3.6-1 MonomialMap</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X86CB265786A878D8">3.6-2 RandomMatrix</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X78127AB787A5C681">3.6-3 GeneratorsOfHomogeneousPart</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X86E9CD307823CC52">3.6-4 SubmoduleGeneratedByHomogeneousPart</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X870CC71A801346E5">3.6-5 RepresentationMapOfRingElement</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X797D315F87081C55">3.6-6 RepresentationMatrixOfKoszulId</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X7DBC9F4F827B4F01">3.6-7 RepresentationMapOfKoszulId</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X7C2D50247FFA3704">3.6-8 KoszulRightAdjoint</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X78E9B52D87FC5F3C">3.6-9 HomogeneousPartOverCoefficientsRing</a></span>
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<div class="ContChap"><a href="chap4.html#X7FE838537D4DF8E7">4 <span class="Heading">The Tate Resolution</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap4.html#X83CE0B0785329667">4.1 <span class="Heading">The Tate Resolution: Operations and Functions</span></a>
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<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X7A9DCED27D5F0D67">4.1-1 TateResolution</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#X81A7E0D380CE7F31">5.1 <span class="Heading">Betti Diagrams</span></a>
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<span class="ContSS"><br /><span class="nocss">  </span><a href="chap5.html#X8441906E83F6845D">5.1-1 <span class="Heading">DE-2.2</span></a>
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<span class="ContSS"><br /><span class="nocss">  </span><a href="chap5.html#X7E32106D7B13B8D9">5.1-2 <span class="Heading">DE-Code</span></a>
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<span class="ContSS"><br /><span class="nocss">  </span><a href="chap5.html#X793A69C4805C6819">5.1-3 <span class="Heading">Schenck-3.2</span></a>
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<span class="ContSS"><br /><span class="nocss">  </span><a href="chap5.html#X7E8F44338461DC08">5.1-4 <span class="Heading">Schenck-8.3</span></a>
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<span class="ContSS"><br /><span class="nocss">  </span><a href="chap5.html#X7B672C498385F92F">5.1-5 <span class="Heading">Schenck-8.3.3</span></a>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap5.html#X85CF19B87D1C375F">5.2 <span class="Heading">Commutative Algebra</span></a>
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<span class="ContSS"><br /><span class="nocss">  </span><a href="chap5.html#X7EA4CC697C01E080">5.2-1 <span class="Heading">Saturate</span></a>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap5.html#X86AF934C83004BF2">5.3 <span class="Heading">Global Section Modules of the Induced Sheaves</span></a>
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<span class="ContSS"><br /><span class="nocss">  </span><a href="chap5.html#X87EE931187E2226C">5.3-1 <span class="Heading">Examples of the ModuleOfGlobalSections Functor and Purity Filtrations</span></a>
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<span class="ContSS"><br /><span class="nocss">  </span><a href="chap5.html#X7DD8F76D7A4206E3">5.3-2 <span class="Heading">Horrocks Mumford bundle</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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