<?xml version="1.0" encoding="UTF-8" ?>DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Strict//EN" "http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd " >html xmlns="http://www.w3.org/1999/xhtml " xml:lang="en" >head >title >GAP (Modules) - Contents</title >meta http-equiv="content-type" content="text/html; charset=UTF-8" />meta name="generator" content="GAPDoc2HTML" />link rel="stylesheet" type="text/css" href="manual.css" />script src="manual.js" type="text/javascript" ></script >script type="text/javascript" >overwriteStyle();</script >head >body class="chap0" onload="jscontent()" >div class="chlinktop" ><span class="chlink1" >Goto Chapter: </span ><a href="chap0.html" >Top</a> <a href="chap1.html" >1</a> <a href="chap2.html" >2</a> <a href="chap3.html" >3</a> <a href="chap4.html" >4</a> <a href="chap5.html" >5</a> <a href="chap6.html" >6</a> <a href="chap7.html" >7</a> <a href="chap8.html" >8</a> <a href="chap9.html" >9</a> <a href="chap10.html" >10</a> <a href="chap11.html" >11</a> <a 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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 />br />Email: <span class="URL" ><a href="mailto:diebold@mathematik.uni-kl.de" >diebold@mathematik.uni-kl.de</a></span >br />Address : <br />Department of Mathematics<br /> University of Kaiserslautern<br /> 67653 Kaiserslautern<br /> Germany<br />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 />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 />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 />"X81488B807F2A1CF1" name="X81488B807F2A1CF1" ></a></p>er the terms and conditions of the GNU Public License Version 2 or (at your option ) any later version.</p>"X82A988D47DFAFCFA" name="X82A988D47DFAFCFA" ></a></p>"X8537FEB07AF2BEC8" name="X8537FEB07AF2BEC8" ></a></p>div class="contents" >"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#X7932D55D786D645A" >1.1 <span class="Heading" >What is the role of the <strong class="pkg" >Modules</strong strong class="pkg" >homalg</strong > project?</span ></a>span >div class="ContSSBlock" >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap1.html#X81F4D4C47828A818" >1.1-1 <span class="Heading" ><strong class="pkg" >Modules</strong > provides ...</span ></a>span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap1.html#X84913827857A1F7B" >1.1-2 <span class="Heading" >Rings supported in a sufficient way</span ></a>span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap1.html#X7C31B1FE786E596E" >1.1-3 <span class="Heading" >Principal limitation</span ></a>span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap1.html#X8583D47D7E570356" >1.1-4 <span class="Heading" >Ring dictionaries (technical)</span ></a>span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap1.html#X7D7570837C21607A" >1.1-5 <span class="Heading" >The advantages of the outsourcing concept</span ></a>span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap1.html#X85C5BCDF797B7954" >1.1-6 <span class="Heading" >Does this mean that <strong class="pkg" >homalg</strong > has only algorithms for the generic case?</span ></a>span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap1.html#X79DFCAF17BD3DDC6" >1.1-7 <span class="Heading" >The principle of least communication (technical)</span ></a>span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap1.html#X7D51BC7A80D43EA0" >1.1-8 <span class="Heading" >Frequently asked questions</span ></a>span >div ></div >div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap1.html#X78DD800B83ABC621" >1.2 <span class="Heading" >This manual</span ></a>span >div >div >div class="ContChap" ><a href="chap2.html#X812F1BC77954DFD3" >2 <span class="Heading" >Installation of the <strong class="pkg" >Modules</strong > Package</span ></a>div >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#X8527706586389E29" >3.1 <span class="Heading" >Why are all examples in this manual over ℤ or <span class="SimpleMath" >ℤ/mℤ</span >?</span ></a>span >div >div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap3.html#X85872612814D30B4" >3.2 <span class="Heading" ><code class="code" >gap> ExamplesForHomalg();</code ></span ></a>span >div >div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap3.html#X7BBB3E988435A713" >3.3 <span class="Heading" >A typical example</span ></a>span >div class="ContSSBlock" >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap3.html#X791E21F47805048A" >3.3-1 <span class="Heading" >HomHom</span ></a>span >div ></div >div >div class="ContChap" ><a href="chap4.html#X7B222197819984A6" >4 <span class="Heading" >Ring Maps</span ></a>div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap4.html#X7EBF1DD67BD0758F" >4.1 <span class="Heading" >Ring Maps: Attributes</span ></a>span >div class="ContSSBlock" >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap4.html#X7E7A0DE685E23202" >4.1-1 KernelSubobject</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap4.html#X82AE604486F48FC4" >4.1-2 KernelEmb</a></span >div ></div >div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap4.html#X7C7401BA7E2221CB" >4.2 <span class="Heading" >Ring Maps: Operations and Functions</span ></a>span >div class="ContSSBlock" >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap4.html#X870F43DE7DDD85A3" >4.2-1 Kernel</a></span >div ></div >div >div class="ContChap" ><a href="chap5.html#X838651287FCCEFD8" >5 <span class="Heading" >Relations</span ></a>div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap5.html#X87DADB7B7CB126DC" >5.1 <span class="Heading" >Relations: Categories and Representations</span ></a>span >div class="ContSSBlock" >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap5.html#X80AD050F7999B7C0" >5.1-1 IsHomalgRelations</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap5.html#X790F68B17A4846DC" >5.1-2 IsHomalgRelationsOfLeftModule</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap5.html#X7FF5A3B180614698" >5.1-3 IsHomalgRelationsOfRightModule</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap5.html#X8322A26C84E80303" >5.1-4 IsRelationsOfFinitelyPresentedModuleRep</a></span >div ></div >div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap5.html#X7CF74FB785F90889" >5.2 <span class="Heading" >Relations: Constructors</span ></a>span >div >div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap5.html#X859231317954D702" >5.3 <span class="Heading" >Relations: Properties</span ></a>span >div class="ContSSBlock" >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap5.html#X798D893B7FBFCF07" >5.3-1 CanBeUsedToDecideZeroEffectively</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap5.html#X7B9398827AEEA2E6" >5.3-2 IsInjectivePresentation</a></span >div ></div >div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap5.html#X7EF7BBCA85851EF1" >5.4 <span class="Heading" >Relations: Attributes</span ></a>span >div >div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap5.html#X7890B5EA80774AB5" >5.5 <span class="Heading" >Relations: Operations and Functions</span ></a>span >div >div >div class="ContChap" ><a href="chap6.html#X7BD5B55C802805B4" >6 <span class="Heading" >Generators</span ></a>div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap6.html#X827B67D27E3B91FC" >6.1 <span class="Heading" >Generators: Categories and Representations</span ></a>span >div class="ContSSBlock" >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap6.html#X79A6BA1280510584" >6.1-1 IsHomalgGenerators</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap6.html#X83E88425797FFC9C" >6.1-2 IsHomalgGeneratorsOfLeftModule</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap6.html#X86E9029487FE58DF" >6.1-3 IsHomalgGeneratorsOfRightModule</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap6.html#X8671AA997D666F04" >6.1-4 IsGeneratorsOfModuleRep</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap6.html#X78512B8A8613FBF1" >6.1-5 IsGeneratorsOfFinitelyGeneratedModuleRep</a></span >div ></div >div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap6.html#X8289206C81622597" >6.2 <span class="Heading" >Generators: Constructors</span ></a>span >div >div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap6.html#X8576E1368448066B" >6.3 <span class="Heading" >Generators: Properties</span ></a>span >div class="ContSSBlock" >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap6.html#X7ED359D87D6B2F79" >6.3-1 IsReduced</a></span >div ></div >div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap6.html#X7E136BCD7F22B571" >6.4 <span class="Heading" >Generators: Attributes</span ></a>span >div class="ContSSBlock" >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap6.html#X7B6F787085536F90" >6.4-1 ProcedureToReadjustGenerators</a></span >div ></div >div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap6.html#X7AC876EC8137AEA4" >6.5 <span class="Heading" >Generators: Operations and Functions</span ></a>span >div >div >div class="ContChap" ><a href="chap7.html#X8183A6857B0C3633" >7 <span class="Heading" >Modules</span ></a>div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap7.html#X7C7EBD2383B99C43" >7.1 <span class="Heading" >Modules: Category and Representations</span ></a>span >div class="ContSSBlock" >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap7.html#X8429977B7FD30F32" >7.1-1 IsHomalgModule</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap7.html#X7FB182707ADDF903" >7.1-2 IsFinitelyPresentedModuleOrSubmoduleRep</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap7.html#X87D53DCC822C8C92" >7.1-3 IsFinitelyPresentedModuleRep</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap7.html#X7BFA03B6820E9E55" >7.1-4 IsFinitelyPresentedSubmoduleRep</a></span >div ></div >div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap7.html#X7DB16C4B87DD115F" >7.2 <span class="Heading" >Modules: Constructors</span ></a>span >div class="ContSSBlock" >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap7.html#X7EC09F6B83CA4068" >7.2-1 LeftPresentation</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap7.html#X7A0C400A8042C284" >7.2-2 RightPresentation</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap7.html#X8067510285E38110" >7.2-3 HomalgFreeLeftModule</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap7.html#X7D9133C6837FDFE7" >7.2-4 HomalgFreeRightModule</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap7.html#X86EA38328275C44E" >7.2-5 HomalgZeroLeftModule</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap7.html#X8796A5957C40155C" >7.2-6 HomalgZeroRightModule</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap7.html#X87BE93798001D733" ><code >7.2-7 \*</code ></a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap7.html#X84C6C8227814E2BC" >7.2-8 Subobject</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap7.html#X811E74A98454101E" >7.2-9 Subobject</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap7.html#X8498CB457DD55DF2" >7.2-10 LeftSubmodule</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap7.html#X8383318085374771" >7.2-11 RightSubmodule</a></span >div ></div >div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap7.html#X83CC1D6079AA2286" >7.3 <span class="Heading" >Modules: Properties</span ></a>span >div class="ContSSBlock" >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap7.html#X7DA27D338374FD28" >7.3-1 IsCyclic</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap7.html#X7B547E8A7969F772" >7.3-2 IsHolonomic</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap7.html#X8393BC5779647A88" >7.3-3 IsReduced</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap7.html#X78020A71848F9FDD" >7.3-4 IsPrimeIdeal</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap7.html#X86160C4E797D12DD" >7.3-5 IsPrimeModule</a></span >div ></div >div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap7.html#X78A9979B862BD51D" >7.4 <span class="Heading" >Modules: Attributes</span ></a>span >div class="ContSSBlock" >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap7.html#X791F809B8432847F" >7.4-1 ResidueClassRing</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap7.html#X7F30DD127FAC8994" >7.4-2 PrimaryDecomposition</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap7.html#X839DA707838F72DC" >7.4-3 RadicalDecomposition</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap7.html#X7D1D919186E5E73A" >7.4-4 ModuleOfKaehlerDifferentials</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap7.html#X7F4B638C7D27C7B2" >7.4-5 RadicalSubobject</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap7.html#X829AFC557B7E49AF" >7.4-6 SymmetricAlgebra</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap7.html#X7AC2D54385C15EBD" >7.4-7 ExteriorAlgebra</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap7.html#X7EB20A71864D46BF" >7.4-8 ElementaryDivisors</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap7.html#X7B83CC6485B028E1" >7.4-9 FittingIdeal</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap7.html#X8671FA1F820AA86D" >7.4-10 NonFlatLocus</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap7.html#X84FECA07854053BE" >7.4-11 LargestMinimalNumberOfLocalGenerators</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap7.html#X7809E0507E882674" >7.4-12 CoefficientsOfUnreducedNumeratorOfHilbertPoincareSeries</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap7.html#X7938E13A7EF4ADB1" >7.4-13 CoefficientsOfNumeratorOfHilbertPoincareSeries</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap7.html#X781E2CDB8743B1C6" >7.4-14 UnreducedNumeratorOfHilbertPoincareSeries</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap7.html#X7C44039382DD5D91" >7.4-15 NumeratorOfHilbertPoincareSeries</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap7.html#X7B93B7D082A50E61" >7.4-16 HilbertPoincareSeries</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap7.html#X87C428A079000336" >7.4-17 AffineDegree</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap7.html#X7F8203B47EF626A5" >7.4-18 DataOfHilbertFunction</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap7.html#X81F1F3EB868D2117" >7.4-19 HilbertFunction</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap7.html#X7AE7FCEA807D189E" >7.4-20 IndexOfRegularity</a></span >div ></div >div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap7.html#X7DDA6B237C17BDBA" >7.5 <span class="Heading" >Modules: Operations and Functions</span ></a>span >div class="ContSSBlock" >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap7.html#X7DDA4A357F4868A0" >7.5-1 HomalgRing</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap7.html#X840D0B4F8798C370" >7.5-2 ByASmallerPresentation</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap7.html#X7AE6575D81856ECB" ><code >7.5-3 \*</code ></a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap7.html#X84D89101872CEA2A" >7.5-4 SubobjectQuotient</a></span >div ></div >div >div class="ContChap" ><a href="chap8.html#X7E8438F77ECB778E" >8 <span class="Heading" >Maps</span div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap8.html#X790FEEBD86F5C143" >8.1 <span class="Heading" >Maps: Categories and Representations</span ></a>span >div class="ContSSBlock" >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap8.html#X7DA293237F14CD74" >8.1-1 IsHomalgMap</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap8.html#X7F34D26882D20FF0" >8.1-2 IsHomalgSelfMap</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap8.html#X813202447B5C8FB3" >8.1-3 IsMapOfFinitelyGeneratedModulesRep</a></span >div ></div >div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap8.html#X8278F43E8373E4A1" >8.2 <span class="Heading" >Maps: Constructors</span ></a>span >div class="ContSSBlock" >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap8.html#X790E02137DBA584C" >8.2-1 HomalgMap</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap8.html#X81489DAF7B0674F3" >8.2-2 HomalgZeroMap</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap8.html#X7BF289B882C9DDF4" >8.2-3 HomalgIdentityMap</a></span >div ></div >div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap8.html#X85C633E77A939735" >8.3 <span class="Heading" >Maps: Properties</span ></a>span >div >div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap8.html#X7EA3B91C78E430BB" >8.4 <span class="Heading" >Maps: Attributes</span ></a>span >div >div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap8.html#X783E9FF8800609EB" >8.5 <span class="Heading" >Maps: Operations and Functions</span ></a>span >div class="ContSSBlock" >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap8.html#X7C8699B282D73E1E" >8.5-1 HomalgRing</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap8.html#X79D029B78624C148" >8.5-2 PreInverse</a></span >div ></div >div >div class="ContChap" ><a href="chap9.html#X7E9BCB99816348F2" >9 <span class="Heading" >Module Elements</span ></a>div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap9.html#X84A51EB87E054D3F" >9.1 <span class="Heading" >Module Elements: Category and Representations</span ></a>span >div class="ContSSBlock" >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap9.html#X784BBB2A782DB774" >9.1-1 IsHomalgElement</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap9.html#X7BF482C77B68ED64" >9.1-2 IsElementOfAModuleGivenByAMorphismRep</a></span >div ></div >div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap9.html#X7CFD0CF27A3FEB9D" >9.2 <span class="Heading" >Module Elements: Constructors</span ></a>span >div >div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap9.html#X7BCBA7E780FE2B14" >9.3 <span class="Heading" >Module Elements: Properties</span ></a>span >div class="ContSSBlock" >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap9.html#X87FA282579406FC0" >9.3-1 IsElementOfIntegers</a></span >div ></div >div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap9.html#X80AE2D1C82A2059C" >9.4 <span class="Heading" >Module Elements: Attributes</span ></a>span >div >div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap9.html#X813DF977812C06B6" >9.5 <span class="Heading" >Module Elements: Operations and Functions</span ></a>span >div class="ContSSBlock" >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap9.html#X8769077379997D89" >9.5-1 HomalgRing</a></span >div ></div >div >div class="ContChap" ><a href="chap10.html#X78D1062D78BE08C1" >10 <span class="Heading" >Functors</span ></a>div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap10.html#X7E41BC437F2B76E1" >10.1 <span class="Heading" >Functors: Category and Representations</span ></a>span >div >div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap10.html#X86EE897086995E47" >10.2 <span class="Heading" >Functors: Constructors</span ></a>span >div >div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap10.html#X7A21845C7C536717" >10.3 <span class="Heading" >Functors: Attributes</span ></a>span >div >div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap10.html#X7D83D0EB87D2D872" >10.4 <span class="Heading" >Basic Functors</span ></a>span >div class="ContSSBlock" >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap10.html#X7B9FE8BF80D47B6E" >10.4-1 functor_Cokernel</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap10.html#X875F177A82BF9B8B" >10.4-2 Cokernel</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap10.html#X7A5B3B307B334706" >10.4-3 functor_ImageObject</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap10.html#X7E3FF900821DCBE6" >10.4-4 ImageObject</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap10.html#X85C128B37E76827F" >10.4-5 Kernel</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap10.html#X7E6CDE7E85F09122" >10.4-6 DefectOfExactness</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap10.html#X7B93718087EFD69B" >10.4-7 Functor_Hom</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap10.html#X80015C78876B4F1E" >10.4-8 Hom</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap10.html#X7A1A077D8268FADE" >10.4-9 Functor_TensorProduct</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap10.html#X87EB0B4A852CF4C6" >10.4-10 TensorProduct</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap10.html#X7D007A7079F7BEE3" >10.4-11 Functor_Ext</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap10.html#X8692578881E71913" >10.4-12 Ext</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap10.html#X821034FE80907E8D" >10.4-13 Functor_Tor</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap10.html#X79821906875CF49E" >10.4-14 Tor</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap10.html#X84C60D997A79524E" >10.4-15 Functor_RHom</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap10.html#X7D8BDC0C817C10AB" >10.4-16 RHom</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap10.html#X806251E3836C00B9" >10.4-17 Functor_LTensorProduct</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap10.html#X7C12DA648798E77E" >10.4-18 LTensorProduct</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap10.html#X7ACC6A7C86E4354C" >10.4-19 Functor_HomHom</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap10.html#X84557A6B79382720" >10.4-20 Functor_LHomHom</a></span >div ></div >div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap10.html#X815BF6DA7FD5D44B" >10.5 <span class="Heading" >Tool Functors</span ></a>span >div >div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap10.html#X879135AC8330C509" >10.6 <span class="Heading" >Other Functors</span ></a>span >div >div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap10.html#X7DACD68E7E5FA324" >10.7 <span class="Heading" >Functors: Operations and Functions</span ></a>span >div >div >div class="ContChap" ><a href="chap11.html#X7E3E740C80F42986" >11 <span class="Heading" >Symmetric Algebra and Koszul Complex</span ></a>div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap11.html#X78E07AD87CE14F53" >11.1 <span class="Heading" >Symmetric Algebra: Constructor</span ></a>span >div class="ContSSBlock" >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap11.html#X79E2C2AF842E8419" >11.1-1 SymmetricPower</a></span >div ></div >div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap11.html#X7A26D0AD7E8F9FB2" >11.2 <span class="Heading" >Symmetric Algebra: Properties and Attributes</span span >div class="ContSSBlock" >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap11.html#X79AECE877A31293F" >11.2-1 IsSymmetricPower</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap11.html#X8750195584FAA0B2" >11.2-2 SymmetricPowerExponent</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap11.html#X7D9225D083DDDB0C" >11.2-3 SymmetricPowerBaseModule</a></span >div ></div >div >div class="ContChap" ><a href="chap12.html#X7BD010F3847B274E" >12 <span class="Heading" >Exterior Algebra and Koszul Complex</span ></a>div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap12.html#X7A005D4E870C281D" >12.1 <span class="Heading" >Exterior Algebra: Constructor</span ></a>span >div class="ContSSBlock" >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap12.html#X787BB7FF85F0AD68" >12.1-1 ExteriorPower</a></span >div ></div >div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap12.html#X7E09B9C5844FC31E" >12.2 <span class="Heading" >Exterior Algebra: Properties and Attributes</span ></a>span >div class="ContSSBlock" >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap12.html#X79C5FE077B58DF82" >12.2-1 IsExteriorPower</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap12.html#X87CF59278702A550" >12.2-2 ExteriorPowerExponent</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap12.html#X8282D0D7800F63CC" >12.2-3 ExteriorPowerBaseModule</a></span >div ></div >div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap12.html#X7A2AC54B87C85695" >12.3 <span class="Heading" >Exterior Algebra: Element Properties</span ></a>span >div class="ContSSBlock" >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap12.html#X7FC4A5DC7B592D04" >12.3-1 IsExteriorPowerElement</a></span >div ></div >div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap12.html#X80D7B36379182854" >12.4 <span class="Heading" >Exterior Algebra: Element Operations</span ></a>span >div class="ContSSBlock" >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap12.html#X7C71C3C77F2E225D" >12.4-1 Wedge</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap12.html#X8236B4167E79F186" >12.4-2 ExteriorPowerElementDual</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap12.html#X85EDBA2783A1E984" >12.4-3 SingleValueOfExteriorPowerElement</a></span >div ></div >div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap12.html#X8050EFB77A600595" >12.5 <span class="Heading" >Koszul complex and Cayley determinant</span ></a>span >div class="ContSSBlock" >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap12.html#X7D84C7AC809B453F" >12.5-1 KoszulCocomplex</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap12.html#X794C601787143D2D" >12.5-2 CayleyDeterminant</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap12.html#X7C72190C8331FADD" >12.5-3 Gcd_UsingCayleyDeterminant</a></span >div ></div >div >div class="ContChap" ><a href="chap13.html#X7A489A5D79DA9E5C" >13 <span class="Heading" >Examples</span ></a>div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap13.html#X7BB9DE017ECE6E86" >13.1 <span class="Heading" >ExtExt</span ></a>span >div >div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap13.html#X7EE63228803A04F1" >13.2 <span class="Heading" >Purity</span ></a>span >div >div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap13.html#X812EF8147AE16E72" >13.3 <span class="Heading" >TorExt-Grothendieck</span ></a>span >div >div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap13.html#X784BC2567875830B" >13.4 <span class="Heading" >TorExt</span ></a>span >div >div >div class="ContChap" ><a href="chapA.html#X7DD2E4EB846C7E75" >A <span class="Heading" >The Mathematical Idea behind <strong class="pkg" >Modules</strong ></span ></a>div >div class="ContChap" ><a href="chapB.html#X8222352C78A19214" >B <span class="Heading" >Logic Subpackages</span ></a>div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chapB.html#X8462717983D4B197" >B.1 <span class="Heading" ><strong class="pkg" >LIMOD</strong >: Logical Implications for Modules</span ></a>span >div >div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chapB.html#X7A553EC57DD2E46E" >B.2 <span class="Heading" ><strong class="pkg" >LIHOM</strong >: Logical Implications for Homomorphisms of Modules</span ></a>span >div >div >div class="ContChap" ><a href="chapC.html#X78684D057C432971" >C <span class="Heading" >Overview of the <strong class="pkg" >Modules</strong > Package Source Code </span ></a>div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chapC.html#X87ED7A1883976BE9" >C.1 <span class="Heading" >Relations and Generators</span ></a>span >div >div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chapC.html#X81DDCFC578069518" >C.2 <span class="Heading" >The Basic Objects</span ></a>span >div >div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chapC.html#X7BDE961D858BC60E" >C.3 <span class="Heading" >The High Level Homological Algorithms</span ></a>span >div >div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chapC.html#X7E8463067BB2F31E" >C.4 <span class="Heading" >Logical Implications for <strong class="pkg" >homalg</strong > Objects</span ></a>span >div >div >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 >br />div >div class="chlinkprevnextbot" > <a href="chap0.html" >[Top of Book]</a> <a href="chap0.html#contents" >[Contents]</a> <a href="chap1.html" >[Next Chapter]</a> </div >div class="chlinkbot" ><span class="chlink1" >Goto Chapter: </span ><a href="chap0.html" >Top</a> <a href="chap1.html" >1</a> <a href="chap2.html" >2</a> <a href="chap3.html" >3</a> <a href="chap4.html" >4</a> <a href="chap5.html" >5</a> <a href="chap6.html" >6</a> <a 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