<?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 >script type="text/javascript" "https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.0/MathJax.js?config=TeX-AMS-MML_HTMLorMML " >script >title >GAP (homalg) - 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_mj.html" >Top</a> <a href="chap1_mj.html" >1</a> <a href="chap2_mj.html" >2</a> <a href="chap3_mj.html" >3</a> <a href="chap4_mj.html" >4</a> <a href="chap5_mj.html" >5</a> <a href="chap6_mj.html" >6</a> <a href="chap7_mj.html" >7</a> <a href="chap8_mj.html" >8</a> <a href="chap9_mj.html" >9</a> <a href="chap10_mj.html" >10</a> <a href="chap11_mj.html" >11</a> <a href="chap12_mj.html" >12</a> <a href="chapA_mj.html" >A</a> <a href="chapB_mj.html" >B</a> <a href="chapC_mj.html" >C</a> <a href="chapD_mj.html" >D</a> <a href="chapE_mj.html" >E</a> <a href="chapF_mj.html" >F</a> <a href="chapBib_mj.html" >Bib</a> <a href="chapInd_mj.html" >Ind</a> </div >div class="chlinkprevnexttop" > <a href="chap0_mj.html" >[Top of Book]</a> <a href="chap0_mj.html#contents" >[Contents]</a> <a href="chap1_mj.html" >[Next Chapter]</a> </div >"mathjaxlink" class="pcenter" ><a href="chap0.html" >[MathJax off]</a></p>"X7D2C85EC87DD46E5" name="X7D2C85EC87DD46E5" ></a></p>div class="pcenter" >h1 >homalg</h1 >div >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 />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 />"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>span class="URL" ><a href="https://www.arangodb.com/speakers/max-neunhoeffer/" >Max Neunhöffer</a></span > not only taught me the philosophy of object-oriented programming in <strong class="pkg" >GAP4</strong >, but also to what extent this philosophy is still unique among programming languages (--> <a href="chapB_mj.html#X7D2A5B127A68AB58" ><span class="RefLink" >Why GAP4?</span ></a>). He, <span class="URL" ><a href="http://www.math.rwth-aachen.de/~Frank.Luebeck/ " >Frank Lübeck</a></span >, and <span class="URL" ><a href="http://www.math.rwth-aachen.de/~Thomas.Breuer/" >Thomas Breuer</a></span > patiently answered trillions of specific questions, even those I was too lazy to look up in the excellent <a href="../../../doc/ref/chap1_mj.html#X874E1D45845007FE" ><span class="RefLink" >reference manual</span ></a>. Without their continuous and tireless help and advice, not only this package but the as a whole <span class="URL" ><a href="https://homalg-project.github.io/homalg_project/" >homalg project</a></span > would have remained on my todo list.</p>"chap1_mj.html#X7DFB63A97E67C0A1" ><span class="RefLink" >ideas</span ></a> that make up this package and the whole <strong class="pkg" >homalg</strong > project came out of intensive discussions with <span class="URL" ><a href="https://www.plymouth.ac.uk/staff/daniel-robertz " >Daniel Robertz</a></span > during our early collaboration, where we developed our philosophy of a meta package for homological algebra and <span class="URL" ><a href="https://algebra.mathematik.uni-siegen.de/barakat/homalg_in_maple/" >implemented</a></span > it in <strong class="pkg" >Maple</strong >. This <strong class="pkg" >Maple</strong > package <strong class="pkg" >homalg</strong > is <span class="URL" ><a href="https://github.com/homalg-project/homalg_project/tree/master/RingsForHomalg/maple" >now part of</a></span > the GAP package <strong class="pkg" >RingsForHomalg</strong >.</p>span class="URL" ><a href="https://www.linkedin.com/in/simongoertzen/ " >Simon Görtzen</a></span > to further pursue and extend these <a href="https://homalg-project.github.io/homalg_project/Modules/doc/chap1_mj.html#X8583D47D7E570356" ><span class="RefLink" >ideas</span ></a> preparing the transition to <strong class="pkg" >GAP4</strong >. With his help <strong class="pkg" >homalg</strong > became an extendable multi-package project.</p>strong class="pkg" >IO</strong > package to start communicating with external computer algebra systems. This was crucial to remedy the yet missing support for important rings in <strong class="pkg" >GAP</strong >. Max provided the first piece of code to access the computer algebra system <strong class="pkg" >Singular</strong >. This was the starting point of the packages <strong class="pkg" >HomalgToCAS</strong > and <strong class="pkg" >IO_ForHomalg</strong >, which were further abstracted by Simon and myself enabling <strong class="pkg" >homalg</strong > to communicate with virtually any external (computer algebra) system.</p>span class="URL" ><a href="https://www.researchgate.net/scientific-contributions/50557632_Thomas_Baechler " >Thomas Bächler</a></span > wrote the package <strong class="pkg" >MapleForHomalg</strong > to directly access <strong class="pkg" >Maple</strong > via its <strong class="pkg" >C</strong >-interface. It offers an alternative to the package <strong class="pkg" >IO_ForHomalg</strong >, which requires <strong class="pkg" >Maple</strong >'s terminal interface cmaple. strong class="pkg" >Sage</strong > was added by Simon, and the support for <strong class="pkg" >Singular</strong > was initiated by <span class="URL" ><a href="https://www.th-owl.de/eecs/fachbereich/team/markus-lange-hegermann/" >Markus Lange-Hegermann</a></span > and continued by him and Simon, while <span class="URL" ><a href="http://www.math.rwth-aachen.de/~Markus.Kirschmer/ " >Markus Kirschmer</a></span > contributed the complete support for <strong class="pkg" >MAGMA</strong >. This formed the beginning of the <strong class="pkg" >RingsForHomalg</strong > package. Recently, Daniel added the support for <strong class="pkg" >Macaulay2</strong >.</p>ms were evaporated with the idea of Thomas Breuer using the so-called <a href="../../../doc/ref/chap86_mj.html#X86390538806F67CF" ><span class="RefLink" >weak pointers</span ></a> in <strong class="pkg" >GAP4</strong > to keep track of all the external objects that became obsolete for <strong class="pkg" >homalg</strong >. This idea took shape in a discussion with him and Frank Lübeck and finally found its way into the package <strong class="pkg" >HomalgToCAS</strong >.</p>eloped their own packages within the <strong class="pkg" >homalg</strong > project (--> Appendix <a href="chapE_mj.html#X849691F37C7AC1B4" ><span class="RefLink" >E</span ></a>). Without their contributions the package <strong class="pkg" >homalg</strong > would have remained a core without a body :</p>ul >li ><p><span class="URL" ><a href="https://www.researchgate.net/scientific-contributions/50557632_Thomas_Baechler " >Thomas Bächler</a></span ></p>li >li ><p>Barbara Bremer</p>li >li ><p><span class="URL" ><a href="http://www.math.rwth-aachen.de/~Thomas.Breuer/ " >Thomas Breuer</a></span ></p>li >li ><p>Anna Fabiańska</p>li >li ><p><span class="URL" ><a href="https://www.linkedin.com/in/simongoertzen/ " >Simon Görtzen</a></span ></p>li >li ><p><span class="URL" ><a href="http://www.math.rwth-aachen.de/~Markus.Kirschmer/ " >Markus Kirschmer</a></span ></p>li >li ><p><span class="URL" ><a href="https://www.th-owl.de/eecs/fachbereich/team/markus-lange-hegermann/ " >Markus Lange-Hegermann</a></span ></p>li >li ><p><span class="URL" ><a href="http://www.math.rwth-aachen.de/~Frank.Luebeck/ " >Frank Lübeck</a></span ></p>li >li ><p><span class="URL" ><a href="https://www.arangodb.com/speakers/max-neunhoeffer/ " >Max Neunhöffer</a></span ></p>li >li ><p><span class="URL" ><a href="https://www.plymouth.ac.uk/staff/daniel-robertz " >Daniel Robertz</a></span ></p>li >ul >span class="URL" ><a href="https://who.rocq.inria.fr/Alban.Quadrat/ " >Alban Quadrat</a></span > for supporting the <strong class="pkg" >homalg</strong > project and for all the wonderful discussions we had. At several places in the code I was happy to add the comment: "I learned this from Alban" .</p>span class="URL" ><a href="https://www.researchgate.net/profile/Wilhelm_Plesken " >Wilhelm Plesken</a></span > remains an inexhaustible source of extremely broad and deep knowledge. Thank you for being such a magnificent person.</p>em >Miriam</em >, <em >Josef</em >, <em >Jonas</em >, and <em >Irene</em > for the endless love and support.</p>"X8537FEB07AF2BEC8" name="X8537FEB07AF2BEC8" ></a></p>div class="contents" >"contents" name="contents" ></a></h3>div class="ContChap" ><a href="chap1_mj.html#X7DFB63A97E67C0A1" >1 <span class="Heading" >Introduction</span ></a>div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap1_mj.html#X7F1B40DB7C1395DD" >1.1 <span class="Heading" >What is the role of the <strong class="pkg" >homalg</strong > package in the <strong class="pkg" >homalg</strong > project?</span ></a>span >div class="ContSSBlock" >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap1_mj.html#X873C99678745ABAF" >1.1-1 <span class="Heading" >Philosophy</span ></a>span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap1_mj.html#X82A79A2A78AED00B" >1.1-2 <span class="Heading" ><strong class="pkg" >homalg</strong > provides ...</span ></a>span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap1_mj.html#X78E6DF7E878D754D" >1.1-3 <span class="Heading" >Building upon the <strong class="pkg" >homalg</strong > package</span ></a>span >div ></div >div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap1_mj.html#X78DD800B83ABC621" >1.2 <span class="Heading" >This manual</span ></a>span >div >div >div class="ContChap" ><a href="chap2_mj.html#X855C716578E12A0B" >2 <span class="Heading" >Installation of the <strong class="pkg" >homalg</strong > Package</span ></a>div >div class="ContChap" ><a href="chap3_mj.html#X78497E777FB3E402" >3 <span class="Heading" >Objects</span ></a>div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap3_mj.html#X7E3651DF87064E72" >3.1 <span class="Heading" >Objects: Category and Representations</span ></a>span >div class="ContSSBlock" >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap3_mj.html#X7E610FA77A49B9EC" >3.1-1 IsHomalgObject</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap3_mj.html#X79FC4A848517AF55" >3.1-2 IsHomalgStaticObject</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap3_mj.html#X7F1BC3F77949E779" >3.1-3 IsFinitelyPresentedObjectRep</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap3_mj.html#X79ED26577A1C2E09" >3.1-4 IsStaticFinitelyPresentedObjectOrSubobjectRep</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap3_mj.html#X7B645ADA876153F2" >3.1-5 IsStaticFinitelyPresentedObjectRep</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap3_mj.html#X837C31E38502E580" >3.1-6 IsStaticFinitelyPresentedSubobjectRep</a></span >div ></div >div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap3_mj.html#X7BD901538362C36E" >3.2 <span class="Heading" >Objects: Constructors</span ></a>span >div class="ContSSBlock" >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap3_mj.html#X810D3BFB7D9FE47E" >3.2-1 Subobject</a></span >div ></div >div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap3_mj.html#X7B3E8C797D15F0B7" >3.3 <span class="Heading" >Objects: Properties</span ></a>span >div class="ContSSBlock" >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap3_mj.html#X7CD2A77778432E7B" >3.3-1 IsFree</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap3_mj.html#X7D49FC85781256AB" >3.3-2 IsStablyFree</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap3_mj.html#X7EC041A77E7E46D2" >3.3-3 IsProjective</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap3_mj.html#X84A8AB217E8F4611" >3.3-4 IsProjectiveOfConstantRank</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap3_mj.html#X7F065FD7822C0A12" >3.3-5 IsInjective</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap3_mj.html#X7FCE608683CCDC6B" >3.3-6 IsInjectiveCogenerator</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap3_mj.html#X8784F151844F01FA" >3.3-7 FiniteFreeResolutionExists</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap3_mj.html#X7A6A34C283332F60" >3.3-8 IsReflexive</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap3_mj.html#X86D92DA17DCE22DD" >3.3-9 IsTorsionFree</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap3_mj.html#X7D8F8A0B81EFD22A" >3.3-10 IsArtinian</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap3_mj.html#X80C6B26284721409" >3.3-11 IsTorsion</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap3_mj.html#X7B894ED27D38E4B5" >3.3-12 IsPure</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap3_mj.html#X8373421F7E085763" >3.3-13 IsCohenMacaulay</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap3_mj.html#X83CBA38E81DC4A72" >3.3-14 IsGorenstein</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap3_mj.html#X7E7AEFBE7801F196" >3.3-15 IsKoszul</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap3_mj.html#X7A20E4597A707218" >3.3-16 HasConstantRank</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap3_mj.html#X7CD026F185A5E41E" >3.3-17 ConstructedAsAnIdeal</a></span >div ></div >div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap3_mj.html#X805B06828294072C" >3.4 <span class="Heading" >Objects: Attributes</span ></a>span >div class="ContSSBlock" >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap3_mj.html#X7E6C8ED2865B6F35" >3.4-1 TorsionSubobject</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap3_mj.html#X82BCEE867CBE84E5" >3.4-2 TheMorphismToZero</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap3_mj.html#X85EFEC127CA408A1" >3.4-3 TheIdentityMorphism</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap3_mj.html#X8236B1D480ED04CD" >3.4-4 FullSubobject</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap3_mj.html#X81679BB58541E235" >3.4-5 ZeroSubobject</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap3_mj.html#X7C16CBCC78C56CDC" >3.4-6 EmbeddingInSuperObject</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap3_mj.html#X7ADC5B647C8E6D8C" >3.4-7 SuperObject</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap3_mj.html#X7FB9A7C3785D92DC" >3.4-8 FactorObject</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap3_mj.html#X7A23EAD67E6B85C1" >3.4-9 UnderlyingSubobject</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap3_mj.html#X7FC5F0AF7CF5DC67" >3.4-10 NatTrIdToHomHom_R</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap3_mj.html#X81889C777A22A5D3" >3.4-11 Annihilator</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap3_mj.html#X809A7C3882912EFD" >3.4-12 EndomorphismRing</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap3_mj.html#X85F3D7CF81E85423" >3.4-13 UnitObject</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap3_mj.html#X7E192147807E66DA" >3.4-14 RankOfObject</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap3_mj.html#X84FDF25D797B874B" >3.4-15 ProjectiveDimension</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap3_mj.html#X807BA3C583D3F1EB" >3.4-16 DegreeOfTorsionFreeness</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap3_mj.html#X7E32A9FC81E0E101" >3.4-17 Grade</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap3_mj.html#X816186E587563E3F" >3.4-18 PurityFiltration</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap3_mj.html#X8021C33D85444081" >3.4-19 CodegreeOfPurity</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap3_mj.html#X84299BAB807A1E13" >3.4-20 HilbertPolynomial</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap3_mj.html#X7BC36CC67CB09858" >3.4-21 AffineDimension</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap3_mj.html#X82A1B55879AB1742" >3.4-22 ProjectiveDegree</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap3_mj.html#X791B772A7E368A88" >3.4-23 ConstantTermOfHilbertPolynomialn</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap3_mj.html#X7FC735717985B092" >3.4-24 ElementOfGrothendieckGroup</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap3_mj.html#X81024DAF8695083E" >3.4-25 ChernPolynomial</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap3_mj.html#X79942F6187DF4434" >3.4-26 ChernCharacter</a></span >div ></div >div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap3_mj.html#X7B4D450B78A86F8B" >3.5 <span class="Heading" >Objects: Operations and Functions</span ></a>span >div class="ContSSBlock" >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap3_mj.html#X87AEDF2985D65DCC" >3.5-1 CurrentResolution</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap3_mj.html#X81FACFAC828CA2F9" >3.5-2 UnderlyingObject</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap3_mj.html#X82AE15AF82136AE0" >3.5-3 Saturate</a></span >div ></div >div >div class="ContChap" ><a href="chap4_mj.html#X7BEB6C617FED52DA" >4 <span class="Heading" >Morphisms</span ></a>div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap4_mj.html#X7DE206257C909BDE" >4.1 <span class="Heading" >Morphisms: Categories and Representations</span ></a>span >div class="ContSSBlock" >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap4_mj.html#X7D0F89828196DFF0" >4.1-1 IsHomalgMorphism</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap4_mj.html#X81458CA5836D582F" >4.1-2 IsHomalgStaticMorphism</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap4_mj.html#X7933C51A842ABA32" >4.1-3 IsHomalgEndomorphism</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap4_mj.html#X823580787F23EB10" >4.1-4 IsMorphismOfFinitelyGeneratedObjectsRep</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap4_mj.html#X84A97E897C74B492" >4.1-5 IsStaticMorphismOfFinitelyGeneratedObjectsRep</a></span >div ></div >div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap4_mj.html#X86A95A9B85D8B58B" >4.2 <span class="Heading" >Morphisms: Constructors</span ></a>span >div >div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap4_mj.html#X7B0B60BD79756A00" >4.3 <span class="Heading" >Morphisms: Properties</span ></a>span >div class="ContSSBlock" >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap4_mj.html#X7F66120A814DC16B" >4.3-1 IsMorphism</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap4_mj.html#X7B7206EC7F584F25" >4.3-2 IsGeneralizedMorphismWithFullDomain</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap4_mj.html#X7AD32A427B247366" >4.3-3 IsGeneralizedEpimorphism</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap4_mj.html#X83C68AEA7FE4AA29" >4.3-4 IsGeneralizedMonomorphism</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap4_mj.html#X83F05F467DA5EA4D" >4.3-5 IsGeneralizedIsomorphism</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap4_mj.html#X814D78347858EC13" >4.3-6 IsOne</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap4_mj.html#X7CB5896082D29173" >4.3-7 IsIdempotent</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap4_mj.html#X78AD1FDD7F53932C" >4.3-8 IsMonomorphism</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap4_mj.html#X8724CEF182DC4064" >4.3-9 IsEpimorphism</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap4_mj.html#X7DFACF1F7D7F7EE9" >4.3-10 IsSplitMonomorphism</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap4_mj.html#X80A66EFA862E56BC" >4.3-11 IsSplitEpimorphism</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap4_mj.html#X7E07BBF57B92BA56" >4.3-12 IsIsomorphism</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap4_mj.html#X7F30E3D37E9D7F37" >4.3-13 IsAutomorphism</a></span >div ></div >div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap4_mj.html#X806EEA4685A4A3F3" >4.4 <span class="Heading" >Morphisms: Attributes</span ></a>span >div class="ContSSBlock" >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap4_mj.html#X7DE8173F80E07AB1" >4.4-1 Source </a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap4_mj.html#X829F76BB80BD55DB" >4.4-2 Range</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap4_mj.html#X7F3927E287087B64" >4.4-3 CokernelEpi</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap4_mj.html#X7D71AE8E838712D7" >4.4-4 CokernelNaturalGeneralizedIsomorphism</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap4_mj.html#X87C00FFB79FA93A8" >4.4-5 KernelSubobject</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap4_mj.html#X82672DB279FAEFCC" >4.4-6 KernelEmb</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap4_mj.html#X82FB6A4687E778D5" >4.4-7 ImageSubobject</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap4_mj.html#X85FA7C19800F72B2" >4.4-8 ImageObjectEmb</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap4_mj.html#X86E3E1BA7BCE4D66" >4.4-9 ImageObjectEpi</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap4_mj.html#X823682157C6B4D63" >4.4-10 MorphismAid</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap4_mj.html#X85F22F4084EA7D31" >4.4-11 InverseOfGeneralizedMorphismWithFullDomain</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap4_mj.html#X8500C49A784C8EDC" >4.4-12 DegreeOfMorphism</a></span >div ></div >div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap4_mj.html#X789623548056F7B7" >4.5 <span class="Heading" >Morphisms: Operations and Functions</span ></a>span >div class="ContSSBlock" >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap4_mj.html#X7B4F9EF27A241520" >4.5-1 ByASmallerPresentation</a></span >div ></div >div >div class="ContChap" ><a href="chap5_mj.html#X79B130FC7906FB4C" >5 <span class="Heading" >Elements</span ></a>div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap5_mj.html#X7FBC2FC77E93856C" >5.1 <span class="Heading" >Elements: Category and Representations</span ></a>span >div class="ContSSBlock" >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap5_mj.html#X784BBB2A782DB774" >5.1-1 IsHomalgElement</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap5_mj.html#X87F3740E85E9AA51" >5.1-2 IsElementOfAnObjectGivenByAMorphismRep</a></span >div ></div >div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap5_mj.html#X8159567F8721ADCA" >5.2 <span class="Heading" >Elements: Constructors</span ></a>span >div >div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap5_mj.html#X7C36DB5C81520E55" >5.3 <span class="Heading" >Elements: Properties</span ></a>span >div class="ContSSBlock" >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap5_mj.html#X7D16A0BB80BA69DC" >5.3-1 IsZero</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap5_mj.html#X7886344B7A8B9304" >5.3-2 IsCyclicGenerator</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap5_mj.html#X80C6B26284721409" >5.3-3 IsTorsion</a></span >div ></div >div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap5_mj.html#X875351A77DEB949A" >5.4 <span class="Heading" >Elements: Attributes</span ></a>span >div class="ContSSBlock" >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap5_mj.html#X8557F75878DEEA58" >5.4-1 Annihilator</a></span >div ></div >div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap5_mj.html#X865C489C7FE750A9" >5.5 <span class="Heading" >Elements: Operations and Functions</span ></a>span >div class="ContSSBlock" >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap5_mj.html#X7A4ED0528640EEFE" >5.5-1 in</a></span >div ></div >div >div class="ContChap" ><a href="chap6_mj.html#X79C388D385DB7CD1" >6 <span class="Heading" >Complexes</span ></a>div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap6_mj.html#X7FF155CB7C4C7CB4" >6.1 <span class="Heading" >Complexes: Category and Representations</span ></a>span >div class="ContSSBlock" >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap6_mj.html#X8166F9FD7BFDA207" >6.1-1 IsHomalgComplex</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap6_mj.html#X825B40448449FFF6" >6.1-2 IsComplexOfFinitelyPresentedObjectsRep</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap6_mj.html#X7B0613FF7A702D48" >6.1-3 IsCocomplexOfFinitelyPresentedObjectsRep</a></span >div ></div >div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap6_mj.html#X7B31FFA97FEE9B80" >6.2 <span class="Heading" >Complexes: Constructors</span ></a>span >div class="ContSSBlock" >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap6_mj.html#X7C0D9D0178477517" >6.2-1 HomalgComplex</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap6_mj.html#X82E0E9D17E29A67B" >6.2-2 HomalgCocomplex</a></span >div ></div >div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap6_mj.html#X80A23E668343440B" >6.3 <span class="Heading" >Complexes: Properties</span ></a>span >div class="ContSSBlock" >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap6_mj.html#X7C668F517AEB1F99" >6.3-1 IsSequence</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap6_mj.html#X856E7B4E8264E8F0" >6.3-2 IsComplex</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap6_mj.html#X847A62A6806046C4" >6.3-3 IsAcyclic</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap6_mj.html#X7F4927337891E086" >6.3-4 IsRightAcyclic</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap6_mj.html#X8673124C83AA8FCC" >6.3-5 IsLeftAcyclic</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap6_mj.html#X78FEA48B7839E683" >6.3-6 IsGradedObject</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap6_mj.html#X793465497B435197" >6.3-7 IsExactSequence</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap6_mj.html#X87ADD4F685457000" >6.3-8 IsShortExactSequence</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap6_mj.html#X7BAF581986905995" >6.3-9 IsSplitShortExactSequence</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap6_mj.html#X84B794FB86C169CF" >6.3-10 IsTriangle</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap6_mj.html#X81E57EE37FC94539" >6.3-11 IsExactTriangle</a></span >div ></div >div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap6_mj.html#X7BC7B49D7F928DF8" >6.4 <span class="Heading" >Complexes: Attributes</span ></a>span >div class="ContSSBlock" >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap6_mj.html#X7DE6E8D8875B515F" >6.4-1 BettiTable</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap6_mj.html#X80EDFDD281834882" >6.4-2 FiltrationByShortExactSequence</a></span >div ></div >div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap6_mj.html#X84E12E9C7A60D9BC" >6.5 <span class="Heading" >Complexes: Operations and Functions</span ></a>span >div class="ContSSBlock" >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap6_mj.html#X7F10893B78FEDEB7" >6.5-1 Add</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap6_mj.html#X79677A407C9EF3A0" >6.5-2 ByASmallerPresentation</a></span >div ></div >div >div class="ContChap" ><a href="chap7_mj.html#X782EF48B7D997E9E" >7 <span class="Heading" >Chain Morphisms</span ></a>div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap7_mj.html#X8703B8017F55336F" >7.1 <span class="Heading" >ChainMorphisms: Categories and Representations</span ></a>span >div class="ContSSBlock" >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap7_mj.html#X7CB62E188027B7C5" >7.1-1 IsHomalgChainMorphism</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap7_mj.html#X853BD37084BFC602" >7.1-2 IsHomalgChainEndomorphism</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap7_mj.html#X7C35D69F7B09BD47" >7.1-3 IsChainMorphismOfFinitelyPresentedObjectsRep</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap7_mj.html#X7DF3EA1D817266C1" >7.1-4 IsCochainMorphismOfFinitelyPresentedObjectsRep</a></span >div ></div >div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap7_mj.html#X83637FBE86C5DDF1" >7.2 <span class="Heading" >Chain Morphisms: Constructors</span ></a>span >div class="ContSSBlock" >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap7_mj.html#X853361547FB213CA" >7.2-1 HomalgChainMorphism</a></span >div ></div >div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap7_mj.html#X789E2EC07C041D78" >7.3 <span class="Heading" >Chain Morphisms: Properties</span ></a>span >div class="ContSSBlock" >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap7_mj.html#X798B6A897FE4FF12" >7.3-1 IsMorphism</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap7_mj.html#X8194427F8423EB00" >7.3-2 IsGeneralizedMorphismWithFullDomain</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap7_mj.html#X84FE6CFD85AB7B73" >7.3-3 IsGeneralizedEpimorphism</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap7_mj.html#X7C7A07FD795C903E" >7.3-4 IsGeneralizedMonomorphism</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap7_mj.html#X7D686DF9832AE258" >7.3-5 IsGeneralizedIsomorphism</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap7_mj.html#X790FC54F7DF8B5B1" >7.3-6 IsOne</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap7_mj.html#X8709A2597FE67C7F" >7.3-7 IsMonomorphism</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap7_mj.html#X7C8E0B1A7A8EE198" >7.3-8 IsEpimorphism</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap7_mj.html#X8724A5E77FD88D49" >7.3-9 IsSplitMonomorphism</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap7_mj.html#X87508506872F4FC3" >7.3-10 IsSplitEpimorphism</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap7_mj.html#X85180A1E83C01BAA" >7.3-11 IsIsomorphism</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap7_mj.html#X856D1F5C7E289064" >7.3-12 IsAutomorphism</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap7_mj.html#X81B2B7BC7B27A1F4" >7.3-13 IsGradedMorphism</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap7_mj.html#X7B5C2D788794699E" >7.3-14 IsQuasiIsomorphism</a></span >div ></div >div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap7_mj.html#X83FBA43B7E5833F0" >7.4 <span class="Heading" >Chain Morphisms: Attributes</span ></a>span >div class="ContSSBlock" >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap7_mj.html#X81A0D7187D28BA34" >7.4-1 Source </a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap7_mj.html#X842454D5851D0C79" >7.4-2 Range</a></span >div ></div >div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap7_mj.html#X7DD92C727DD630DA" >7.5 <span class="Heading" >Chain Morphisms: Operations and Functions</span ></a>span >div class="ContSSBlock" >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap7_mj.html#X875F27D07EB78998" >7.5-1 ByASmallerPresentation</a></span >div ></div >div >div class="ContChap" ><a href="chap8_mj.html#X7CEDAD61826170CF" >8 <span class="Heading" >Bicomplexes</span ></a>div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap8_mj.html#X7CBAE2807BD16E7E" >8.1 <span class="Heading" >Bicomplexes: Category and Representations</span ></a>span >div class="ContSSBlock" >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap8_mj.html#X80B7C45A850F4C3E" >8.1-1 IsHomalgBicomplex</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap8_mj.html#X7892BBCD783ABE16" >8.1-2 IsBicomplexOfFinitelyPresentedObjectsRep</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap8_mj.html#X7A82F6DC7C4C7761" >8.1-3 IsBicocomplexOfFinitelyPresentedObjectsRep</a></span >div ></div >div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap8_mj.html#X842D047F7E00F774" >8.2 <span class="Heading" >Bicomplexes: Constructors</span ></a>span >div class="ContSSBlock" >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap8_mj.html#X86D50FE285F49BF6" >8.2-1 HomalgBicomplex</a></span >div ></div >div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap8_mj.html#X854AA4C379C813AC" >8.3 <span class="Heading" >Bicomplexes: Properties</span ></a>span >div class="ContSSBlock" >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap8_mj.html#X7912E2147849BA74" >8.3-1 IsBisequence</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap8_mj.html#X87886CA9828D0B4A" >8.3-2 IsBicomplex</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap8_mj.html#X85363EC87E54554C" >8.3-3 IsTransposedWRTTheAssociatedComplex</a></span >div ></div >div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap8_mj.html#X7E2F2E387A4EF533" >8.4 <span class="Heading" >Bicomplexes: Attributes</span ></a>span >div class="ContSSBlock" >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap8_mj.html#X7C805D967E803BEF" >8.4-1 TotalComplex</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap8_mj.html#X7E672CA37AA3D34C" >8.4-2 SpectralSequence</a></span >div ></div >div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap8_mj.html#X81357E7A7C6D31F5" >8.5 <span class="Heading" >Bicomplexes: Operations and Functions</span ></a>span >div class="ContSSBlock" >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap8_mj.html#X7CE9470285B819BC" >8.5-1 UnderlyingComplex</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap8_mj.html#X7D4B66E08666B142" >8.5-2 ByASmallerPresentation</a></span >div ></div >div >div class="ContChap" ><a href="chap9_mj.html#X86C997977B62C726" >9 <span class="Heading" >Bigraded Objects</span ></a>div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap9_mj.html#X82C303E27EA6C844" >9.1 <span class="Heading" >BigradedObjects: Categories and Representations</span ></a>span >div class="ContSSBlock" >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap9_mj.html#X795C082E83748032" >9.1-1 IsHomalgBigradedObject</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap9_mj.html#X7ADBEEA47D650EF2" >9.1-2 IsHomalgBigradedObjectAssociatedToAnExactCouple</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap9_mj.html#X7994D63E7F77C704" >9.1-3 IsHomalgBigradedObjectAssociatedToAFilteredComplex</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap9_mj.html#X8007507A79E54A1A" >9.1-4 IsHomalgBigradedObjectAssociatedToABicomplex</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap9_mj.html#X7AE4EB99817C4508" >9.1-5 IsBigradedObjectOfFinitelyPresentedObjectsRep</a></span >div ></div >div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap9_mj.html#X7A37F65D79540DFE" >9.2 <span class="Heading" >Bigraded Objects: Constructors</span ></a>span >div class="ContSSBlock" >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap9_mj.html#X79DCB6FF7E6FFA8B" >9.2-1 HomalgBigradedObject</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap9_mj.html#X7D0A240684BD8FC3" >9.2-2 AsDifferentialObject</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap9_mj.html#X783AA6E3817BFC0F" >9.2-3 DefectOfExactness</a></span >div ></div >div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap9_mj.html#X83F0D79981589A42" >9.3 <span class="Heading" >Bigraded Objects: Properties</span ></a>span >div class="ContSSBlock" >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap9_mj.html#X82DD24197D46CB80" >9.3-1 IsEndowedWithDifferential</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap9_mj.html#X8466E4747DF9DDF4" >9.3-2 IsStableSheet</a></span >div ></div >div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap9_mj.html#X7A5828337CE2F4F2" >9.4 <span class="Heading" >Bigraded Objects: Operations and Functions</span ></a>span >div class="ContSSBlock" >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap9_mj.html#X7A70FD7C82C0C837" >9.4-1 ByASmallerPresentation</a></span >div ></div >div >div class="ContChap" ><a href="chap10_mj.html#X87330D8C82E71B50" >10 <span class="Heading" >Spectral Sequences</span ></a>div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap10_mj.html#X85E1174B7BDA291E" >10.1 <span class="Heading" >SpectralSequences: Categorie and Representations</span ></a>span >div class="ContSSBlock" >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap10_mj.html#X795DCCD88630BA47" >10.1-1 IsHomalgSpectralSequence</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap10_mj.html#X7F2858CB84D2FF7F" >10.1-2 IsHomalgSpectralSequenceAssociatedToAnExactCouple</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap10_mj.html#X7A6FDA637E4D77CA" >10.1-3 IsHomalgSpectralSequenceAssociatedToAFilteredComplex</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap10_mj.html#X7E7F02B379ABFBF6" >10.1-4 IsHomalgSpectralSequenceAssociatedToABicomplex</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap10_mj.html#X81B2C07D7BBD25A9" >10.1-5 IsSpectralSequenceOfFinitelyPresentedObjectsRep</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap10_mj.html#X7ACDC0C97D8F072A" >10.1-6 IsSpectralCosequenceOfFinitelyPresentedObjectsRep</a></span >div ></div >div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap10_mj.html#X84F3E1DF86C576A3" >10.2 <span class="Heading" >Spectral Sequences: Constructors</span ></a>span >div class="ContSSBlock" >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap10_mj.html#X840EE4DE7D84F72D" >10.2-1 HomalgSpectralSequence</a></span >div ></div >div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap10_mj.html#X8014BAE984177944" >10.3 <span class="Heading" >Spectral Sequences: Attributes</span ></a>span >div class="ContSSBlock" >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap10_mj.html#X862BD6E2875BC376" >10.3-1 GeneralizedEmbeddingsInTotalObjects</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap10_mj.html#X7B84FE76787EAD55" >10.3-2 GeneralizedEmbeddingsInTotalDefects</a></span >div ></div >div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap10_mj.html#X7BD192607D03A699" >10.4 <span class="Heading" >Spectral Sequences: Operations and Functions</span ></a>span >div class="ContSSBlock" >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap10_mj.html#X8775988481D1579F" >10.4-1 ByASmallerPresentation</a></span >div ></div >div >div class="ContChap" ><a href="chap11_mj.html#X78D1062D78BE08C1" >11 <span class="Heading" >Functors</span ></a>div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap11_mj.html#X7E41BC437F2B76E1" >11.1 <span class="Heading" >Functors: Category and Representations</span ></a>span >div class="ContSSBlock" >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap11_mj.html#X7EB19E0787C99FF2" >11.1-1 IsHomalgFunctor</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap11_mj.html#X87ECF5AF7A154723" >11.1-2 IsHomalgFunctorRep</a></span >div ></div >div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap11_mj.html#X86EE897086995E47" >11.2 <span class="Heading" >Functors: Constructors</span ></a>span >div class="ContSSBlock" >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap11_mj.html#X79407A4E78D628FF" >11.2-1 CreateHomalgFunctor</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap11_mj.html#X79454910823BD09F" >11.2-2 InsertObjectInMultiFunctor</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap11_mj.html#X7E0DE63378A5E204" >11.2-3 RightSatelliteOfCofunctor</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap11_mj.html#X87448A45780737AE" >11.2-4 LeftSatelliteOfFunctor</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap11_mj.html#X79EBC65E7DB3FDFB" >11.2-5 RightDerivedCofunctor</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap11_mj.html#X7AC81ED178F2ECB7" >11.2-6 LeftDerivedFunctor</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap11_mj.html#X7B0F972B850EB3CF" >11.2-7 ComposeFunctors</a></span >div ></div >div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap11_mj.html#X7A21845C7C536717" >11.3 <span class="Heading" >Functors: Attributes</span ></a>span >div class="ContSSBlock" >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap11_mj.html#X845E5EF17BBBF64C" >11.3-1 NameOfFunctor</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap11_mj.html#X796A383A7AEDA56E" >11.3-2 OperationOfFunctor</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap11_mj.html#X7BCB7F008620570C" >11.3-3 Genesis</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap11_mj.html#X83DB28187E1A4E92" >11.3-4 ProcedureToReadjustGenerators</a></span >div ></div >div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap11_mj.html#X7D83D0EB87D2D872" >11.4 <span class="Heading" >Basic Functors</span ></a>span >div class="ContSSBlock" >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap11_mj.html#X7E1FD2EA8358FEA7" >11.4-1 functor_Kernel</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap11_mj.html#X795B435785C96DFD" >11.4-2 functor_DefectOfExactness</a></span >div ></div >div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap11_mj.html#X815BF6DA7FD5D44B" >11.5 <span class="Heading" >Tool Functors</span ></a>span >div >div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap11_mj.html#X879135AC8330C509" >11.6 <span class="Heading" >Other Functors</span ></a>span >div >div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap11_mj.html#X7DACD68E7E5FA324" >11.7 <span class="Heading" >Functors: Operations and Functions</span ></a>span >div class="ContSSBlock" >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap11_mj.html#X7EAE59AC7D402D5A" >11.7-1 InstallFunctor</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap11_mj.html#X7BD3887982B2663E" >11.7-2 InstallDeltaFunctor</a></span >div ></div >div >div class="ContChap" ><a href="chap12_mj.html#X7A489A5D79DA9E5C" >12 <span class="Heading" >Examples</span ></a>div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap12_mj.html#X7BB9DE017ECE6E86" >12.1 <span class="Heading" >ExtExt</span ></a>span >div >div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap12_mj.html#X7EE63228803A04F1" >12.2 <span class="Heading" >Purity</span ></a>span >div >div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap12_mj.html#X812EF8147AE16E72" >12.3 <span class="Heading" >TorExt-Grothendieck</span ></a>span >div >div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap12_mj.html#X784BC2567875830B" >12.4 <span class="Heading" >TorExt</span ></a>span >div >div >div class="ContChap" ><a href="chapA_mj.html#X7FBD6C8A83D64BE4" >A <span class="Heading" >The Mathematical Idea behind <strong class="pkg" >homalg</strong ></span ></a>div >div class="ContChap" ><a href="chapB_mj.html#X816F972F826BE589" >B <span class="Heading" >Development</span ></a>div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chapB_mj.html#X85B7FB29805B14C9" >B.1 <span class="Heading" >Why was <strong class="pkg" >homalg</strong > discontinued in <span class="URL" ><a href="http://www.maplesoft.com/ " >Maple</a></span >?</span ></a>span >div >div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chapB_mj.html#X7D2A5B127A68AB58" >B.2 <span class="Heading" >Why <span class="URL" ><a href="http://www.gap-system.org/ " >GAP4</a></span >?</span ></a>span >div class="ContSSBlock" >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chapB_mj.html#X8148BBD87B272E84" >B.2-1 <span class="Heading" ><strong class="pkg" >GAP</strong > is free and open software</span ></a>span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chapB_mj.html#X799403717D4414BA" >B.2-2 <span class="Heading" ><strong class="pkg" >GAP</strong > has an area of expertise</span ></a>span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chapB_mj.html#X863002078157C105" >B.2-3 <span class="Heading" ><strong class="pkg" >GAP4</strong > can communicate</span ></a>span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chapB_mj.html#X78EB6CAE7C7F2F7C" >B.2-4 <span class="Heading" ><strong class="pkg" >GAP4</strong > is a <em >mathematical</em > object-oriented programming language</span ></a>span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chapB_mj.html#X7E1B9133835F33FD" >B.2-5 <span class="Heading" ><strong class="pkg" >GAP4</strong > packages are easily extendible</span ></a>span >div ></div >div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chapB_mj.html#X7F7CF4AE830E312B" >B.3 <span class="Heading" >Why not <span class="URL" ><a href="http://www.sagemath.org/ " >Sage</a></span >?</span ></a>span >div >div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chapB_mj.html#X81A6C501841B20C4" >B.4 <span class="Heading" >How does <strong class="pkg" >homalg</strong > compare to <strong class="pkg" >Sage</strong >?</span ></a>span >div class="ContSSBlock" >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chapB_mj.html#X82A6EC2C850909B1" >B.4-1 <span class="Heading" >They differ in objectives and scale</span ></a>span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chapB_mj.html#X8346070A86BF5CC5" >B.4-2 <span class="Heading" >They differ in the programming language</span ></a>span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chapB_mj.html#X8040F9357F008C1E" >B.4-3 <span class="Heading" >They differ in the way they communicate with the outer world</span ></a>span >div ></div >div >div class="ContChap" ><a href="chapC_mj.html#X8222352C78A19214" >C <span class="Heading" >Logic Subpackages</span ></a>div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chapC_mj.html#X84B5336279AF7DE4" >C.1 <span class="Heading" ><strong class="pkg" >LIOBJ</strong >: Logical Implications for Objects of Abelian Categories</span ></a>span >div >div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chapC_mj.html#X8744FAF47E59C422" >C.2 <span class="Heading" ><strong class="pkg" >LIMOR</strong >: Logical Implications for Morphisms of Abelian Categories</span ></a>span >div >div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chapC_mj.html#X8024C3D08006C35A" >C.3 <span class="Heading" ><strong class="pkg" >LICPX</strong >: Logical Implications for Complexes in Abelian Categories</span ></a>span >div >div >div class="ContChap" ><a href="chapD_mj.html#X7DC71AA679A8CB8D" >D <span class="Heading" >Debugging <strong class="pkg" >homalg</strong ></span ></a>div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chapD_mj.html#X8062637283DD739D" >D.1 <span class="Heading" >Increase the assertion level</span ></a>span >div >div >div class="ContChap" ><a href="chapE_mj.html#X849691F37C7AC1B4" >E <span class="Heading" >The Core Packages and the Idea behind their Splitting</span ></a>div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chapE_mj.html#X7DCE809486D10888" >E.1 <span class="Heading" >The 6=2+4 split</span ></a>span >div class="ContSSBlock" >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chapE_mj.html#X818F306582DFF370" >E.1-1 <span class="Heading" >Logically independent</span ></a>span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chapE_mj.html#X7EE794358500309A" >E.1-2 <span class="Heading" >Black boxes</span ></a>span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chapE_mj.html#X786DDEFD85AD19F4" >E.1-3 <span class="Heading" >Summing up</span ></a>span >div ></div >div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chapE_mj.html#X865D30A382B80584" >E.2 <span class="Heading" >The 4=1+1+1+1 split</span ></a>span >div class="ContSSBlock" >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chapE_mj.html#X80FB5BB57BBE5B17" >E.2-1 <span class="Heading" ><strong class="pkg" >HomalgToCAS</strong ></span ></a>span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chapE_mj.html#X86F316DA837A4FE4" >E.2-2 <span class="Heading" ><strong class="pkg" >IO_ForHomalg</strong > and Alternatives</span ></a>span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chapE_mj.html#X78E3E28C81E8DC12" >E.2-3 <span class="Heading" ><strong class="pkg" >RingsForHomalg</strong ></span ></a>span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chapE_mj.html#X7AD8DBE280C58EE9" >E.2-4 <span class="Heading" >Your own <strong class="pkg" >RingsForHomalg</strong ></span ></a>span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chapE_mj.html#X7A0A10B585C49632" >E.2-5 <span class="Heading" ><strong class="pkg" >ExamplesForHomalg</strong ></span ></a>span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chapE_mj.html#X7F4F8D6F7CD6B765" >E.2-6 <span class="Heading" >Documentation</span ></a>span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chapE_mj.html#X814A1DC581E36F66" >E.2-7 <span class="Heading" >Crediting</span ></a>span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chapE_mj.html#X78397E8681145827" >E.2-8 <span class="Heading" >Stability</span ></a>span >div ></div >div >div class="ContChap" ><a href="chapF_mj.html#X84555A0687FBAE33" >F <span class="Heading" >Overview of the <strong class="pkg" >homalg</strong > Package Source Code </span ></a>div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chapF_mj.html#X81DDCFC578069518" >F.1 <span class="Heading" >The Basic Objects</span ></a>span >div >div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chapF_mj.html#X7BDE961D858BC60E" >F.2 <span class="Heading" >The High Level Homological Algorithms</span ></a>span >div >div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chapF_mj.html#X7E8463067BB2F31E" >F.3 <span class="Heading" >Logical Implications for <strong class="pkg" >homalg</strong > Objects</span ></a>span >div >div >div class="ContChap" ><a href="chapBib_mj.html" ><span class="Heading" >References</span ></a></div >div class="ContChap" ><a href="chapInd_mj.html" ><span class="Heading" >Index</span ></a></div >br />div >div class="chlinkprevnextbot" > <a href="chap0_mj.html" >[Top of Book]</a> <a href="chap0_mj.html#contents" >[Contents]</a> <a href="chap1_mj.html" >[Next Chapter]</a> </div >div class="chlinkbot" ><span class="chlink1" >Goto Chapter: </span ><a href="chap0_mj.html" >Top</a> <a href="chap1_mj.html" >1</a> <a href="chap2_mj.html" >2</a> <a href="chap3_mj.html" >3</a> <a href="chap4_mj.html" >4</a> <a href="chap5_mj.html" >5</a> <a href="chap6_mj.html" >6</a> <a href="chap7_mj.html" >7</a> <a href="chap8_mj.html" >8</a> <a href="chap9_mj.html" >9</a> <a href="chap10_mj.html" >10</a> <a href="chap11_mj.html" >11</a> <a href="chap12_mj.html" >12</a> <a href="chapA_mj.html" >A</a> <a href="chapB_mj.html" >B</a> <a href="chapC_mj.html" >C</a> <a href="chapD_mj.html" >D</a> <a href="chapE_mj.html" >E</a> <a href="chapF_mj.html" >F</a> <a href="chapBib_mj.html" >Bib</a> <a href="chapInd_mj.html" >Ind</a> </div >hr />"foot" >generated by <a href="http://www.math.rwth-aachen.de/~Frank.Luebeck/GAPDoc " >GAPDoc2HTML</a></p>body >html >quality 100%
¤ Dauer der Verarbeitung: 0.23 Sekunden
¤ *© Formatika GbR, Deutschland
