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<?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" src="https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.0/MathJax.js?config=TeX- </script> <title>GAP (homalg) - Chapter 6: Complexes</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="chap6" onload="jscontent()"> <div class="chlinktop"><span class="chlink1">Goto Chapter: </span><a href="chap0_mj.html">Top< <div class="chlinkprevnexttop"> <a href="chap0_mj.html">[Top of Book]</a> <a href="chap0_mj.ht <p id="mathjaxlink" class="pcenter"><a href="chap6.html">[MathJax off]</a></p> <p><a id="X79C388D385DB7CD1" name="X79C388D385DB7CD1"></a></p> <div class="ChapSects"><a href="chap6_mj.html#X79C388D385DB7CD1">6 <span class="Heading">C <div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap6_mj.html </span> <div class="ContSSBlock"> <span class="ContSS"><br /><span class="nocss"> </span><a href="chap6_mj.html#X8166F9FD7BFDA20 <span class="ContSS"><br /><span class="nocss"> </span><a href="chap6_mj.html#X825B40448449FFF <span class="ContSS"><br /><span class="nocss"> </span><a href="chap6_mj.html#X7B0613FF7A702D4 </div></div> <div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap6_mj.html </span> <div class="ContSSBlock"> <span class="ContSS"><br /><span class="nocss"> </span><a href="chap6_mj.html#X7C0D9D017847751 <span class="ContSS"><br /><span class="nocss"> </span><a href="chap6_mj.html#X82E0E9D17E29A67 </div></div> <div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap6_mj.html </span> <div class="ContSSBlock"> <span class="ContSS"><br /><span class="nocss"> </span><a href="chap6_mj.html#X7C668F517AEB1F9 <span class="ContSS"><br /><span class="nocss"> </span><a href="chap6_mj.html#X856E7B4E8264E8F <span class="ContSS"><br /><span class="nocss"> </span><a href="chap6_mj.html#X847A62A6806046C <span class="ContSS"><br /><span class="nocss"> </span><a href="chap6_mj.html#X7F4927337891E08 <span class="ContSS"><br /><span class="nocss"> </span><a href="chap6_mj.html#X8673124C83AA8FC <span class="ContSS"><br /><span class="nocss"> </span><a href="chap6_mj.html#X78FEA48B7839E68 <span class="ContSS"><br /><span class="nocss"> </span><a href="chap6_mj.html#X793465497B43519 <span class="ContSS"><br /><span class="nocss"> </span><a href="chap6_mj.html#X87ADD4F68545700 <span class="ContSS"><br /><span class="nocss"> </span><a href="chap6_mj.html#X7BAF58198690599 <span class="ContSS"><br /><span class="nocss"> </span><a href="chap6_mj.html#X84B794FB86C169C <span class="ContSS"><br /><span class="nocss"> </span><a href="chap6_mj.html#X81E57EE37FC9453 </div></div> <div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap6_mj.html </span> <div class="ContSSBlock"> <span class="ContSS"><br /><span class="nocss"> </span><a href="chap6_mj.html#X7DE6E8D8875B515 <span class="ContSS"><br /><span class="nocss"> </span><a href="chap6_mj.html#X80EDFDD28183488 </div></div> <div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap6_mj.html </span> <div class="ContSSBlock"> <span class="ContSS"><br /><span class="nocss"> </span><a href="chap6_mj.html#X7F10893B78FEDEB <span class="ContSS"><br /><span class="nocss"> </span><a href="chap6_mj.html#X79677A407C9EF3A </div></div> </div> <h3>6 <span class="Heading">Complexes</span></h3> <p><a id="X7FF155CB7C4C7CB4" name="X7FF155CB7C4C7CB4"></a></p> <h4>6.1 <span class="Heading">Complexes: Category and Representations</span></h4> <p><a id="X8166F9FD7BFDA207" name="X8166F9FD7BFDA207"></a></p> <h5>6.1-1 IsHomalgComplex</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Is <p>Returns: <code class="code">true</code> or <code class="code">false</code></p> <p>The <strong class="pkg">GAP</strong> category of <strong class="pkg">homalg</strong> (co)compl <p>(It is a subcategory of the <strong class="pkg">GAP</strong> category <code class="code">IsHoma <p><a id="X825B40448449FFF6" name="X825B40448449FFF6"></a></p> <h5>6.1-2 IsComplexOfFinitelyPresentedObjectsRep</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Is <p>Returns: <code class="code">true</code> or <code class="code">false</code></p> <p>The <strong class="pkg">GAP</strong> representation of complexes of finitley presented <stron <p>(It is a representation of the <strong class="pkg">GAP</strong> category <code class="func">IsH <p><a id="X7B0613FF7A702D48" name="X7B0613FF7A702D48"></a></p> <h5>6.1-3 IsCocomplexOfFinitelyPresentedObjectsRep</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Is <p>Returns: <code class="code">true</code> or <code class="code">false</code></p> <p>The <strong class="pkg">GAP</strong> representation of cocomplexes of finitley presented <str <p>(It is a representation of the <strong class="pkg">GAP</strong> category <code class="func">IsH <p><a id="X7B31FFA97FEE9B80" name="X7B31FFA97FEE9B80"></a></p> <h4>6.2 <span class="Heading">Complexes: Constructors</span></h4> <p><a id="X7C0D9D0178477517" name="X7C0D9D0178477517"></a></p> <h5>6.2-1 HomalgComplex</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Ho <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Ho <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Ho <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Ho <p>Returns: a <strong class="pkg">homalg</strong> complex</p> <p>The first syntax creates a complex (i.e. chain complex) with the single <strong class="pkg">ho <p>The second syntax creates a complex with the single <strong class="pkg">homalg</strong> morphi <p>The third syntax creates a complex (i.e. chain complex) with the single <strong class="pkg">ho <p>The fourth syntax creates a complex with the single <strong class="pkg">homalg</strong> (co)ch <p>If <var class="Arg">d</var> is not provided it defaults to zero in all cases. <br /> To add a morphism ( <div class="example"><pre> <span class="GAPprompt">gap></span> <span class="GAPinput">zz := HomalgRingOfIntegers( );</spa Z <span class="GAPprompt">gap></span> <span class="GAPinput">M := HomalgMatrix( "[ 2, 3, 4, 5 <A 2 x 3 matrix over an internal ring> <span class="GAPprompt">gap></span> <span class="GAPinput">M := LeftPresentation( M );</span> <A non-torsion left module presented by 2 relations for 3 generators> <span class="GAPprompt">gap></span> <span class="GAPinput">N := HomalgMatrix( "[ 2, 3, 4, 5, <A 2 x 4 matrix over an internal ring> <span class="GAPprompt">gap></span> <span class="GAPinput">N := LeftPresentation( N );</span> <A non-torsion left module presented by 2 relations for 4 generators> <span class="GAPprompt">gap></span> <span class="GAPinput">mat := HomalgMatrix( "[ \ <span class="GAPprompt">></span> <span class="GAPinput">0, 3, 6, 9, \</span> <span class="GAPprompt">></span> <span class="GAPinput">0, 2, 4, 6, \</span> <span class="GAPprompt">></span> <span class="GAPinput">0, 3, 6, 9 \</span> <span class="GAPprompt">></span> <span class="GAPinput">]", 3, 4, zz ); <A 3 x 4 matrix over an internal ring> <span class="GAPprompt">gap></span> <span class="GAPinput">phi := HomalgMap( mat, M, N );</span> <A "homomorphism" of left modules> <span class="GAPprompt">gap></span> <span class="GAPinput">IsMorphism( phi );</span> true <span class="GAPprompt">gap></span> <span class="GAPinput">phi;</span> <A homomorphism of left modules> </pre></div> <p>The first possibility:</p> <div class="example"><pre> <span class="GAPprompt">gap></span> <span class="GAPinput">C := HomalgComplex( N );</span> <A non-zero graded homology object consisting of a single left module at degre\ e 0> <span class="GAPprompt">gap></span> <span class="GAPinput">Add( C, phi );</span> <span class="GAPprompt">gap></span> <span class="GAPinput">C;</span> <A complex containing a single morphism of left modules at degrees [ 0 .. 1 ]> </pre></div> <p>The second possibility:</p> <div class="example"><pre> <span class="GAPprompt">gap></span> <span class="GAPinput">C := HomalgComplex( phi );</span> <A non-zero acyclic complex containing a single morphism of left modules at de\ grees [ 0 .. 1 ]> </pre></div> <p><a id="X82E0E9D17E29A67B" name="X82E0E9D17E29A67B"></a></p> <h5>6.2-2 HomalgCocomplex</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Ho <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Ho <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Ho <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Ho <p>Returns: a <strong class="pkg">homalg</strong> complex</p> <p>The first syntax creates a cocomplex (i.e. cochain complex) with the single <strong class="pk <p>The second syntax creates a cocomplex with the single <strong class="pkg">homalg</strong> morp <p>The third syntax creates a cocomplex (i.e. cochain complex) with the single <strong class="pk <p>The fourth syntax creates a cocomplex with the single <strong class="pkg">homalg</strong> (co) <p>If <var class="Arg">d</var> is not provided it defaults to zero in all cases. <br /> To add a morphism ( <div class="example"><pre> <span class="GAPprompt">gap></span> <span class="GAPinput">zz := HomalgRingOfIntegers( );</spa Z <span class="GAPprompt">gap></span> <span class="GAPinput">M := HomalgMatrix( "[ 2, 3, 4, 5 <A 2 x 3 matrix over an internal ring> <span class="GAPprompt">gap></span> <span class="GAPinput">M := RightPresentation( Involution <A non-torsion right module on 3 generators satisfying 2 relations> <span class="GAPprompt">gap></span> <span class="GAPinput">N := HomalgMatrix( "[ 2, 3, 4, 5, <A 2 x 4 matrix over an internal ring> <span class="GAPprompt">gap></span> <span class="GAPinput">N := RightPresentation( Involution <A non-torsion right module on 4 generators satisfying 2 relations> <span class="GAPprompt">gap></span> <span class="GAPinput">mat := HomalgMatrix( "[ \ <span class="GAPprompt">></span> <span class="GAPinput">0, 3, 6, 9, \</span> <span class="GAPprompt">></span> <span class="GAPinput">0, 2, 4, 6, \</span> <span class="GAPprompt">></span> <span class="GAPinput">0, 3, 6, 9 \</span> <span class="GAPprompt">></span> <span class="GAPinput">]", 3, 4, zz ); <A 3 x 4 matrix over an internal ring> <span class="GAPprompt">gap></span> <span class="GAPinput">phi := HomalgMap( Involution( mat ), <A "homomorphism" of right modules> <span class="GAPprompt">gap></span> <span class="GAPinput">IsMorphism( phi );</span> true <span class="GAPprompt">gap></span> <span class="GAPinput">phi;</span> <A homomorphism of right modules> </pre></div> <p>The first possibility:</p> <div class="example"><pre> <span class="GAPprompt">gap></span> <span class="GAPinput">C := HomalgCocomplex( M );</span> <A non-zero graded cohomology object consisting of a single right module at de\ gree 0> <span class="GAPprompt">gap></span> <span class="GAPinput">Add( C, phi );</span> <span class="GAPprompt">gap></span> <span class="GAPinput">C;</span> <A cocomplex containing a single morphism of right modules at degrees [ 0 .. 1 ]> </pre></div> <p>The second possibility:</p> <div class="example"><pre> <span class="GAPprompt">gap></span> <span class="GAPinput">C := HomalgCocomplex( phi );</span> <A non-zero acyclic cocomplex containing a single morphism of right modules at\ degrees [ 0 .. 1 ]> </pre></div> <p><a id="X80A23E668343440B" name="X80A23E668343440B"></a></p> <h4>6.3 <span class="Heading">Complexes: Properties</span></h4> <p><a id="X7C668F517AEB1F99" name="X7C668F517AEB1F99"></a></p> <h5>6.3-1 IsSequence</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Is <p>Returns: <code class="code">true</code> or <code class="code">false</code></p> <p>Check if all maps in <var class="Arg">C</var> are well-defined.</p> <p><a id="X856E7B4E8264E8F0" name="X856E7B4E8264E8F0"></a></p> <h5>6.3-2 IsComplex</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Is <p>Returns: <code class="code">true</code> or <code class="code">false</code></p> <p>Check if <var class="Arg">C</var> is complex.</p> <p><a id="X847A62A6806046C4" name="X847A62A6806046C4"></a></p> <h5>6.3-3 IsAcyclic</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Is <p>Returns: <code class="code">true</code> or <code class="code">false</code></p> <p>Check if the <strong class="pkg">homalg</strong> complex <var class="Arg">C</var> is acyclic, i.e <p><a id="X7F4927337891E086" name="X7F4927337891E086"></a></p> <h5>6.3-4 IsRightAcyclic</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Is <p>Returns: <code class="code">true</code> or <code class="code">false</code></p> <p>Check if the <strong class="pkg">homalg</strong> complex <var class="Arg">C</var> is acyclic, i.e <p><a id="X8673124C83AA8FCC" name="X8673124C83AA8FCC"></a></p> <h5>6.3-5 IsLeftAcyclic</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Is <p>Returns: <code class="code">true</code> or <code class="code">false</code></p> <p>Check if the <strong class="pkg">homalg</strong> complex <var class="Arg">C</var> is acyclic, i.e <p><a id="X78FEA48B7839E683" name="X78FEA48B7839E683"></a></p> <h5>6.3-6 IsGradedObject</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Is <p>Returns: <code class="code">true</code> or <code class="code">false</code></p> <p>Check if the <strong class="pkg">homalg</strong> complex <var class="Arg">C</var> is a graded obje <p><a id="X793465497B435197" name="X793465497B435197"></a></p> <h5>6.3-7 IsExactSequence</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Is <p>Returns: <code class="code">true</code> or <code class="code">false</code></p> <p>Check if the <strong class="pkg">homalg</strong> complex <var class="Arg">C</var> is exact.</p> <p><a id="X87ADD4F685457000" name="X87ADD4F685457000"></a></p> <h5>6.3-8 IsShortExactSequence</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Is <p>Returns: <code class="code">true</code> or <code class="code">false</code></p> <p>Check if the <strong class="pkg">homalg</strong> complex <var class="Arg">C</var> is a short exact <p><a id="X7BAF581986905995" name="X7BAF581986905995"></a></p> <h5>6.3-9 IsSplitShortExactSequence</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Is <p>Returns: <code class="code">true</code> or <code class="code">false</code></p> <p>Check if the <strong class="pkg">homalg</strong> complex <var class="Arg">C</var> is a split short <p><a id="X84B794FB86C169CF" name="X84B794FB86C169CF"></a></p> <h5>6.3-10 IsTriangle</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Is <p>Returns: <code class="code">true</code> or <code class="code">false</code></p> <p>Set to true if the <strong class="pkg">homalg</strong> complex <var class="Arg">C</var> is a triang <p><a id="X81E57EE37FC94539" name="X81E57EE37FC94539"></a></p> <h5>6.3-11 IsExactTriangle</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Is <p>Returns: <code class="code">true</code> or <code class="code">false</code></p> <p>Check if the <strong class="pkg">homalg</strong> complex <var class="Arg">C</var> is an exact tria <p><a id="X7BC7B49D7F928DF8" name="X7BC7B49D7F928DF8"></a></p> <h4>6.4 <span class="Heading">Complexes: Attributes</span></h4> <p><a id="X7DE6E8D8875B515F" name="X7DE6E8D8875B515F"></a></p> <h5>6.4-1 BettiTable</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Be <p>Returns: a <strong class="pkg">homalg</strong> diagram</p> <p>The Betti diagram of the <strong class="pkg">homalg</strong> complex <var class="Arg">C</var> of g <p><a id="X80EDFDD281834882" name="X80EDFDD281834882"></a></p> <h5>6.4-2 FiltrationByShortExactSequence</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Fi <p>Returns: a <strong class="pkg">homalg</strong> diagram</p> <p>The filtration induced by the short exact sequence <var class="Arg">C</var> on its middle object <p><a id="X84E12E9C7A60D9BC" name="X84E12E9C7A60D9BC"></a></p> <h4>6.5 <span class="Heading">Complexes: Operations and Functions</span></h4> <p><a id="X7F10893B78FEDEB7" name="X7F10893B78FEDEB7"></a></p> <h5>6.5-1 Add</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Ad <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Ad <p>Returns: a <strong class="pkg">homalg</strong> complex</p> <p>In the first syntax the morphism <var class="Arg">phi</var> is added to the (co)chain complex <var <p>In the second syntax the matrix <var class="Arg">mat</var> is interpreted as the matrix of the new < <div class="example"><pre> <span class="GAPprompt">gap></span> <span class="GAPinput">zz := HomalgRingOfIntegers( );</spa Z <span class="GAPprompt">gap></span> <span class="GAPinput">mat := HomalgMatrix( "[ 0, 1, 0, <A 2 x 2 matrix over an internal ring> <span class="GAPprompt">gap></span> <span class="GAPinput">phi := HomalgMap( mat );</span> <A homomorphism of left modules> <span class="GAPprompt">gap></span> <span class="GAPinput">C := HomalgComplex( phi );</span> <A non-zero acyclic complex containing a single morphism of left modules at de\ grees [ 0 .. 1 ]> <span class="GAPprompt">gap></span> <span class="GAPinput">Add( C, mat );</span> <span class="GAPprompt">gap></span> <span class="GAPinput">C;</span> <A sequence containing 2 morphisms of left modules at degrees [ 0 .. 2 ]> <span class="GAPprompt">gap></span> <span class="GAPinput">Display( C );</span> ------------------------- at homology degree: 2 Z^(1 x 2) ------------------------- [ [ 0, 1 ], [ 0, 0 ] ] the map is currently represented by the above 2 x 2 matrix ------------v------------ at homology degree: 1 Z^(1 x 2) ------------------------- [ [ 0, 1 ], [ 0, 0 ] ] the map is currently represented by the above 2 x 2 matrix ------------v------------ at homology degree: 0 Z^(1 x 2) ------------------------- <span class="GAPprompt">gap></span> <span class="GAPinput">IsComplex( C );</span> true <span class="GAPprompt">gap></span> <span class="GAPinput">IsAcyclic( C );</span> true <span class="GAPprompt">gap></span> <span class="GAPinput">IsExactSequence( C );</span> false <span class="GAPprompt">gap></span> <span class="GAPinput">C;</span> <A non-zero acyclic complex containing 2 morphisms of left modules at degrees [ 0 .. 2 ]> </pre></div> <p><a id="X79677A407C9EF3A0" name="X79677A407C9EF3A0"></a></p> <h5>6.5-2 ByASmallerPresentation</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ By <p>Returns: a <strong class="pkg">homalg</strong> complex</p> <p>It invokes <code class="code">ByASmallerPresentation</code> for <strong class="pkg">homalg</ <div class="example"><pre> InstallMethod( ByASmallerPresentation, "for homalg complexes", [ IsHomalgComplex ], function( C ) List( ObjectsOfComplex( C ), ByASmallerPresentation ); if Length( ObjectDegreesOfComplex( C ) ) > 1 then List( MorphismsOfComplex( C ), DecideZero ); fi; IsZero( C ); return C; end ); </pre></div> <p>This method performs side effects on its argument <var class="Arg">C</var> and returns it.</p> <div class="example"><pre> <span class="GAPprompt">gap></span> <span class="GAPinput">zz := HomalgRingOfIntegers( );</spa Z <span class="GAPprompt">gap></span> <span class="GAPinput">M := HomalgMatrix( "[ 2, 3, 4, 5 <A 2 x 3 matrix over an internal ring> <span class="GAPprompt">gap></span> <span class="GAPinput">M := LeftPresentation( M );</span> <A non-torsion left module presented by 2 relations for 3 generators> <span class="GAPprompt">gap></span> <span class="GAPinput">N := HomalgMatrix( "[ 2, 3, 4, 5, <A 2 x 4 matrix over an internal ring> <span class="GAPprompt">gap></span> <span class="GAPinput">N := LeftPresentation( N );</span> <A non-torsion left module presented by 2 relations for 4 generators> <span class="GAPprompt">gap></span> <span class="GAPinput">mat := HomalgMatrix( "[ \ <span class="GAPprompt">></span> <span class="GAPinput">0, 3, 6, 9, \</span> <span class="GAPprompt">></span> <span class="GAPinput">0, 2, 4, 6, \</span> <span class="GAPprompt">></span> <span class="GAPinput">0, 3, 6, 9 \</span> <span class="GAPprompt">></span> <span class="GAPinput">]", 3, 4, zz ); <A 3 x 4 matrix over an internal ring> <span class="GAPprompt">gap></span> <span class="GAPinput">phi := HomalgMap( mat, M, N );</span> <A "homomorphism" of left modules> <span class="GAPprompt">gap></span> <span class="GAPinput">IsMorphism( phi );</span> true <span class="GAPprompt">gap></span> <span class="GAPinput">phi;</span> <A homomorphism of left modules> <span class="GAPprompt">gap></span> <span class="GAPinput">C := HomalgComplex( phi );</span> <A non-zero acyclic complex containing a single morphism of left modules at de\ grees [ 0 .. 1 ]> <span class="GAPprompt">gap></span> <span class="GAPinput">Display( C );</span> ------------------------- at homology degree: 1 [ [ 2, 3, 4 ], [ 5, 6, 7 ] ] Cokernel of the map Z^(1x2) --> Z^(1x3), currently represented by the above matrix ------------------------- [ [ 0, 3, 6, 9 ], [ 0, 2, 4, 6 ], [ 0, 3, 6, 9 ] ] the map is currently represented by the above 3 x 4 matrix ------------v------------ at homology degree: 0 [ [ 2, 3, 4, 5 ], [ 6, 7, 8, 9 ] ] Cokernel of the map Z^(1x2) --> Z^(1x4), currently represented by the above matrix ------------------------- </pre></div> <p>And now:</p> <div class="example"><pre> <span class="GAPprompt">gap></span> <span class="GAPinput">ByASmallerPresentation( C );</span> <A non-zero acyclic complex containing a single morphism of left modules at de\ grees [ 0 .. 1 ]> <span class="GAPprompt">gap></span> <span class="GAPinput">Display( C );</span> ------------------------- at homology degree: 1 Z/< 3 > + Z^(1 x 1) ------------------------- [ [ 0, 0, 0 ], [ 2, 0, 0 ] ] the map is currently represented by the above 2 x 3 matrix ------------v------------ at homology degree: 0 Z/< 4 > + Z^(1 x 2) ------------------------- </pre></div> <div class="chlinkprevnextbot"> <a href="chap0_mj.html">[Top of Book]</a> <a href="chap0_mj.ht <div class="chlinkbot"><span class="chlink1">Goto Chapter: </span><a href="chap0_mj.html">Top< <hr /> <p class="foot">generated by <a href="http://www.math.rwth-aachen.de/~Frank.Luebeck/GAP </body> </html>
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