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<!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 (GaussForHomalg) - Chapter 4: Example</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="chap4" onload="jscontent()"> <div class="chlinktop"><span class="chlink1">Goto Chapter: </span><a href="chap0.html">Top</a> <a <div class="chlinkprevnexttop"> <a href="chap0.html">[Top of Book]</a> <a href="chap0.html#con <p id="mathjaxlink" class="pcenter"><a href="chap4_mj.html">[MathJax on]</a></p> <p><a id="X85861B017AEEC50B" name="X85861B017AEEC50B"></a></p> <div class="ChapSects"><a href="chap4.html#X85861B017AEEC50B">4 <span class="Heading">Exam <div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap4.html#X7 </span> </div> </div> <h3>4 <span class="Heading">Example</span></h3> <p><a id="X791E21F47805048A" name="X791E21F47805048A"></a></p> <h4>4.1 <span class="Heading">HomHom</span></h4> <p>The following example is taken from Section 2 of <a href="chapBib.html#biBBREACA">[BR06]</a>. < <p>Here we compute the (infinite) long exact homology sequence of the covariant functor <span cla <div class="example"><pre> <span class="GAPprompt">gap></span> <span class="GAPinput">LoadPackage( "Modules", ">= 2023.1 true <span class="GAPprompt">gap></span> <span class="GAPinput">R := HomalgRingOfIntegers( 2^8 );</s Z/256Z <span class="GAPprompt">gap></span> <span class="GAPinput">Display( R );</span> <An internal ring> <span class="GAPprompt">gap></span> <span class="GAPinput">M := LeftPresentation( [ 2^5 ], R );</s <A cyclic left module presented by 1 relation for a cyclic generator> <span class="GAPprompt">gap></span> <span class="GAPinput">Display( M );</span> Z/256Z/< ZmodnZObj(32,256) > <span class="GAPprompt">gap></span> <span class="GAPinput">M;</span> <A cyclic left module presented by 1 relation for a cyclic generator> <span class="GAPprompt">gap></span> <span class="GAPinput">_M := LeftPresentation( [ 2^3 ], R );</ <A cyclic left module presented by 1 relation for a cyclic generator> <span class="GAPprompt">gap></span> <span class="GAPinput">Display( _M );</span> Z/256Z/< ZmodnZObj(8,256) > <span class="GAPprompt">gap></span> <span class="GAPinput">_M;</span> <A cyclic left module presented by 1 relation for a cyclic generator> <span class="GAPprompt">gap></span> <span class="GAPinput">alpha2 := HomalgMap( [ 1 ], M, _M );</sp <A "homomorphism" of left modules> <span class="GAPprompt">gap></span> <span class="GAPinput">IsMorphism( alpha2 );</span> true <span class="GAPprompt">gap></span> <span class="GAPinput">alpha2;</span> <A homomorphism of left modules> <span class="GAPprompt">gap></span> <span class="GAPinput">Display( alpha2 );</span> 1 the map is currently represented by the above 1 x 1 matrix <span class="GAPprompt">gap></span> <span class="GAPinput">M_ := Kernel( alpha2 );</span> <A cyclic left module presented by yet unknown relations for a cyclic generato\ r> <span class="GAPprompt">gap></span> <span class="GAPinput">alpha1 := KernelEmb( alpha2 );</span> <A monomorphism of left modules> <span class="GAPprompt">gap></span> <span class="GAPinput">seq := HomalgComplex( alpha2 );</spa <An acyclic complex containing a single morphism of left modules at degrees [ 0 .. 1 ]> <span class="GAPprompt">gap></span> <span class="GAPinput">Add( seq, alpha1 );</span> <span class="GAPprompt">gap></span> <span class="GAPinput">seq;</span> <A sequence containing 2 morphisms of left modules at degrees [ 0 .. 2 ]> <span class="GAPprompt">gap></span> <span class="GAPinput">IsShortExactSequence( seq );</span> true <span class="GAPprompt">gap></span> <span class="GAPinput">seq;</span> <A short exact sequence containing 2 morphisms of left modules at degrees [ 0 .. 2 ]> <span class="GAPprompt">gap></span> <span class="GAPinput">Display( seq );</span> ------------------------- at homology degree: 2 Z/256Z/< ZmodnZObj(4,256) > ------------------------- 8 the map is currently represented by the above 1 x 1 matrix ------------v------------ at homology degree: 1 Z/256Z/< ZmodnZObj(32,256) > ------------------------- 1 the map is currently represented by the above 1 x 1 matrix ------------v------------ at homology degree: 0 Z/256Z/< ZmodnZObj(8,256) > ------------------------- <span class="GAPprompt">gap></span> <span class="GAPinput">K := LeftPresentation( [ 2^7 ], R );</s <A cyclic left module presented by 1 relation for a cyclic generator> <span class="GAPprompt">gap></span> <span class="GAPinput">L := RightPresentation( [ 2^4 ], R );</ <A cyclic right module on a cyclic generator satisfying 1 relation> <span class="GAPprompt">gap></span> <span class="GAPinput">triangle := LHomHom( 4, seq, K, L, "t" <An exact triangle containing 3 morphisms of left complexes at degrees [ 1, 2, 3, 1 ]> <span class="GAPprompt">gap></span> <span class="GAPinput">lehs := LongSequence( triangle );</s <A sequence containing 14 morphisms of left modules at degrees [ 0 .. 14 ]> <span class="GAPprompt">gap></span> <span class="GAPinput">ByASmallerPresentation( lehs );</s <A non-zero sequence containing 14 morphisms of left modules at degrees [ 0 .. 14 ]> <span class="GAPprompt">gap></span> <span class="GAPinput">IsExactSequence( lehs );</span> false <span class="GAPprompt">gap></span> <span class="GAPinput">lehs;</span> <A non-zero left acyclic complex containing 14 morphisms of left modules at degrees [ 0 .. 14 ]> <span class="GAPprompt">gap></span> <span class="GAPinput">Assert( 0, IsLeftAcyclic( lehs ) );</ <span class="GAPprompt">gap></span> <span class="GAPinput">Display( lehs );</span> ------------------------- at homology degree: 14 Z/256Z/< ZmodnZObj(4,256) > ------------------------- 4 the map is currently represented by the above 1 x 1 matrix ------------v------------ at homology degree: 13 Z/256Z/< ZmodnZObj(8,256) > ------------------------- 2 the map is currently represented by the above 1 x 1 matrix ------------v------------ at homology degree: 12 Z/256Z/< ZmodnZObj(8,256) > ------------------------- 2 the map is currently represented by the above 1 x 1 matrix ------------v------------ at homology degree: 11 Z/256Z/< ZmodnZObj(4,256) > ------------------------- 4 the map is currently represented by the above 1 x 1 matrix ------------v------------ at homology degree: 10 Z/256Z/< ZmodnZObj(8,256) > ------------------------- 2 the map is currently represented by the above 1 x 1 matrix ------------v------------ at homology degree: 9 Z/256Z/< ZmodnZObj(8,256) > ------------------------- 2 the map is currently represented by the above 1 x 1 matrix ------------v------------ at homology degree: 8 Z/256Z/< ZmodnZObj(4,256) > ------------------------- 4 the map is currently represented by the above 1 x 1 matrix ------------v------------ at homology degree: 7 Z/256Z/< ZmodnZObj(8,256) > ------------------------- 2 the map is currently represented by the above 1 x 1 matrix ------------v------------ at homology degree: 6 Z/256Z/< ZmodnZObj(8,256) > ------------------------- 2 the map is currently represented by the above 1 x 1 matrix ------------v------------ at homology degree: 5 Z/256Z/< ZmodnZObj(4,256) > ------------------------- 4 the map is currently represented by the above 1 x 1 matrix ------------v------------ at homology degree: 4 Z/256Z/< ZmodnZObj(8,256) > ------------------------- 2 the map is currently represented by the above 1 x 1 matrix ------------v------------ at homology degree: 3 Z/256Z/< ZmodnZObj(8,256) > ------------------------- 2 the map is currently represented by the above 1 x 1 matrix ------------v------------ at homology degree: 2 Z/256Z/< ZmodnZObj(4,256) > ------------------------- 8 the map is currently represented by the above 1 x 1 matrix ------------v------------ at homology degree: 1 Z/256Z/< ZmodnZObj(16,256) > ------------------------- 1 the map is currently represented by the above 1 x 1 matrix ------------v------------ at homology degree: 0 Z/256Z/< ZmodnZObj(8,256) > ------------------------- </pre></div> <div class="chlinkprevnextbot"> <a href="chap0.html">[Top of Book]</a> <a href="chap0.html#con <div class="chlinkbot"><span class="chlink1">Goto Chapter: </span><a href="chap0.html">Top</a> <a <hr /> <p class="foot">generated by <a href="https://www.math.rwth-aachen.de/~Frank.Luebeck/GA </body> </html> ¤ Dauer der Verarbeitung: 0.12 Sekunden (vorverarbeitet) ¤*© Formatika GbR, Deutschland |
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