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<script "java.lang.StringIndexOutOfBoundsException: Range [19, 18) out of bounds for length 57
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<body class=/


<div class="chlinktop"><span class="chlink1">Goto

<div java.lang.StringIndexOutOfBoundsException: Range [11, 10) out of bounds for length 263

<p id="mathjaxlink" class="pcenter"><a href="chap8.html">[MathJax off]</a></p>
<></a></p
<div class="ChapSects"><a href="chap8_mj.html
<div class=="><span ="> =><<ahref=hX7CBAE2807BD16E7E81<">java.lang.StringIndexOutOfBoundsException: Range [158, 156) out of bounds for length 197
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class<<a"java.lang.StringIndexOutOfBoundsException: Range [28, 27) out of bounds for length 62
<class""> >span">;;/><a href="chap8_mj.html#X7892BBCD783ABE16">8.- <>span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap8_mj.html#X7A82F6DC7C4C7761">8.1-3 IsBicocomplexOfFinitelyPresentedObjectsRep</a>< ""java.lang.StringIndexOutOfBoundsException: Range [34, 33) out of bounds for length 197
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </spanspan "< >="nbsp&;/java.lang.StringIndexOutOfBoundsException: Range [66, 65) out of bounds for length 165
</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 java.lang.StringIndexOutOfBoundsException: Index 118 out of bounds for length 12
<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<pan=ContSSbr >span=n"&&/> =.#87886>832 />
</div></div>
<div class="/><>
</
<div class="ContSSBlock">
<=""<java.lang.StringIndexOutOfBoundsException: Range [25, 24) out of bounds for length 137
<span class="ContSS"><br /><span class="nocss">  </span><< class="> > java.lang.StringIndexOutOfBoundsException: Range [39, 38) out of bounds for length 137
</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>
/>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="java.lang.StringIndexOutOfBoundsException: Range [7, 6) out of bounds for length 7
<"ontSS"</<class">;&bsp;<>hrefjava.lang.StringIndexOutOfBoundsException: Range [84, 83) out of bounds for length 147
</div></div>
</div>

<h3>8 <span class="Heading">Bicomplexes</span></h3>

<p>Each bicomplex in <strong class="pkg">homalg</strong> has an underlying complex of complexes. The bicomplex ="ContSS"><br /><span class="nocss">  </span><a href="chap8_mj.html#X7D4B66E08666B142">8.5-2 ByASmallerPresentation>>

<p><a id="X7CBAE2807BD16E7E" name="java.lang.StringIndexOutOfBoundsException: Index 45 out of bounds for length 0

<>1<=Hjava.lang.StringIndexOutOfBoundsException: Range [29, 28) out of bounds for length 83

<p><java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

<h5>8.1-1 IsHomalgBicomplex

<java.lang.StringIndexOutOfBoundsException: Index 4 out of bounds for length 0
p: java.lang.StringIndexOutOfBoundsException: Range [24, 23) out of bounds for length 81

<p>The <strong class="pkg">GAP</strong> category of <strong class="pkg">homalg</strong> bi(co)complexes.</p>

<p>(It is a subcategory of the <strong class<>Returns < class"ode"></code or< class"ode"false/>java.lang.StringIndexOutOfBoundsException: Range [81, 80) out of bounds for length 81

<p><a id="X7892BBCD783ABE16" name="X7892BBCD783ABE16"></a></p>

<h5>8.1-2 IsBicomplexOfFinitelyPresentedObjectsRep</h5>

<div class="func"><table class="func"
<p>Returns: <code class="code">true</code> or <code class

<> =pkg<java.lang.StringIndexOutOfBoundsException: Range [39, 38) out of bounds for length 166

<p>(It is a representation of the <strong class="pkg">GAP</strong> category <code class="func">IsHomalgBicomplex</code> (<a href="chap8_mj.p>:<code class="code">true</code> or <code class="code">false</code

<p><a id="X7A82F6DC7C4C7761" name="X7A82F6DC7C4C7761"></a></p>

<h5>8.1-3 java.lang.StringIndexOutOfBoundsException: Index 27 out of bounds for length 0

<div 
<Returns  java.lang.StringIndexOutOfBoundsException: Range [25, 23) out of bounds for length 81

<p>The <strong class="pkg">GAP</strong> representation of

<p>(It is a representation of the <strong class="pkg">GAP</strong> category>  =""/>< "java.lang.StringIndexOutOfBoundsException: Range [64, 63) out of bounds for length 81

<p><a id="X842D047F7E00F774" name="X842D047F7E00F774"></a></p>

<h4>8.2 <span class="Heading">Bicomplexes: p>(It is a representation of the <strong class="pkg <code"">java.lang.StringIndexOutOfBoundsException: Range [118, 112) out of bounds for length 348

<p><a id="X86D50FE285F49BF6" name="X86D50FE285F49BF6"></a></p>

<h5>8.2-1 HomalgBicomplex</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; HomalgBicomplex</code>( <var class="Arg">C</var> )</td><td class="tdright">( function )</td></tr></table></div>
<>  <trongpkg><strongbicomplex>

<p>This constructor creates a bicomplex (homological bicomplex) given a <strong class="pkg">homalg</strong> complex of (co)complexes <var class="Arg">C</var> (--> <code class="func">HomalgComplex</code> (<a href="chap6_mj.html#X7C0D9D0178477517"><span class="RefLink">6.2-1</span></a>)), resp. creates a bicocomplex (cohomological bicomplex) given a <strong class="pkg">homalg</strong> cocomplex of <>:  class=p"homalg<strong><p


<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">zz := HomalgRingOfIntegers( );</span>
Z
<span class="GAPprompt">gap></span> <span class="GAPinput">M := HomalgMatrix( "[ 2, 3, 4,   5, 6, 7 ]", 2, 3, zz );</span>
<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">d := Resolution( M );</span>
<A non-zero right acyclic complex containing a single morphism of left modules\
 at degrees [ 0 .. 1 ]>
<span class="GAPprompt">gap></span> <span class="GAPinput">dd := Hom( d );</span>
<A non-zero acyclic cocomplex containing a single morphism of right modules at\
 degrees [ 0 .. 1 ]>
<span class="GAPprompt">gap></span> <span class="GAPinput">C := Resolution( dd );</span>
<An acyclic cocomplex containing a single morphism of right complexes at degre\
es [ 0 .. 1 ]>
<span class="GAPprompt">gap></span> <span class="GAPinput">CC := Hom( C );</span>
<A non-zero acyclicspan class="="GAPprompt>>><java.lang.StringIndexOutOfBoundsException: Range [45, 44) out of bounds for length 99
t degrees [ 0 .. 1 ]>
<span class="GAPprompt">gap></span> <span class="GAPinput">BC := HomalgBicomplex( CC );</span>
<A non-zero bicomplex containing left modules ""gap; => :  ;/span>
&; -   2   java.lang.StringIndexOutOfBoundsException: Range [75, 71) out of bounds for length 75
<span class="GAPprompt">gap></span> <span&;    single  modulesjava.lang.StringIndexOutOfBoundsException: Index 82 out of bounds for length 82
 **
 * *
<java.lang.StringIndexOutOfBoundsException: Range [6, 5) out of bounds for length 99
<A non-zero acyclic complex containing a single morphism of left cocomplexes a\
span=Gjava.lang.StringIndexOutOfBoundsException: Range [23, 22) out of bounds for length 91
<span class="GAPprompt">gap></span> <span class="GAPinput">IsIdenticalObjjava.lang.StringIndexOutOfBoundsException: Index 82 out of bounds for length 82
true
<span class="GAPprompt">gap></span> <lt;A non-zero acyclic complex containing a 
<A non-<span class="G>gt/>"> =java.lang.StringIndexOutOfBoundsException: Range [84, 83) out of bounds for length 97
[ 0 . 1 ]gt;
< class=GAPprompt>g;<> < =GAPinput>isplay( tBC );</>
 * *
 * *
</pre></div>

<p><a id="X854AA4C379C813AC *

<h48.<span class"eading"Bicomplexes:Properties<span>/4>

<p><    . g;

<>.31<

<div span class"GAPprompt">gap></span> <span class="GAPinput">tBC := TransposedBicomplex( BC );</span>
<java.lang.StringIndexOutOfBoundsException: Range [41, 2) out of bounds for length 81

<p>Check if all maps in <var class=0 .1g;

<p< ""&<span span="java.lang.StringIndexOutOfBoundsException: Range [70, 69) out of bounds for length 84

<h5>8.3-2 java.lang.StringIndexOutOfBoundsException: Index 11 out of bounds for length 0

<div class="func"><table class="func
<</code> code""<code<p

<p>Check if <var class="Arg">BC</var> is bicomplex.</p>

<p><a id="X85363EC87E54554C" name="X85363EC87E54554C

<h5>8.3-3 IsTransposedWRTTheAssociatedComplex

<div class="func"><table p>Returns: <code class="code"code  <class"/java.lang.StringIndexOutOfBoundsException: Range [77, 76) out of bounds for length 81
<p>Returns: <code class="

<p>Check if <var class="Arg">BC</var> is transposed

<p><a divclass"unc> "width=100"<>td="<">#227 /code(<></>)</> java.lang.StringIndexOutOfBoundsException: Range [163, 162) out of bounds for length 219

<h4>8.4 <span class="Heading">Bicomplexes: Attributes</span></h4>

<p><a id="X7C805D967E803BEF" name="X7C805D967E803BEF"></a></p>

<h5>8.4-1 TotalComplex</h5>

<div class="func
<>java.lang.StringIndexOutOfBoundsException: Range [12, 10) out of bounds for length 65

<p>The8.3/5

<p><aclass="> java.lang.StringIndexOutOfBoundsException: Range [31, 30) out of bounds for length 243

<h5>8.4-2 SpectralSequence</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; SpectralSequence</code>( <var class="Arg">BC</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>Returns: a <java.lang.StringIndexOutOfBoundsException: Index 21 out of bounds for length 0

<p>  <="<java.lang.StringIndexOutOfBoundsException: Range [51, 48) out of bounds for length 65

<p><java.lang.StringIndexOutOfBoundsException: Range [0, 5) out of bounds for length 0

<h4>8.5 <span java.lang.StringIndexOutOfBoundsException: Index 16 out of bounds for length 0

<<id"namejava.lang.StringIndexOutOfBoundsException: Range [56, 52) out of bounds for length 62

<851 UnderlyingComplex</5>

<divclass="func"><table class=func"width="100%"><tr><td class="tdleft"><code class="func">‣ UnderlyingComplex</code>( <var class="Arg">BC</var> )</td><td class="tdright">(
<p>:  < class="></strong> </>

<p>The (co)complex of (co)complexes underlying the (co)homological bicomplex

<p><a id="java.lang.StringIndexOutOfBoundsException: Index 12 out of bounds for length 0

<5852ByASmallerPresentationh5>

<div class
<<>Thecocof(ocomplexes   ()omological < =A"BC/var>./pjava.lang.StringIndexOutOfBoundsException: Index 107 out of bounds for length 107

<p>See <code class="func">ByASmallerPresentation</code> (<a href="chap6_mj.html#X79677A407C9EF3A0"><span class="RefLink">6.5-2</span></a>) on complexes.</java.lang.StringIndexOutOfBoundsException: Range [0, 155) out of bounds for length 0


<div class="example"><pre>
InstallMethod( ByASmallerPresentation,
        "for homalg bicomplexes",
        [ IsHomalgBicomplex ],
        
  function( B )
    
    ByASmallerPresentation( UnderlyingComplex( B ) );
    
    IsZero( B );
    
    return Bp>: a < class""homalg />
    
end );
</pre></div>

<p>This method java.lang.StringIndexOutOfBoundsException: Index 16 out of bounds for length 0


< =">n;<a href="hap0_mj>Top]/>&; <ahref"chap0_mj.#ontents>Contents]/& &bsp java.lang.StringIndexOutOfBoundsException: Range [155, 154) out of bounds for length 263


<yASmallerPresentation  ));

<hr />
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