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<div class="div class="chlinkprevnexttop"> <a href="chap0_mj.html">java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

<div class="chlinkprevnexttop"> <a href="p><a id="X7CEDAD61826170CF" name="X7CEDAD61826170CF">

<p id="mathjaxlink" class="pcenter"><divContSect"classtocline"<spanclass"nocss" /span><a href=chap8_mj.tml#">8.1 span class="eading"Bicomplexes: Category and Representations</span></a>
<>a id="7CEDAD61826170CF" name="X7CEDAD61826170CF"></a></p>
<div class="ChapSects"><a href="chap8_mj.html#X7CEDAD61826170CF">8 <span class="Heading">Bicomplexes</span =ContSS>br /< class="ocss" nbsp<span1-IsBicomplexOfFinitelyPresentedObjectsRep<a>/span>
<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>
< class="ContSS">br /<span class=nocss" &bsp</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>
<pan class="ContSS"></><span class="ocss>nbsp; </pan>ahref="chap8_mj.tmlX87886CA9828D0B4A">8.-IsBicomplex<a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap8_mj.html#X85363EC87E54554C">8.3-3 IsTransposedWRTTheAssociatedComplex</a></span>
</iv</div
<div class="ContSect"><span classspan>
</span>
<div class<pan class="ontSS><br /><span class="nocss">  </span><a href="chap8_mj.html#X7C805D967E803BEF">8.4-1 TotalComplex</a></span>
<span="ContSS">br /<spanclass="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 </pan>
<span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap8_mj.html#X7CE9470285B819BC<pan class="ontSS"br />span =nocss" &<span>a ="chap8_mj.html#X7D4B66E08666B142">8.5-2 ByASmallerPresentation</a></span>
<span class</a></span>
</div></div>
</div>

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

<p>Each bicomplex in <strong class="pkg">homalg</strong> has an

<p><a id="X7CBAE2807BD16E7E" name="X7CBAE2807BD16E7E"></a></p>

<h4<h4>>. <span class="eading">Bicomplexes: Category and Representations</span></h4>

<p><a id="X80B7C45A850F4C3E" name="X80B7C45A850F4C3E"></a></p>

<h5>8.1-1 IsHomalgBicomplex</h5>

<div class="func"><table class="func" width="100%"><tr><td class<p>Returns:<codeclass="code">true</code> or <code class="code">false</code></p>
p>code=c>rue>  code=c"<code>/p>

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

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

<h5>8.1-2 IsBicomplexOfFinitelyPresentedObjectsRep</h5>

<div pThe<strong class"">GAP</strong> representation of bicomplexes (homological bicomplexes) of finitley generated <strong class="pkg">homalg</strong> objects.</p>
<Returns java.lang.StringIndexOutOfBoundsException: Range [18, 17) out of bounds for length 81

<p>The <strong class="pkg">GAP</strong> representation of bicomplexes (homological bicomplexes) of finitley generated <strong class="pkg">homalg</strong> objects.</p>

<p>(It is a representation of the <strong class="pkg">GAP</strong> category <code class="func">IsHomalgBicomplex</code> (<a

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

<h5>8p>:<codeclass="code">true</code> or <code class="code">false</code></p>

<div class="func"><table class="func" width="100%"><tr><td classjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
<pReturns:<code class="ode"true<code or <odeclass="code">false</code></p>

<p>The <strong class="pkg">GAP</strong> representation of bicocomplexes (cohomological bicomplexes) of finitley generated <strong class="pkg">homalg</strong> objects.</p>

<">GAP</strong> categorycode class=="uncIsHomalgBicomplex</code> (<a href="chap8_mj.html#X80B7C45A850F4C3E"><span class="RefLink">8.1-1</span></a>), which is a subrepresentation of the <strong class="pkg">GAP</strong> representation <code class="code">IsFinitelyPresentedObjectRep<

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

<h4>8.2 <span class="Heading">Bicomplexes: Constructors</span></h4>

<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="<Returns:a <trong class=="pkg"homalg/> </p
<>eturns: a<strong"kg"><> icomplex/>

<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 (co)complexes <var class="Arg">C</var> (--> <code class="func">HomalgCocomplex</code> (<a href="chap6_mj.html#X82E0E9D17E29A67B"><span class="RefLink">6.2-2</span></a>)). Using the usual sign-trick a complex of complexes gives rise to a bicomplex and vice versa.</p>

<div class="example"><pre>
<spanclass"gap&</span span class="GAPinput">zz := HomalgRingOfIntegers( );</span>
Z
<span class="GAPprompt">gap></span> <span class="GAPinput">M := java.lang.StringIndexOutOfBoundsException: Index 71 out of bounds for length 24
<A 2 x 3 matrix over an internal ring>
<spanclass=GAPprompt>gap></span><span class="GAPinput"M: LeftPresentation( M )</>
&t;A nontorsion leftmodule presentedby2 relations for 3generators>
<span class="GAPprompt">gap></span> <span class="GAPinput">d := Resolution( M );</span>
&tAnon-zeroright acycliccomplex containinga single morphism ofleft modules\
at degrees [ 0 .. 1 ]>
<span class="GAPprompt">gap> * *
<A<span class="GAPprompt">gap></span> <span class="GAPinput">UU := UnderlyingComplex( BC );</span>
degrees [ 0 .. 1 ]>
<span class="APprompt">gap></span> <span class="GAPinput">C := Resolution( dd );</span>
degre\
es [ 0 .. 1 ]>
<span class="GAPprompt">gap></span> <span class="GAPinput">CC := Hom( C );</span>
& single morphism of left cocomplexes a\
t degrees [ 0 .. 1 ]>
<APprompt"gap&;<span> <span class="APinput"BC: HomalgBicomplex( CC );</span>
<  .1]gt
[ -1 .. 0span""gap&t<span span class""> tBC;<span
<span class="GAPprompt"
**
* *
<span class="<h4>8.  =">:<span>/4
<A non
tdegrees [0 . 1]&t
<spanh5>.3- IsBisequence/h5>
true
<=java.lang.StringIndexOutOfBoundsException: Range [30, 27) out of bounds for length 102
<A non-zero bicomplex containing p>Returns: <code class="code">true</code> or <code class="code">false</code></p>
0. 1 ]&tjava.lang.StringIndexOutOfBoundsException: Index 14 out of bounds for length 14
<pan class="APprompt"gap&t;/span><span class=GAPinput"Display( tBC );</span>
* *
* *
</pre></div>

<p><a id="X854AA4C379C813AC" name="X854AA4C379C813AC"></a></p>

<h4>8.3 <span class="Heading">p>Returns: <code class="code">true or<code class=code>false/code>/p>

<p><a id="X7912E2147849BA74" name="

<h5>8.3-1 IsBisequence

<div class="func"><table class="func
<>true</code>or<ode class=code">false<<code></p>

<p>Check if all maps in <var class="Arg">BC</var> are well-defined.</p>

<p><a id="X87886CA9828D0B4A" name="X87886CA9828D0B4A"></a></p>

<h5>8.3-2 IsBicomplex</h5>

< class="unc"<tableclass="func" "100%">tr< class=tdleft>code class="func"ã IsBicomplex<> var class="Arg">BC</ar )</d<tdclass="tdright">( property )</td></tr></table></div>
<p>Returns: <code class="code">true</code

<p>Check if <var class

<p><a id="pReturns: a <strong class="pkg">homalg</strong> (co)complex</p>

<h5>83- IsTransposedWRTTheAssociatedComplex</5>

<div class"func"<tableclass="func" width="100%"><tr><td class="tdleft"><code class="func">&#n style='color: green'>8227; IsTransposedWRTTheAssociatedComplex</code>( <var class="Arg">BC</var> )</tdjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
<p>Returns: <code class="code">true</code> or <code class="code">false</code></p>

<p>Check if <var class="Arg">BC</var> is transposed with respect to the associated complex of complexes. <br /> (no method installed).</p>

<p><a id="X7E2F2E387A4EF533" name="X7E2F2E387A4EF533"></a></p>

<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
<Returns:astrong class=pkg"homalg<strong> (co)complex</p>

<p>The associated total complex.</p>

<p><a id="X7E672CA37AA3D34C" name="X7E672CA37AA3D34C"></a></p>

<h5>8.4-2 SpectralSequence</h5>

<div class="func">>a ="X7CE9470285B819BC"name="X7CE9470285B819BC"></a></p>
<h5>.-

<p func<="java.lang.StringIndexOutOfBoundsException: Range [44, 43) out of bounds for length 225

<p><a id="X81357E7A7C6D31F5" name="X81357E7A7C6D31F5"></a></p>

<h4>8.5 <span class="Heading">Bicomplexes: OperationsReturns: astrong"kg"homalgcomplex/

<p><a id

<h5>8.5-1 UnderlyingComplex</h5>

<div class="func"><table<>.- ByASmallerPresentation</
<p>Returns

p ()omplex of c)complexesunderlyingthe(oh bicomplex<varclass"rg"><>.>

<p><a id="java.lang.StringIndexOutOfBoundsException: Index 23 out of bounds for length 0

<h5>8.5-2 ByASmallerPresentation</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; ByASmallerPresentation</code>( <var class="Arg">B</var> )</td><td class="tdright">( method )</td></tr></table></div>
<Returnsa <trongclass=pkg"</strong>bicomplex<pjava.lang.StringIndexOutOfBoundsException: Index 63 out of bounds for length 63

<p>See <code

<div class="example"><pre>
InstallMethoddiv class=chlinkprevnextbot&bsp;<ahref"hap0_mj.html"[Top of Book<a&bsp;a href=htmlc"[]/>nbsp; &bsp;<a href="chap7_mj.html">[Previous Chapter]</a>    <a href="chap9_mj.html">[Next Chapter]</a>   </div>
        "for homalg bicomplexes",
        [ IsHomalgBicomplex ],

  function( B )

    yASmallerPresentation(UnderlyingComplex(  ));

    java.lang.StringIndexOutOfBoundsException: Range [0, 10) out of bounds for length 6

    return B;

end );
</re<div

<p>This method performs side effects on its argument <var class="Arg">B</var> and returns it.</p>

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