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<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>

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<p id="mathjaxlink" class="pcenter"><a href="chap8.html">[MathJax off]</a></p>
<p><a id="X7CEDAD61826170CF" name="X7CEDAD61826170CF"></a></p>
<div class="ChapSects"><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>

<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 structure is simply the addition of the known sign trick which induces the obvious equivalence between the category of bicomplexes and the category of complexes with complexes as objects and chain morphisms as morphisms. The majority of filtered complexes in algebra and geometry (unlike topology) arise as the total complex of a bicomplex. Hence, most spectral sequences in algebra are spectral sequences of bicomplexes. Indeed, bicomplexes in <strong class="pkg">homalg</strong> are mainly used as an input for the spectral sequence machinery.</p>

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

<h4>8.1 <span class="Heading">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="tdleft"><code class="func">‣ IsHomalgBicomplex</code>( <var class="Arg">BC</var> )</td><td class="tdright">( category )</td></tr></table></div>
<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> bi(co)complexes.</p>

<p>(It is a subcategory of the <strong class="pkg">GAP</strong> category <code class="code">IsHomalgObject</code>.)</p>

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

<h5>8.1-2 IsBicomplexOfFinitelyPresentedObjectsRep</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsBicomplexOfFinitelyPresentedObjectsRep</code>( <var class="Arg">BC</var> )</td><td class="tdright">( representation )</td></tr></table></div>
<p>Returns: <code class="code">true</code> or <code class="code">false</code></p>

<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 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</code>.)</p>

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

<h5>8.1-3 IsBicocomplexOfFinitelyPresentedObjectsRep</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsBicocomplexOfFinitelyPresentedObjectsRep</code>( <var class="Arg">BC</var> )</td><td class="tdright">( representation )</td></tr></table></div>
<p>Returns: <code class="code">true</code> or <code class="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>

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

<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">‣ HomalgBicomplex</code>( <var class="Arg">C</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Returns: a <strong class="pkg">homalg</strong> bicomplex</p>

<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>
<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 acyclic complex containing a single morphism of left cocomplexes a\
t degrees [ 0 .. 1 ]>
<span class="GAPprompt">gap></span> <span class="GAPinput">BC := HomalgBicomplex( CC );</span>
<A non-zero bicomplex containing left modules at bidegrees [ 0 .. 1 ]x
[ -1 .. 0 ]>
<span class="GAPprompt">gap></span> <span class="GAPinput">Display( BC );</span>
 * *
 * *
<span class="GAPprompt">gap></span> <span class="GAPinput">UU := UnderlyingComplex( BC );</span>
<A non-zero acyclic complex containing a single morphism of left cocomplexes a\
t degrees [ 0 .. 1 ]>
<span class="GAPprompt">gap></span> <span class="GAPinput">IsIdenticalObj( UU, CC );</span>
true
<span class="GAPprompt">gap></span> <span class="GAPinput">tBC := TransposedBicomplex( BC );</span>
<A non-zero bicomplex containing left modules at bidegrees [ -1 .. 0 ]x
[ 0 .. 1 ]>
<span class="GAPprompt">gap></span> <span class="GAPinput">Display( tBC );</span>
 * *
 * *
</pre></div>

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

<h4>8.3 <span class="Heading">Bicomplexes: Properties</span></h4>

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

<h5>8.3-1 IsBisequence</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsBisequence</code>( <var class="Arg">BC</var> )</td><td class="tdright">( property )</td></tr></table></div>
<p>Returns: <code class="code">true</code> or <code 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>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsBicomplex</code>( <var class="Arg">BC</var> )</td><td class="tdright">( property )</td></tr></table></div>
<p>Returns: <code class="code">true</code> or <code class="code">false</code></p>

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

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

<h5>8.3-3 IsTransposedWRTTheAssociatedComplex</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsTransposedWRTTheAssociatedComplex</code>( <var class="Arg">BC</var> )</td><td class="tdright">( property )</td></tr></table></div>
<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="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ TotalComplex</code>( <var class="Arg">BC</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>Returns: a <strong 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"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ SpectralSequence</code>( <var class="Arg">BC</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>Returns: a <strong class="pkg">homalg</strong> (co)homological spectral sequence</p>

<p>The associated spectral sequence.</p>

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

<h4>8.5 <span class="Heading">Bicomplexes: Operations and Functions</span></h4>

<p><a id="X7CE9470285B819BC" name="X7CE9470285B819BC"></a></p>

<h5>8.5-1 UnderlyingComplex</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ UnderlyingComplex</code>( <var class="Arg">BC</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Returns: a <strong class="pkg">homalg</strong> complex</p>

<p>The (co)complex of (co)complexes underlying the (co)homological bicomplex <var class="Arg">BC</var>.</p>

<p><a id="X7D4B66E08666B142" name="X7D4B66E08666B142"></a></p>

<h5>8.5-2 ByASmallerPresentation</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ ByASmallerPresentation</code>( <var class="Arg">B</var> )</td><td class="tdright">( method )</td></tr></table></div>
<p>Returns: a <strong class="pkg">homalg</strong> bicomplex</p>

<p>See <code class="func">ByASmallerPresentation</code> (<a href="chap6_mj.html#X79677A407C9EF3A0"><span class="RefLink">6.5-2</span></a>) on complexes.</p>


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

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


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