<p>Spectral sequences are regarded as the computational sledgehammer in homological algebra. Quoting the last lines of Rotman's book [Rot79]:
<p>"The reader should now be convinced that virtually every purely homological result may be proved with spectral sequences. Even though "elementary" proofs may exist for many of these results, spectral sequences offer a systematic approach in place of sporadic success."</p>
<p>The <strong class="pkg">GAP</strong> category of <strong class="pkg">homalg</strong> associated to a filtered complex.</p>
<p>(It is a subcategory of the <strong class="pkg">GAP</strong> category <code class="code">IsHomalgSpectralSequence</code>.) <br /> <br /> The <span class="SimpleMath">\(0\)</span>-th spectral sheet <span class="SimpleMath">\(E_0\)</span> stemming from a filtration is a bigraded (differential) object, which, in general, does not stem from an exact couple (although <span class="SimpleMath">\(E_1\)</span>, <span class="SimpleMath">\(E_2\)</span>, ... do).</p>
<p>The <strong class="pkg">GAP</strong> category of <strong class="pkg">homalg</strong> associated to a bicomplex.</p>
<p>(It is a subcategory of the <strong class="pkg">GAP</strong> category <br /> <code class="code">IsHomalgSpectralSequenceAssociatedToAFilteredComplex</code>.)</p>
<p>The <strong class="pkg">GAP</strong> representation of homological spectral sequences 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">IsHomalgSpectralSequence</code> (<a href="chap10_mj.html#X795DCCD88630BA47"><span class="RefLink">10.1-1</span></a>), which is a subrepresentation of the <strong class="pkg">GAP</strong> representation <code class="code">IsFinitelyPresentedObjectRep</code>.)</p>
<p>The <strong class="pkg">GAP</strong> representation of cohomological spectral sequences 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">IsHomalgSpectralSequence</code> (<a href="chap10_mj.html#X795DCCD88630BA47"><span class="RefLink">10.1-1</span></a>), which is a subrepresentation of the <strong class="pkg">GAP</strong> representation <code class="code">IsFinitelyPresentedObjectRep</code>.)</p>
<p>The first syntax is the main constructor. It creates the homological (resp. cohomological) spectral sequence associated to the homological (resp. cohomological) bicomplex <var class="Arg">B</var> starting at level <span class="SimpleMath">\(0\)</span> and ending at level <var class="Arg">r</var><span class="SimpleMath">\(\geq 0\)</span> (regardless if the spectral sequence stabilizes earlier). The generalized embeddings into the objects of 0-th sheet are always computed for each higher sheet <span class="SimpleMath">\(Er\)</span> and stored as a record under the component <span class="SimpleMath">\(Er\)</span>!.absolute_embeddings. If <var class="Arg">a</var> is greater than <span class="SimpleMath">\(0\)</span> the generalized embeddings into the objects of the <var class="Arg">a</var>-th sheet also get computed for each higher sheet <span class="SimpleMath">\(Er\)</span> and stored as a record under the component <span class="SimpleMath">\(Er\)</span>!.relative_embeddings. The level <var class="Arg">a</var> at which the spectral sequence becomes intrinsic is a natural candidate for <var class="Arg">a</var>. The <var class="Arg">a</var>-th sheet is called the <em>special</em> sheet.</p>
<p>If <var class="Arg">r</var><span class="SimpleMath">\(=-1\)</span> it computes all the sheets of the spectral sequence until the sequence stabilizes, i.e. until all higher arrows become zero.</p>
<p>If <var class="Arg">a</var><span class="SimpleMath">\(=-1\)</span> no special sheet is specified.</p>
<p>In the second syntax <var class="Arg">a</var> is set to <span class="SimpleMath">\(-1\)</span>.</p>
<p>In the third syntax <var class="Arg">r</var> is set to <span class="SimpleMath">\(-1\)</span>.</p>
<p>In the fourth syntax both <var class="Arg">r</var> and <var class="Arg">a</var> are set to <span class="SimpleMath">\(-1\)</span>.</p>
<p>The following example demonstrates the computation of a <span class="SimpleMath">\(Tor-Ext\)</span> spectral sequence:</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>
<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">dM := 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">CC := Hom( dM, dM );</span>
<A non-zero acyclic cocomplex containing a single morphism of right complexes \
at degrees [ 0 .. 1 ]>
<span class="GAPprompt">gap></span> <span class="GAPinput">B := HomalgBicomplex( CC );</span>
<A non-zero bicocomplex containing right modules at bidegrees [ 0 .. 1 ]x
[ -1 .. 0 ]>
</pre></div>
<p>Now we construct the spectral sequence associated to the bicomplex <span class="SimpleMath">\(B\)</span>, also called the <em>first</em> spectral sequence:</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">I_E := HomalgSpectralSequence( 2, B );</span>
<A stable cohomological spectral sequence with sheets at levels
[ 0 .. 2 ] each consisting of right modules at bidegrees [ 0 .. 1 ]x
[ -1 .. 0 ]>
<span class="GAPprompt">gap></span> <span class="GAPinput">Display( I_E );</span>
a cohomological spectral sequence at bidegrees
[ [ 0 .. 1 ], [ -1 .. 0 ] ]
---------
Level 0:
* *
* *
---------
Level 1:
* *
. .
---------
Level 2:
s s
. .
</pre></div>
<p>Legend:</p>
<ul>
<li><p>A star <var class="Arg">*</var> stands for a nonzero object.</p>
</li>
<li><p>A dot <var class="Arg">.</var> stands for a zero object.</p>
</li>
<li><p>The letter <var class="Arg">s</var> stands for a nonzero object that became stable.</p>
</li>
</ul>
<p>The <em>second</em> spectral sequence of the bicomplex is, by definition, the spectral sequence associated to the transposed bicomplex:</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">tB := TransposedBicomplex( B );</span>
<A non-zero bicocomplex containing right modules at bidegrees [ -1 .. 0 ]x
[ 0 .. 1 ]>
<span class="GAPprompt">gap></span> <span class="GAPinput">II_E := HomalgSpectralSequence( tB, 2 );</span>
<A stable cohomological spectral sequence with sheets at levels
[ 0 .. 2 ] each consisting of right modules at bidegrees [ -1 .. 0 ]x
[ 0 .. 1 ]>
<span class="GAPprompt">gap></span> <span class="GAPinput">Display( II_E );</span>
a cohomological spectral sequence at bidegrees
[ [ -1 .. 0 ], [ 0 .. 1 ] ]
---------
Level 0:
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