Editing Spectral sequence (section)

=== The spectral sequence of a filtered complex ===

A very common type of spectral sequence comes from a [[Filtration (abstract algebra)|filtered]] cochain complex, as it naturally induces a bigraded object. Consider a cochain complex <math> (C^{\bullet}, d) </math> together with a descending filtration,  <math display="inline"> ... \supset\, F^{-2}C^{\bullet} \,\supset\, F^{-1}C^{\bullet} \supset F^{0}C^{\bullet} \,\supset\, F^{1}C^{\bullet} \,\supset\, F^{2}C^{\bullet} \,\supset\, F^{3}C^{\bullet} \,\supset... \, </math> . We require that the boundary map is compatible with the filtration, i.e. <math display="inline"> d(F^pC^n) \subset F^pC^{n+1}</math>, and that the filtration is ''exhaustive'', that is, the union of the set of all <math display="inline">F^pC^{\bullet}</math> is the entire chain complex <math display="inline">C^{\bullet}</math>. Then there exists a spectral sequence with <math display="inline"> E_0^{p,q} = F^{p}C^{p+q}/F^{p+1}C^{p+q} </math> and <math display="inline"> E_1^{p,q} = H^{p+q}(F^{p}C^{\bullet}/F^{p+1}C^{\bullet}) </math>.<ref>{{citation|surname1=[[Serge Lang]]|title=Algebra|edition=Überarbeitete 3.|series=Graduate Texts in Mathematics 211|publisher=[[Springer Science+Business Media|Springer-Verlag]]|publication-place=New York|isbn=038795385X|date=2002|language=German
}}</ref> Later, we will also assume that the filtration is ''Hausdorff'' or ''separated'', that is, the intersection of the set of all <math display="inline">F^pC^{\bullet}</math> is zero.

The filtration is useful because it gives a measure of nearness to zero: As ''p'' increases, <math display="inline">F^pC^{\bullet}</math>  gets closer and closer to zero.  We will construct a spectral sequence from this filtration where coboundaries and cocycles in later sheets get closer and closer to coboundaries and cocycles in the original complex.  This spectral sequence is doubly graded by the filtration degree ''p'' and the complementary degree {{math|1=''q'' = ''n'' &minus; ''p''}}.

==== Construction ====
<math> C^{\bullet} </math> has only a single grading and a filtration, so we first construct a doubly graded object for the first page of the spectral sequence. To get the second grading, we will take the associated graded object with respect to the filtration.  We will write it in an unusual way which will be justified at the <math> E_1 </math> step:

:<math>Z_{-1}^{p,q} = Z_0^{p,q} = F^p C^{p+q}</math>
:<math>B_0^{p,q} = 0</math>
:<math>E_0^{p,q} = \frac{Z_0^{p,q}}{B_0^{p,q} + Z_{-1}^{p+1,q-1}} = \frac{F^p C^{p+q}}{F^{p+1} C^{p+q}}</math>
:<math>E_0 = \bigoplus_{p,q\in\mathbf{Z}} E_0^{p,q}</math>

Since we assumed that the boundary map was compatible with the filtration, <math> E_0 </math> is a doubly graded object and there is a natural doubly graded boundary map <math> d_0 </math> on <math> E_0 </math>.  To get <math> E_1 </math>, we take the homology of <math> E_0 </math>.

:<math>\bar{Z}_1^{p,q} = \ker d_0^{p,q} : E_0^{p,q} \rightarrow E_0^{p,q+1} = \ker d_0^{p,q} : F^p C^{p+q}/F^{p+1} C^{p+q} \rightarrow F^p C^{p+q+1}/F^{p+1} C^{p+q+1}</math>
:<math>\bar{B}_1^{p,q} = \mbox{im } d_0^{p,q-1} : E_0^{p,q-1} \rightarrow E_0^{p,q} = \mbox{im } d_0^{p,q-1} : F^p C^{p+q-1}/F^{p+1} C^{p+q-1} \rightarrow F^p C^{p+q}/F^{p+1} C^{p+q}</math>
:<math>E_1^{p,q} = \frac{\bar{Z}_1^{p,q}}{\bar{B}_1^{p,q}} = \frac{\ker d_0^{p,q} : E_0^{p,q} \rightarrow E_0^{p,q+1}}{\mbox{im } d_0^{p,q-1} : E_0^{p,q-1} \rightarrow E_0^{p,q}}</math>
:<math>E_1 = \bigoplus_{p,q\in\mathbf{Z}} E_1^{p,q} = \bigoplus_{p,q\in\mathbf{Z}} \frac{\bar{Z}_1^{p,q}}{\bar{B}_1^{p,q}}</math>

Notice that <math>\bar{Z}_1^{p,q}</math> and <math>\bar{B}_1^{p,q}</math> can be written as the images in <math>E_0^{p,q}</math> of
 
:<math>Z_1^{p,q} = \ker d_0^{p,q} : F^p C^{p+q} \rightarrow C^{p+q+1}/F^{p+1} C^{p+q+1}</math>
:<math>B_1^{p,q} = (\mbox{im } d_0^{p,q-1} : F^p C^{p+q-1} \rightarrow C^{p+q}) \cap F^p C^{p+q}</math>

and that we then have

:<math>E_1^{p,q} = \frac{Z_1^{p,q}}{B_1^{p,q} + Z_0^{p+1,q-1}}.</math>

<math>Z_1^{p,q}</math> are exactly the elements which the differential pushes up one level in the filtration, and <math>B_1^{p,q}</math> are exactly the image of the elements which the differential pushes up zero levels in the filtration.  This suggests that we should choose <math>Z_r^{p,q}</math> to be the elements which the differential pushes up ''r'' levels in the filtration and <math>B_r^{p,q}</math> to be image of the elements which the differential pushes up ''r-1'' levels in the filtration. In other words, the spectral sequence should satisfy

:<math>Z_r^{p,q} = \ker d_0^{p,q} : F^p C^{p+q} \rightarrow C^{p+q+1}/F^{p+r} C^{p+q+1}</math>
:<math>B_r^{p,q} = (\mbox{im } d_0^{p-r+1,q+r-2} : F^{p-r+1} C^{p+q-1} \rightarrow C^{p+q}) \cap F^p C^{p+q}</math>
:<math>E_r^{p,q} = \frac{Z_r^{p,q}}{B_r^{p,q} + Z_{r-1}^{p+1,q-1}}</math>

and we should have the relationship

:<math>B_r^{p,q} = d_0^{p,q}(Z_{r-1}^{p-r+1,q+r-2}).</math>

For this to make sense, we must find a differential <math> d_r </math> on each <math> E_r </math> and verify that it leads to homology isomorphic to <math> E_{r+1} </math>.  The differential 
 
:<math>d_r^{p,q} : E_r^{p,q} \rightarrow E_r^{p+r,q-r+1}</math>

is defined by restricting the original differential <math> d </math> defined on <math>C^{p+q}</math> to the subobject <math>Z_r^{p,q}</math>. It is straightforward to check that the homology of <math> E_r </math> with respect to this differential is <math> E_{r+1} </math>, so this gives a spectral sequence. Unfortunately, the differential is not very explicit. Determining differentials or finding ways to work around them is one of the main challenges to successfully applying a spectral sequence.

==== Spectral sequences constructed with this method ====

* [[Hodge–de Rham spectral sequence]]
* Spectral sequence of a double complex
* Can be used to construct Mixed Hodge structures<ref>{{cite arXiv|last1=Elzein|first1=Fouad|last2=Trang|first2=Lê Dung|date=2013-02-23|title=Mixed Hodge Structures|eprint=1302.5811|pages=40, 4.0.2|class=math.AG}}</ref>