Editing Spectral sequence (section)

== Constructions of spectral sequences ==
Spectral sequences can be constructed by various ways. In algebraic topology, an exact couple is perhaps the most common tool for the construction. In algebraic geometry, spectral sequences are usually constructed from filtrations of cochain complexes.

=== Spectral sequence of an exact couple ===
{{main|Exact couple}}
[[Image:Exact couple.png|right]] 
Another technique for constructing spectral sequences is [[William Schumacher Massey|William Massey]]'s method of exact couples.  Exact couples are particularly common in algebraic topology. Despite this they are unpopular in abstract algebra, where most spectral sequences come from filtered complexes.

To define exact couples, we begin again with an abelian category. As before, in practice this is usually the category of doubly graded modules over a ring.  An '''exact couple''' is a pair of objects (''A'', ''C''), together with three homomorphisms between these objects: ''f'' : ''A'' → ''A'', ''g'' : ''A'' → ''C'' and ''h'' : ''C'' → ''A'' subject to certain exactness conditions:

*[[Image (mathematics)|Image]] ''f'' = [[Kernel (algebra)|Kernel]] ''g''
*Image ''g'' = Kernel ''h''
*Image ''h'' = Kernel ''f''

We will abbreviate this data by (''A'', ''C'', ''f'', ''g'', ''h'').  Exact couples are usually depicted as triangles.  We will see that ''C'' corresponds to the ''E''<sub>0</sub> term of the spectral sequence and that ''A'' is some auxiliary data.

To pass to the next sheet of the spectral sequence, we will form the '''derived couple'''.  We set:

*''d'' = ''g'' <small>o</small> ''h''
*''A<nowiki>'</nowiki>'' = ''f''(''A'')
*''C<nowiki>'</nowiki>'' = Ker ''d'' / Im ''d''
*''f&thinsp;<nowiki>'</nowiki>'' = ''f''|<sub>''A<nowiki>'</nowiki>''</sub>, the restriction of ''f'' to ''A<nowiki>'</nowiki>''
*''h<nowiki>'</nowiki>'' : ''C<nowiki>'</nowiki>'' → ''A<nowiki>'</nowiki>'' is induced by ''h''.  It is straightforward to see that ''h'' induces such a map.
*''g<nowiki>'</nowiki>'' : ''A<nowiki>'</nowiki>'' → ''C<nowiki>'</nowiki>'' is defined on elements as follows: For each ''a'' in ''A<nowiki>'</nowiki>'', write ''a'' as ''f''(''b'') for some ''b'' in ''A''.  ''g<nowiki>'</nowiki>''(''a'') is defined to be the image of ''g''(''b'') in ''C<nowiki>'</nowiki>''. In general, ''g<nowiki>'</nowiki>'' can be constructed using one of the embedding theorems for abelian categories.

From here it is straightforward to check that (''A<nowiki>'</nowiki>'', ''C<nowiki>'</nowiki>'', ''f&thinsp;<nowiki>'</nowiki>'', ''g<nowiki>'</nowiki>'', ''h<nowiki>'</nowiki>'') is an exact couple. ''C<nowiki>'</nowiki>'' corresponds to the ''E<sub>1</sub>'' term of the spectral sequence. We can iterate this procedure to get exact couples (''A''<sup>(''n'')</sup>, ''C''<sup>(''n'')</sup>, ''f''<sup>(''n'')</sup>, ''g''<sup>(''n'')</sup>, ''h''<sup>(''n'')</sup>).

In order to construct a spectral sequence, let ''E<sub>n</sub>'' be ''C''<sup>(''n'')</sup> and ''d<sub>n</sub>'' be ''g''<sup>(''n'')</sup> <small>o</small> ''h''<sup>(''n'')</sup>.

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

* [[Serre spectral sequence]]{{sfn|May}} - used to compute (co)homology of a fibration
* [[Atiyah–Hirzebruch spectral sequence]] - used to compute (co)homology of extraordinary cohomology theories, such as [[K-theory]]
* [[Bockstein spectral sequence]].
* Spectral sequences of filtered complexes

=== 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>

=== The spectral sequence of a double complex ===

Another common spectral sequence is the spectral sequence of a double complex. A '''double complex''' is a collection of objects ''C<sub>i,j</sub>'' for all integers ''i'' and ''j'' together with two differentials, {{mvar|d}}{{i sup|{{rn|I}}}} and {{mvar|d}}{{i sup|{{rn|II}}}}.  {{mvar|d}}{{i sup|{{rn|I}}}} is assumed to decrease ''i'', and {{mvar|d}}{{i sup|{{rn|II}}}} is assumed to decrease ''j''. Furthermore, we assume that the differentials ''anticommute'', so that ''d''{{i sup|{{rn|I}}}} ''d''{{i sup|{{rn|II}}}} + ''d''{{i sup|{{rn|II}}}} ''d''{{i sup|{{rn|I}}}} = 0. Our goal is to compare the iterated homologies <math>H^\textrm{I}_i(H^\textrm{II}_j(C_{\bullet,\bullet}))</math> and <math>H^\textrm{II}_j(H^\textrm{I}_i(C_{\bullet,\bullet}))</math>. We will do this by filtering our double complex in two different ways.  Here are our filtrations:

:<math>(C_{i,j}^\textrm{I})_p = \begin{cases}
0 & \text{if } i < p \\
C_{i,j} & \text{if } i \ge p \end{cases}</math>
:<math>(C_{i,j}^\textrm{II})_p = \begin{cases}
0 & \text{if } j < p \\
C_{i,j} & \text{if } j \ge p \end{cases}</math>

To get a spectral sequence, we will reduce to the previous example. We define the ''total complex'' ''T''(''C''<sub>•,•</sub>) to be the complex whose {{prime|''n''}}th term is <math>\bigoplus_{i+j=n} C_{i,j}</math> and whose differential is {{mvar|d}}{{i sup|{{rn|I}}}} + {{mvar|d}}{{i sup|{{rn|II}}}}.  This is a complex because {{mvar|d}}{{i sup|{{rn|I}}}} and {{mvar|d}}{{i sup|{{rn|II}}}} are anticommuting differentials. The two filtrations on ''C<sub>i,j</sub>'' give two filtrations on the total complex:

:<math>T_n(C_{\bullet,\bullet})^\textrm{I}_p = \bigoplus_{i+j=n \atop i > p-1} C_{i,j}</math>
:<math>T_n(C_{\bullet,\bullet})^\textrm{II}_p = \bigoplus_{i+j=n \atop j > p-1} C_{i,j}</math>

To show that these spectral sequences give information about the iterated homologies, we will work out the ''E''{{i sup|0}}, ''E''{{i sup|1}}, and ''E''{{i sup|2}} terms of the {{rn|I}} filtration on ''T''(''C''<sub>•,•</sub>).  The ''E''{{i sup|0}} term is clear:

:<math>{}^\textrm{I}E^0_{p,q} =
T_n(C_{\bullet,\bullet})^\textrm{I}_p / T_n(C_{\bullet,\bullet})^\textrm{I}_{p+1} =
\bigoplus_{i+j=n \atop i > p-1} C_{i,j} \Big/
\bigoplus_{i+j=n \atop i > p} C_{i,j} =
C_{p,q},</math>
where {{nowrap|''n'' {{=}} ''p'' + ''q''}}.

To find the ''E''{{i sup|1}} term, we need to determine {{mvar|d}}{{i sup|{{rn|I}}}} + {{mvar|d}}{{i sup|{{rn|II}}}} on ''E''{{i sup|0}}. Notice that the differential must have degree &minus;1 with respect to ''n'', so we get a map

:<math>d^\textrm{I}_{p,q} + d^\textrm{II}_{p,q} :
T_n(C_{\bullet,\bullet})^\textrm{I}_p / T_n(C_{\bullet,\bullet})^\textrm{I}_{p+1} =
C_{p,q} \rightarrow
T_{n-1}(C_{\bullet,\bullet})^\textrm{I}_p / T_{n-1}(C_{\bullet,\bullet})^\textrm{I}_{p+1} =
C_{p,q-1}</math>

Consequently, the differential on ''E''{{i sup|0}} is the map ''C''<sub>''p'',''q''</sub> → ''C''<sub>''p'',''q''&minus;1</sub> induced by {{mvar|d}}{{i sup|{{rn|I}}}} + {{mvar|d}}{{i sup|{{rn|II}}}}.  But  {{mvar|d}}{{i sup|{{rn|I}}}} has the wrong degree to induce such a map, so {{mvar|d}}{{i sup|{{rn|I}}}} must be zero on ''E''{{i sup|0}}. That means the differential is exactly {{mvar|d}}{{i sup|{{rn|II}}}}, so we get

:<math>{}^\textrm{I}E^1_{p,q} = H^\textrm{II}_q(C_{p,\bullet}).</math>

To find ''E''{{i sup|2}}, we need to determine

:<math>d^\textrm{I}_{p,q} + d^\textrm{II}_{p,q} :
H^\textrm{II}_q(C_{p,\bullet}) \rightarrow
H^\textrm{II}_q(C_{p+1,\bullet})</math>

Because ''E''{{i sup|1}} was exactly the homology with respect to {{mvar|d}}{{i sup|{{rn|II}}}}, {{mvar|d}}{{i sup|{{rn|II}}}} is zero on ''E''{{i sup|1}}. Consequently, we get

:<math>{}^\textrm{I}E^2_{p,q} = H^\textrm{I}_p(H^\textrm{II}_q(C_{\bullet,\bullet})).</math>

Using the other filtration gives us a different spectral sequence with a similar ''E''{{i sup|2}} term:

:<math>{}^\textrm{II}E^2_{p,q} = H^\textrm{II}_q(H^{I}_p(C_{\bullet,\bullet})).</math>

What remains is to find a relationship between these two spectral sequences. It will turn out that as ''r'' increases, the two sequences will become similar enough to allow useful comparisons.