Editing Spectral sequence (section)

=== 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