Exact couple


In mathematics, an exact couple, due to, is a general source of spectral sequences. It is common especially in algebraic topology; for example, Serre spectral sequence can be constructed by first constructing an exact couple.
For the definition of an exact couple and the construction of a spectral sequence from it, see spectral sequence#Exact couples. For a basic example, see Bockstein spectral sequence. The present article covers additional materials.

Exact couple of a filtered complex

Let R be a ring, which is fixed throughout the discussion. Note if R is Z, then modules over R are the same thing as abelian groups.
Each filtered chain complex of modules determines an exact couple, which in turn determines a spectral sequence, as follows. Let C be a chain complex graded by integers and suppose it is given an increasing filtration: for each integer p, there is an inclusion of complexes:
From the filtration one can form the associated graded complex:
which is doubly-graded and which is the zero-th page of the spectral sequence:
To get the first page, for each fixed p, we look at the short exact sequence of complexes:
from which we obtain a long exact sequence of homologies:
With the notation, the above reads:
which is precisely an exact couple and is a complex with the differential. The derived couple of this exact couple gives the second page and we iterate. In the end, one obtains the complexes with the differential d:
The next lemma gives a more explicit formula for the spectral sequence; in particular, it shows the spectral sequence constructed above is the same one in more traditional direct construction, in which one uses the formula below as definition.
Sketch of proof: Remembering, it is easy to see:
where they are viewed as subcomplexes of.
We will write the bar for. Now, if, then for some. On the other hand, remembering k is a connecting homomorphism, where x is a representative living in. Thus, we can write: for some. Hence, modulo, yielding.
Next, we note that a class in is represented by a cycle x such that. Hence, since j is induced by,.
We conclude: since,
Proof: See the last section of May.

Exact couple of a double complex

A double complex determines two exact couples; whence, the two spectral sequences, as follows. Let be a double complex. With the notation, for each with fixed p, we have the exact sequence of cochain complexes:
Taking cohomology of it gives rise to an exact couple:
where we used the notation
By symmetry, that is, by switching first and second indexes, one also obtains the other exact couple.

Example: Serre spectral sequence

The Serre spectral sequence arises from a fibration:
For the sake of transparency, we only consider the case when the spaces are CW complexes, F is connected and B is simply connected; the general case involves more technicality.