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 (which is immediate), 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: ("p" is still fixed)
With the notation formula_6, the above reads:
which is precisely an exact couple and formula_8 is a complex with the differential formula_9. The derived couple of this exact couple gives the second page and we iterate. In the end, one obtains the complexes formula_10 with the differential "d":
Sketch of proof: Remembering formula_9, it is easy to see:
where they are viewed as subcomplexes of formula_8.
We will write the bar for formula_14. Now, if formula_15, then formula_16 for some formula_17. On the other hand, remembering "k" is a connecting homomorphism, formula_18 where "x" is a representative living in formula_19. Thus, we can write: formula_20 for some formula_21. Hence, formula_22 modulo formula_23, yielding formula_24.
Next, we note that a class in formula_25 is represented by a cycle "x" such that formula_26. Hence, since "j" is induced by formula_27, formula_28.
We conclude: since formula_29,
Proof: See the last section of May. formula_31
Exact couple of a double complex.
A double complex determines two exact couples; whence, the two spectral sequences, as follows. (Some authors call the two spectral sequences horizontal and vertical.) Let formula_32 be a double complex. With the notation formula_33, 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 (namely, local coefficient system).