In mathematics, an exact couple, due to William S. Massey (1952), 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 § Spectral sequence of an exact couple.
For a basic example, see Bockstein spectral sequence.
The present article covers additional materials.
Let R be a ring, which is fixed throughout the discussion.
, 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
, 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,[1] in which one uses the formula below as definition (cf.
Spectral sequence#The spectral sequence of a filtered complex).
{\displaystyle A_{p}^{r}=\{c\in F_{p}C\mid d(c)\in F_{p-r}C\}}
Then for each p Sketch of proof:[2][3] Remembering
, it is easy to see: where they are viewed as subcomplexes of
We will write the bar for
On the other hand, remembering k is a connecting homomorphism,
where x is a representative living in
Next, we note that a class in
is represented by a cycle x such that
, then the spectral sequence Er converges to
Proof: See the last section of May.
A double complex determines two exact couples; whence, the two spectral sequences, as follows.
(Some authors call the two spectral sequences horizontal and vertical.)
be a double complex.
, for each with fixed p, we have the exact sequence of cochain complexes: Taking cohomology of it gives rise to an exact couple: By symmetry, that is, by switching first and second indexes, one also obtains the other exact couple.
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).