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.
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.
