Let R be a ring, which is fixed throughout the discussion. Note if R is , 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:
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, in which one uses the formula below as definition (cf. Spectral sequence#The spectral sequence of a filtered complex).
Lemma — Let , which inherits -grading from . Then for each p
Sketch of proof:[1][2] 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 ,
Theorem — If and for each n there is an integer such that , then the spectral sequence Er converges to ; that is, .
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. (Some authors call the two spectral sequences horizontal and vertical.) Let be a double complex.[3] 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:
By symmetry, that is, by switching first and second indexes, one also obtains the other exact couple.
Example: Serre spectral sequence
This section needs expansion. You can help by adding to it. (August 2020)