Exact couple
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#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 , 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:
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).
Notes
- May, Proof of (7.3)
- Weibel 1994, Theorem 5.9.4.
- We prefer cohomological notation here since the applications are often in algebraic geometry.
References
- May, J. Peter, A primer on spectral sequences (PDF)
- Massey, William S. (1952), "Exact couples in algebraic topology. I, II", Annals of Mathematics, Second Series, 56: 363–396, doi:10.2307/1969805, MR 0052770.
- Weibel, Charles A. (1994), An introduction to homological algebra, Cambridge Studies in Advanced Mathematics, 38, Cambridge: Cambridge University Press, doi:10.1017/CBO9781139644136, ISBN 0-521-43500-5, MR 1269324.