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 . 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:
:$F_ C \subset F_p C.$
From the filtration one can form the associated graded complex:
:$\operatorname C = \bigoplus_^\infty F_p C/F_ C,$
which is doubly-graded and which is the zero-th page of the spectral sequence:
:$E^0_ = (\operatorname C)_ = (F_p C / F_ C)_.$

To get the first page, for each fixed ''p'', we look at the short exact sequence of complexes:
:$0 \to F_ C \to F_p C \to (\operatornameC)_p \to 0$
from which we obtain a long exact sequence of homologies: (''p'' is still fixed)
:$\cdots \to H_n(F_ C) \overset\to H_n(F_p C) \overset \to H_n(\operatorname(C)_p) \overset\to H_(F_ C) \to \cdots$
With the notation $D_ = H_ (F_p C), \, E^1_ = H_ (\operatorname(C)_p)$, the above reads:
:$\cdots \to D_ \overset\to D_ \overset \to E^1_ \overset\to D_ \to \cdots,$
which is precisely an exact couple and $E^1$ is a complex with the differential $d = j \circ k$. The derived couple of this exact couple gives the second page and we iterate. In the end, one obtains the complexes $E^r_$ with the differential ''d'':
:$E^r_ \overset\to D^r_ \overset\to E^r_.$

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).

Sketch of proof: Remembering $d = j \circ k$, it is easy to see:
:$Z^r= k^ (\operatorname i^r), \, B^r = j (\operatorname i^r),$
where they are viewed as subcomplexes of $E^1$.

We will write the bar for $F_p C \to F_p C / F_ C$. Now, if $$overline\in Z^_ \subset E^1_, then $k($overline = i^( for some \in D_ = H_(F_p C). On the other hand, remembering ''k'' is a connecting homomorphism, $k($overline = (x)/math> where ''x'' is a representative living in $(F_p C)_$. Thus, we can write: $d(x) - i^(y) = d(x')$ for some $x' \in F_C$. Hence, $$overline\in Z^r_p \Leftrightarrow x \in A^r_p modulo $F_ C$, yielding $Z_p^r \simeq (A^r_p + F_C)/F_ C$.

Next, we note that a class in $\operatorname(i^: H_(F_pC) \to H_(F_ C))$ is represented by a cycle ''x'' such that $x \in d(F_ C)$. Hence, since ''j'' is induced by $\overline$, $B^_p = j (\operatorname i^) \simeq (d(A^_) + F_ C)/F_ C$.

We conclude: since $A^r_p \cap F_ C = A^_$,
:$E^r_ = \simeq \simeq . \qquad \square$

Proof: See the last section of May. $\square$

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 $K^$ be a double complex.We prefer cohomological notation here since the applications are often in algebraic geometry. With the notation $G^p = \bigoplus_ K^$, for each with fixed ''p'', we have the exact sequence of cochain complexes:

:$0 \to G^ \to G^p \to K^ \to 0.$

Taking cohomology of it gives rise to an exact couple:

:$\cdots \to D^ \overset\to E_1^ \overset\to \cdots$

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:
:$F \to E \to B.$
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

References

*
*.
*{{citation
, last = Weibel , first = Charles A. , author-link = Charles Weibel
, doi = 10.1017/CBO9781139644136
, isbn = 0-521-43500-5
, location = Cambridge
, mr = 1269324
, publisher = Cambridge University Press
, series = Cambridge Studies in Advanced Mathematics
, title = An introduction to homological algebra
, volume = 38
, year = 1994

Spectral sequences