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 § Spectral Sequence of an exact couple. 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 [math]\displaystyle{ \Z }[/math], 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:

[math]\displaystyle{ F_{p-1} C \subset F_p C. }[/math]

From the filtration one can form the associated graded complex:

[math]\displaystyle{ \operatorname{gr} C = \bigoplus_{-\infty}^\infty F_p C/F_{p-1} C, }[/math]

which is doubly-graded and which is the zero-th page of the spectral sequence:

[math]\displaystyle{ E^0_{p, q} = (\operatorname{gr} C)_{p, q} = (F_p C / F_{p-1} C)_{p+q}. }[/math]

To get the first page, for each fixed p, we look at the short exact sequence of complexes:

[math]\displaystyle{ 0 \to F_{p-1} C \to F_p C \to (\operatorname{gr}C)_p \to 0 }[/math]

from which we obtain a long exact sequence of homologies: (p is still fixed)

[math]\displaystyle{ \cdots \to H_n(F_{p-1} C) \overset{i}\to H_n(F_p C) \overset{j} \to H_n(\operatorname{gr}(C)_p) \overset{k}\to H_{n-1}(F_{p-1} C) \to \cdots }[/math]

With the notation [math]\displaystyle{ D_{p, q} = H_{p+q} (F_p C), \, E^1_{p, q} = H_{p + q} (\operatorname{gr}(C)_p) }[/math], the above reads:

[math]\displaystyle{ \cdots \to D_{p - 1, q + 1} \overset{i}\to D_{p, q} \overset{j} \to E^1_{p, q} \overset{k}\to D_{p - 1, q} \to \cdots, }[/math]

which is precisely an exact couple and [math]\displaystyle{ E^1 }[/math] is a complex with the differential [math]\displaystyle{ d = j \circ k }[/math]. The derived couple of this exact couple gives the second page and we iterate. In the end, one obtains the complexes [math]\displaystyle{ E^r_{*, *} }[/math] with the differential d:

[math]\displaystyle{ E^r_{p, q} \overset{k}\to D^r_{p - 1, q} \overset{{}^r j}\to E^r_{p - r, q + r - 1}. }[/math]

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).

Lemma — Let [math]\displaystyle{ A^r_p = \{ c \in F_p C \mid d(c) \in F_{p-r} C \} }[/math], which inherits [math]\displaystyle{ \mathbb{Z} }[/math]-grading from [math]\displaystyle{ F_pC }[/math]. Then for each p

[math]\displaystyle{ E^r_{p, *} \simeq {A^r_p \over d(A^{r-1}_{p+r-1}) + A^{r-1}_{p-1}}. }[/math]

Sketch of proof: Remembering [math]\displaystyle{ d = j \circ k }[/math], it is easy to see:

[math]\displaystyle{ Z^r= k^{-1} (\operatorname{im} i^r), \, B^r = j (\operatorname{ker} i^r), }[/math]

where they are viewed as subcomplexes of [math]\displaystyle{ E^1 }[/math].

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

Next, we note that a class in [math]\displaystyle{ \operatorname{ker}(i^{r-1}: H_{p+q}(F_pC) \to H_{p+q}(F_{p + r - 1} C)) }[/math] is represented by a cycle x such that [math]\displaystyle{ x \in d(F_{p+r-1} C) }[/math]. Hence, since j is induced by [math]\displaystyle{ \overline{\cdot} }[/math], [math]\displaystyle{ B^{r-1}_p = j (\operatorname{ker} i^{r-1}) \simeq (d(A^{r-1}_{p+r-1}) + F_{p-1} C)/F_{p-1} C }[/math].

We conclude: since [math]\displaystyle{ A^r_p \cap F_{p-1} C = A^{r-1}_{p-1} }[/math],

[math]\displaystyle{ E^r_{p, *} = {Z^{r-1}_p \over B^{r-1}_p} \simeq {A^r_p + F_{p-1} C \over d(A^{r-1}_{p+r-1}) + F_{p-1}C} \simeq {A^r_p \over d(A^{r-1}_{p+r-1}) + A^{r-1}_{p-1}}. \qquad \square }[/math]

Theorem — If [math]\displaystyle{ C = \cup_p F_p C }[/math] and for each n there is an integer [math]\displaystyle{ s(n) }[/math] such that [math]\displaystyle{ F_{s(n)} C_n = 0 }[/math], then the spectral sequence E^r converges to [math]\displaystyle{ H_*(C) }[/math]; that is, [math]\displaystyle{ E^\infty_{p, q} = F_p H_{p+q}(C)/F_{p-1} H_{p+q}(C) }[/math].

Proof: See the last section of May. [math]\displaystyle{ \square }[/math]

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 [math]\displaystyle{ K^{p,q} }[/math] be a double complex. With the notation [math]\displaystyle{ G^p = \bigoplus_{i \ge p} K^{i, *} }[/math], for each with fixed p, we have the exact sequence of cochain complexes:

[math]\displaystyle{ 0 \to G^{p+1} \to G^p \to K^{p, *} \to 0. }[/math]

Taking cohomology of it gives rise to an exact couple:

[math]\displaystyle{ \cdots \to D^{p, q} \overset{j}\to E_1^{p, q} \overset{k}\to \cdots }[/math]

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:

[math]\displaystyle{ F \to E \to B. }[/math]

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).

References

• May, J. Peter, A primer on spectral sequences, http://www.math.uchicago.edu/~may/MISC/SpecSeqPrimer.pdf 
• "Exact couples in algebraic topology. I, II", Annals of Mathematics, Second Series 56: 363–396, 1952, doi:10.2307/1969805 .
• An introduction to homological algebra, Cambridge Studies in Advanced Mathematics, 38, Cambridge: Cambridge University Press, 1994, doi:10.1017/CBO9781139644136, ISBN 0-521-43500-5