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:
 * $$F_{p-1} C \subset F_p C.$$

From the filtration one can form the associated graded complex:
 * $$\operatorname{gr} C = \bigoplus_{-\infty}^\infty F_p C/F_{p-1} C,$$

which is doubly-graded and which is the zero-th page of the spectral sequence:
 * $$E^0_{p, q} = (\operatorname{gr} C)_{p, q} = (F_p C / F_{p-1} C)_{p+q}.$$

To get the first page, for each fixed p, we look at the short exact sequence of complexes:
 * $$0 \to F_{p-1} C \to F_p C \to (\operatorname{gr}C)_p \to 0$$

from which we obtain a long exact sequence of homologies: (p is still fixed)
 * $$\cdots \to H_n(F_{p-1} C) \overset{i}\to H_n(F_p C) \overset{j} \to H_n(\operatorname{gr}(C)_p) \overset{k}\to H_{n-1}(F_{p-1} C) \to \cdots$$

With the notation $$D_{p, q} = H_{p+q} (F_p C), \, E^1_{p, q} = H_{p + q} (\operatorname{gr}(C)_p)$$, the above reads:
 * $$\cdots \to D_{p - 1, q + 1} \overset{i}\to D_{p, q} \overset{j} \to E^1_{p, q} \overset{k}\to D_{p - 1, q} \to \cdots,$$

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:
 * $$E^r_{p, q} \overset{k}\to D^r_{p - 1, q} \overset{{}^r j}\to E^r_{p - r, q + r - 1}.$$

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).

Sketch of proof: Remembering $$d = j \circ k$$, it is easy to see:
 * $$Z^r= k^{-1} (\operatorname{im} i^r), \, B^r = j (\operatorname{ker} i^r),$$

where they are viewed as subcomplexes of $$E^1$$.

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

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

We conclude: since $$A^r_p \cap F_{p-1} C = A^{r-1}_{p-1}$$,
 * $$E^r_{p, *} = {Z^{r-1}_p \over B^{r-1}_p} \simeq {A^r_p + F_{p-1} C \over d(A^{r-1}_{p+r-1}) + F_{p-1}C} \simeq {A^r_p \over d(A^{r-1}_{p+r-1}) + A^{r-1}_{p-1}}. \qquad \square$$

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


 * $$0 \to G^{p+1} \to G^p \to K^{p, *} \to 0.$$

Taking cohomology of it gives rise to an exact couple:


 * $$\cdots \to D^{p, q} \overset{j}\to E_1^{p, q} \overset{k}\to \cdots$$

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).