Exact couple

In mathematics, an exact couple, due to 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 sequence 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:

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

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#The spectral sequence of a filtered complex).

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

$E^r_{p, *} \simeq {A^r_p \over d(A^{r-1}_{p+r-1}) + A^{r-1}_{p-1}}.$

Sketch of proof:^[1][2] 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 $A^r_p \cap F_{p-1} C = A^{r-1}_{p-1}$,

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

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

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

Exact couple of a double complex

A double complex determines two exact couples; whence, the two spectra sequences, as follows. (Some authors call the two spectral sequences horizontal and vertical.) Let $K^{p,q}$ be a double complex.^[3] 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:

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

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:

$F \to E \to B.$

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

1. May, Proof of (7.3)
2. Weibel 1994, Theorem 5.9.4.
3. We prefer cohomological notation here since the applications are often in algebraic geometry.

References

• May, J. P., A primer on spectral sequences (PDF)
• Massey, W. 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.