Exact couple explained

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 = o plus ^{inf ty} _{-inf ty} 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,} = (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)… \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 … With the notation D _{p,} = H _{p+q} (F _{p} C), E ^{1} _{p, q} = H _{p} (\operatorname{gr}(C) _{p)} , the above reads:… \to D _{p} \overset{i}\to D _{p,} \overset{j} \to E ^{1} _{p, q} \overset{k}\to D _{p} \to … , 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).

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} = 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} . 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 ^{r} _{p} \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} C)) is represented by a cycle x such that x \in d(F _{p+r-1} C) . Hence, since j is induced by \overline{ ⋅ } , 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.^[1] With the notation G ^{p} = o plus _{i} 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:

… \to D ^{p,} \overset{j}\to E ^{p, q} _{1} \overset{k}\to …

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).

References

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Notes and References

1. We prefer cohomological notation here since the applications are often in algebraic geometry.