Spectral Sequence - Exact Couples

Exact Couples

The most powerful technique for the construction of spectral sequences is William Massey's method of exact couples. Exact couples are particularly common in algebraic topology, where there are many spectral sequences for which no other construction is known. In fact, all known spectral sequences can be constructed using exact couples. Despite this they are unpopular in abstract algebra, where most spectral sequences come from filtered complexes. To define exact couples, we begin again with an abelian category. As before, in practice this is usually the category of doubly graded modules over a ring. An exact couple is a pair of objects A and C, together with three homomorphisms between these objects: f : A → A, g : A → C and h : C → A subject to certain exactness conditions:

• Image f = Kernel g
• Image g = Kernel h
• Image h = Kernel f

We will abbreviate this data by (A, C, f, g, h). Exact couples are usually depicted as triangles. We will see that C corresponds to the E₀ term of the spectral sequence and that A is some auxiliary data.

To pass to the next sheet of the spectral sequence, we will form the derived couple. We set:

• d = g h
• A' = f(A)
• C' = Ker d / Im d
• f' = f|_A', the restriction of f to A'
• h' : C' → A' is induced by h. It is straightforward to see that h induces such a map.
• g' : A' → C' is defined on elements as follows: For each a in A', write a as f(b) for some b in A. g'(a) is defined to be the image of g(b) in C'. In general, g' can be constructed using one of the embedding theorems for abelian categories.

From here it is straightforward to check that (A', C', f', g', h') is an exact couple. C' corresponds to the E₁ term of the spectral sequence. We can iterate this procedure to get exact couples (A(n), C(n), f(n), g(n), h(n)). We let E_n be C(n) and d_n be g(n) h(n). This gives a spectral sequence.