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.

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:

From the filtration one can form the associated graded complex:

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

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

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

With the notation ${\displaystyle D_{p,q}=H_{p+q}(F_{p}C),\,E_{p,q}^{1}=H_{p+q}(\operatorname {gr} (C)_{p})}$, the above reads:

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