and
nonabelian homological algebra
Exact couples are a tool for constructing spectral sequences.
Given an abelian category $\mathcal{C}$, an exact couple in $\mathcal{C}$ is a cyclic exact sequence of three morphisms among two objects of the form
This being cyclic, it is usually depeicted as a triangle
The archetypical example from which this and the following definition draw their meaning is the following.
Let $X$ be a topological space or chain complex or spectrum or similar, and assume that it is equipped with a resolution of the form
where each hook is a fiber sequence. Then the induced long exact sequences of homotopy groups
for all $s$ give an exact couple by taking $E$ and $D$ to be the bigraded abelian groups
and taking $\phi$ and $k$ to be given by the functoriality of the homotopy groups $\pi_{\bullet}$ and finally taking $j$ to be given by the connecting homomorphism.
For instance for the original diagram an Adams resolution then this spectral sequence is the Adams spectral sequence.
The spectral sequences induced by an exact couple is the one built by repeating the following two-step process:
first, observe that the composite $d=k j \colon E\to E$ is nilpotent: $d^2=0$
second, the homology $E'$ of $(E,d)$ supports a map $j':E'\to \varphi D$, and receives a map $k':\varphi D\to E'$. Setting $D'=\varphi D$, by general nonsense
is again an exact couple, called the derived exact couple.
The sequence of complexes $(E,d),(E',d'),\dots$ obtained this way is then a spectral sequence, by construction. This is the spectral sequence induced by the exact couple.
The exact couple recipe for spectral sequences is notable in that it doesnβt mention any grading on the objects $D,E$; trivially, an exact couple can be specified by a short exact sequence $\coker \varphi\to E\to \ker\varphi$, although this obscures the focus usually given to $E$. In applications, a bi-grading is usually induced by the context, which also specifies bidegrees for the initial maps $j,k,\varphi$, leading to the conventions mentioned earlier.
Examples of exact couples can be constructed in a number of ways. Importantly, any short exact sequence involving two distinct chain complexes provides an exact couple among their total homology complexes, via the Mayer-Vietoris long exact sequence; in particular, applying this procedure to the relative homology of a filtered complex gives precisely the spectral sequence of a filtered complex.
For another example, choosing a chain complex of flat modules $(C^\dot,d)$, tensoring with the short exact sequence
gives the exact couple
in which $\beta$ is the mod-$p$ Bockstein homomorphism.
An early paper is:
A standard textbook account is section 5.9 of
A review with an eye towards application to the Adams spectral sequence is in