# nLab exact couple

### Context

#### Homological algebra

homological algebra

and

nonabelian homological algebra

diagram chasing

# Contents

## Idea

Exact couples are a tool for constructing spectral sequences.

## Definition

### Exact couples

###### Definition

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

$\cdots \stackrel{k}{\to} E \overset{j}{\to} D \overset{\varphi}{\to} D \overset{k}{\to} E \overset{j}{\to} \cdots \,.$
###### Remark

This being cyclic, it is usually depeicted as a triangle

$\array{ D && \stackrel{\varphi}{\longrightarrow} && D \\ & {}_{\mathllap{j}}\nwarrow && \swarrow_{\mathrlap{k}} \\ && E }$

The archetypical example from which this and the following definition draw their meaning is the following.

###### Example

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

$\array{ X = X_0 &\stackrel{g_0}{\leftarrow}& X_1 &\stackrel{g_1}{\leftarrow}& X_2 &\stackrel{g_2}{\leftarrow}& X_3 &\stackrel{}{\leftarrow}& \cdots \\ \downarrow^{\mathrlap{f_0}} && \downarrow^{\mathrlap{f_1}} && \downarrow^{\mathrlap{f_2}} && \downarrow^{\mathrlap{f_3}} && \\ K_0 && K_1 && K_2 && K_3 }$

where each hook is a fiber sequence. Then the induced long exact sequences of homotopy groups

$\cdots \pi_\bullet(X_{s+1}) \longrightarrow \pi_\bullet(X_s) \longrightarrow \pi_\bullet(K_s) \longrightarrow \cdots$

for all $s$ give an exact couple by taking $E$ and $D$ to be the bigraded abelian groups

$D \coloneqq \pi_\bullet(X_\bullet)$
$E \coloneqq \pi_\bullet(K_\bullet) \,.$

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.

### Spectral sequences from exact couples

###### Definition

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

$E' \overset{j'}{\to} D' \overset{\varphi}{\to} D' \overset{k'}{\to} E' \overset{j'}{\to}.$

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.

###### Remark

The exact couple recipe for spectral sequences is notable in that it doesn’t 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

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

$\mathbb{Z}/p\mathbb{Z} \to \mathbb{Z}/p^2\mathbb{Z} \to \mathbb{Z}/p\mathbb{Z}$

gives the exact couple

$H^\bullet(d,\mathbb{Z}/p^2\mathbb{Z}) \overset{[\cdot]}{\to} H^\bullet(d,\mathbb{Z}/p\mathbb{Z}) \overset{\beta}{\to} H^\bullet(d,\mathbb{Z}/p\mathbb{Z}) \overset{p}{\to}H^\bullet(d,\mathbb{Z}/p^2\mathbb{Z})\cdots$

in which $\beta$ is the mod-$p$ Bockstein homomorphism.

## References

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

Revised on November 17, 2013 02:16:03 by Urs Schreiber (82.113.98.128)