# 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 $𝒞$, an exact couple in $𝒞$ is a cyclic exact sequence of three morphisms among two objects of the form

$\cdots \stackrel{k}{\to }E\stackrel{j}{\to }D\stackrel{\phi }{\to }D\stackrel{k}{\to }E\stackrel{j}{\to }\cdots \phantom{\rule{thinmathspace}{0ex}}.$\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

$\begin{array}{ccccc}D& & \stackrel{\phi }{⟶}& & D\\ & {}_{j}↖& & {↙}_{k}\\ & & E\end{array}$\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

$\begin{array}{ccccccccc}X={X}_{0}& \stackrel{{g}_{0}}{←}& {X}_{1}& \stackrel{{g}_{1}}{←}& {X}_{2}& \stackrel{{g}_{2}}{←}& {X}_{3}& \stackrel{}{←}& \cdots \\ {↓}^{{f}_{0}}& & {↓}^{{f}_{1}}& & {↓}^{{f}_{2}}& & {↓}^{{f}_{3}}& & \\ {K}_{0}& & {K}_{1}& & {K}_{2}& & {K}_{3}\end{array}$\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

$\cdots {\pi }_{•}\left({X}_{s+1}\right)⟶{\pi }_{•}\left({X}_{s}\right)⟶{\pi }_{•}\left({K}_{s}\right)⟶\cdots$\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

$D≔{\pi }_{•}\left({X}_{•}\right)$D \coloneqq \pi_\bullet(X_\bullet)
$E≔{\pi }_{•}\left({K}_{•}\right)\phantom{\rule{thinmathspace}{0ex}}.$E \coloneqq \pi_\bullet(K_\bullet) \,.

and taking $\varphi$ and $k$ to be given by the functoriality of the homotopy groups ${\pi }_{•}$ 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 $d=kj:E\to E$ is nilpotent: ${d}^{2}=0$

• second, the homology $E\prime$ of $\left(E,d\right)$ supports a map $j\prime :E\prime \to \phi D$, and receives a map $k\prime :\phi D\to E\prime$. Setting $D\prime =\phi D$, by general nonsense

$E\prime \stackrel{j\prime }{\to }D\prime \stackrel{\phi }{\to }D\prime \stackrel{k\prime }{\to }E\prime \stackrel{j\prime }{\to }.$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 $\left(E,d\right),\left(E\prime ,d\prime \right),\dots$ 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 $coker\phi \to E\to \mathrm{ker}\phi$, 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 $j,k,\phi$, 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 $\left({C}^{\stackrel{˙}{,}}d\right)$, tensoring with the short exact sequence

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

gives the exact couple

${H}^{•}\left(d,ℤ/{p}^{2}ℤ\right)\stackrel{\left[\cdot \right]}{\to }{H}^{•}\left(d,ℤ/pℤ\right)\stackrel{\beta }{\to }{H}^{•}\left(d,ℤ/pℤ\right)\stackrel{p}{\to }{H}^{•}\left(d,ℤ/{p}^{2}ℤ\right)\cdots$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)