# 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

An exact couple is an exact sequence of three morphisms among two objects

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

### Spectral sequences from exact couples

These construct spectral sequences by a two-step process:

• first, the composite $d=\mathrm{kj}:E\to E$ 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.

The sequence of complexes $\left(E,d\right),\left(E\prime ,d\prime \right),\dots$ is a spectral sequence, by construction.

###### 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

Section 5.9 of

Created on August 26, 2012 18:46:09 by Urs Schreiber (89.204.137.239)