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exact couple

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 $(E,d)$ 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 $(E,d),(E\prime ,d\prime ),\dots$ 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 $coker\phi \to E\to \ker \phi$, 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 $(C^{\dot{,}}d)$, tensoring with the short exact sequence

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

gives the exact couple

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

• Charles Weibel, An Introduction to Homological Algebra