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exact couple

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Idea

Exact couples are a tool for constructing spectral sequences.

Definition

Exact couples

Definition

Given an abelian category π’ž\mathcal{C}, an exact couple in π’ž\mathcal{C} is a cyclic exact sequence of three morphisms among two objects of the form

⋯→kE→jD→φD→kE→j⋯. \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

D βŸΆΟ† D jβ†– ↙ k E \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 XX be a topological space or chain complex or spectrum or similar, and assume that it is equipped with a resolution of the form

X=X 0 ←g 0 X 1 ←g 1 X 2 ←g 2 X 3 ← β‹― ↓ f 0 ↓ f 1 ↓ f 2 ↓ f 3 K 0 K 1 K 2 K 3 \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

β‹―Ο€ β€’(X s+1)βŸΆΟ€ β€’(X s)βŸΆΟ€ β€’(K s)βŸΆβ‹― \cdots \pi_\bullet(X_{s+1}) \longrightarrow \pi_\bullet(X_s) \longrightarrow \pi_\bullet(K_s) \longrightarrow \cdots

for all ss give an exact couple by taking EE and DD to be the bigraded abelian groups

D≔π β€’(X β€’) D \coloneqq \pi_\bullet(X_\bullet)
E≔π β€’(K β€’). E \coloneqq \pi_\bullet(K_\bullet) \,.

and taking Ο•\phi and kk to be given by the functoriality of the homotopy groups Ο€ β€’\pi_{\bullet} and finally taking jj 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:Eβ†’Ed=k j \colon E\to E is nilpotent: d 2=0d^2=0

  • second, the homology Eβ€²E' of (E,d)(E,d) supports a map jβ€²:Eβ€²β†’Ο†Dj':E'\to \varphi D, and receives a map kβ€²:Ο†Dβ†’Eβ€²k':\varphi D\to E'. Setting Dβ€²=Ο†DD'=\varphi D, by general nonsense

    Eβ€²β†’jβ€²Dβ€²β†’Ο†Dβ€²β†’kβ€²Eβ€²β†’jβ€². 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),(Eβ€²,dβ€²),…(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,ED,E; trivially, an exact couple can be specified by a short exact sequence cokerΟ†β†’Eβ†’kerΟ†\coker \varphi\to E\to \ker\varphi, although this obscures the focus usually given to EE. In applications, a bi-grading is usually induced by the context, which also specifies bidegrees for the initial maps j,k,Ο†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 ,Λ™d)(C^\dot,d), tensoring with the short exact sequence

β„€/pβ„€β†’β„€/p 2β„€β†’β„€/pβ„€ \mathbb{Z}/p\mathbb{Z} \to \mathbb{Z}/p^2\mathbb{Z} \to \mathbb{Z}/p\mathbb{Z}

gives the exact couple

H β€’(d,β„€/p 2β„€)β†’[β‹…]H β€’(d,β„€/pβ„€)β†’Ξ²H β€’(d,β„€/pβ„€)β†’pH β€’(d,β„€/p 2β„€)β‹― 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 Ξ²\beta is the mod-pp 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)