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spectral sequence of a filtered complex

Context

Homological algebra

homological algebra

and

nonabelian homological algebra

Context

Basic definitions

Stable homotopy theory notions

Constructions

Lemmas

diagram chasing

Homology theories

Theorems

Contents

Idea

The spectral sequence of a filtered chain complex

F p1C F pC C \cdots \hookrightarrow F_{p-1}C_\bullet \hookrightarrow F_p C_\bullet \hookrightarrow \cdots \hookrightarrow C_\bullet

is a tool for computing the chain homology of C C_\bullet from the chain homologies of the associated graded objects

G pCF pC /F p1C ; G_p C \coloneqq F_p C_\bullet/F_{p-1}C_\bullet \,;

which is in general simpler. This is a special case of the spectral sequence of a filtered stable homotopy type.

The sequence asymptotes to the homology of C C_\bullet by approximating cycles and boundaries of CC by their ”rr-approximation”: an rr-almost cycle is a chain in filtering degree pp whose differential vanishes only up to terms that are rr steps lower in filtering degree, and an rr-almost boundary in filtering degree pp is a cycle that is the differential of a chain which may be (only) up to rr-degrees higher in filtering degree. The corresponding rr-almost homology of C C_\bullet in filtering degree pp is the term E p, rE^r_{p,\bullet} of the spectral sequence.

If the filtering is bounded then rr-almost homology for sufficiently large rr (”\infty-almost homology”) is the clearly the genuine homology, and so the spectral sequence converges to the correct homology. But the point is that typically it reaches the correct value already at some low finite degree rr (it “collapses”), and so allows one to obtain the genuine homology from some finite rr-almost homology.

One may also regard the spectral sequence of a filtered complex as a tool for organizing data derivable from the families of long exact sequence in homology

H q(F p1C )H q(F pC )H q(F pC /F q1C )H q1(F p1C ) \cdots \to H_q (F_{p-1} C_\bullet) \to H_q (F_p C_\bullet) \to H_q(F_p C_\bullet / F_{q-1} C_\bullet) \to H_{q-1}(F_{p-1} C_\bullet) \to \cdots

which are induced by the short exact sequences

0F p1C qF pC qF pC q/F p1C q0 0 \to F_{p-1}C_q \hookrightarrow F_p C_q \to F_p C_q / F_{p-1} C_q \to 0

coming from the filtering.

Definition

We give the definition

and

Via relative homology

Let RR be a ring and write 𝒜=R\mathcal{A} = RMod for its category of modules.

Let

F p1C F pC C \cdots \hookrightarrow F_{p-1}C_\bullet \hookrightarrow F_p C_\bullet \hookrightarrow \cdots \hookrightarrow C_\bullet

be a filtered chain complex in 𝒜\mathcal{A}, with associated graded complex denoted G C G_\bullet C_\bullet.

Remark

In more detail this means that

  1. [ nC n n1]C n1][\cdots \stackrel{\partial_{n}}{\to} C_n \stackrel{\partial_{n-1}}{\to}] C_{n-1} \to \cdots] is a chain complex, hence {C n}\{C_n\} are objects in 𝒜\mathcal{A} (RR-modules) and { n}\{\partial_n\} are morphisms (module homomorphisms) with n n+1=0\partial_n \circ \partial_{n+1} = 0;

  2. For each nn \in \mathbb{Z} there is a filtering F C nF_\bullet C_n on C nC_n and all these filterings are compatible with the differentials in that

    (F pC n)F pC n1 \partial(F_p C_n) \subset F_p C_{n-1}
  3. The grading associated to the filtering is such that the pp-graded elements are those in the quotient

    G pC nF pC nF p1C n. G_p C_n \coloneqq \frac{F_p C_n}{ F_{p-1} C_n} \,.

    Since the differentials respect the grading we have chain complexes G pC G_p C_\bullet in each filtering degree pp.

We use element-notation in the following as if 𝒜\mathcal{A} were a category of modules.

rr-Almost cycles and boundaries

Definition

Given a filtered chain complex F C F_\bullet C_\bullet as above we say for all r,p,qr, p, q \in \mathbb{Z} that

  1. G pC p+qG_p C_{p+q} is the module of (p,q)(p,q)-chains or of (p+q)(p+q)-chains in filtering degree pp;

  2. Z p,q r {cG pC p+qc=0modF prC } ={cF pC p+q(c)F prC p+q1}/F p1C p+q\begin{aligned} Z^r_{p,q} & \coloneqq \left\{ c \in G_p C_{p+q} | \partial c = 0 \, mod F_{p-r} C_{\bullet} \right\} \\ & = \left\{ c \in F_p C_{p+q} | \partial(c) \in F_{p-r} C_{p+q-1} \right\}/ F_{p-1}C_{p+q} \end{aligned}

    is the module of rr-almost (p,q)(p,q)-cycles (the (p+q)(p+q)-chains whose differential vanishes modulo terms of filtering degree prp-r);

  3. B p,q r(F p+r1C p+q+1),B^{r}_{p,q} \coloneqq \partial(F_{p+r-1} C_{p+q+1}) \,,

    is the module of rr-almost (p,q)(p,q)-boundaries.

Similarly we set

Z p,q {cF pC p+qc=0}/F p1C p+q=Z(G pC p+q) Z^\infty_{p,q} \coloneqq \{c \in F_p C_{p + q} | \partial c = 0 \}/F_{p-1}C_{p+q} = Z(G_p C_{p+q})
B p,q =(F pC p+q+1). B^\infty_{p,q} = \partial( F_p C_{p+q+1} ) \,.

From this definition we immediately have that the differentials :C p+qC p+q1\partial \colon C_{p+q} \to C_{p+q-1} restrict to the rr-almost cycles as follows:

Proposition

The differentials of C C_\bullet restrict on rr-almost cycles to morphisms of the form

r:Z p,q rZ pr,q+r1 r. \partial^r \colon Z^r_{p,q} \to Z^r_{p-r, q+r-1} \,.

These are still differentials: 2=0\partial^2 = 0.

Proof

By the very definition of Z p,q rZ^r_{p,q} it consists of elements in filtering degree pp on which \partial decreases the filtering degree to prp-r. Also by definition of differential on a chain complex, \partial decreases the actual degree p+qp+q by one. This explains that \partial restricted to Z p,q rZ^r_{p,q} lands in Z pr,q+r1 Z^\bullet_{p-r,q+r-1}.

Now the image constists indeed of actual boundaries, not just rr-almost boundaries. But since actual boundaries are in particular rr-almost boundaries, we may take the codomain to be Z pr,q+r1 rZ^r_{p-r,q+r-1}.

Proposition

We have a sequence of canonical inclusions

B p,q 0B p,q 1B p,q Z p,q Z p,q 1Z p,q 0. B^0_{p,q} \hookrightarrow B^1_{p,q} \hookrightarrow \cdots B^\infty_{p,q} \hookrightarrow Z^\infty_{p,q} \hookrightarrow \cdots \hookrightarrow Z^1_{p,q} \hookrightarrow Z^0_{p,q} \,.
Proposition

The (r+1)(r+1)-almost cycles are the r\partial^r-kernel inside the rr-almost cycles:

Z p,q r+1=ker(Z p,q r rZ pr,q+r1 r). Z^{r+1}_{p,q} = ker( Z^r_{p,q} \stackrel{\partial^r}{\to} Z^r_{p-r, q+r-1} ) \,.
Proof

An element cF pC p+qc \in F_p C_{p+q} represents

  1. an element in Z p,q rZ^r_{p,q} if cF prC p+q1\partial c \in F_{p-r} C_{p+q-1}

  2. an element in Z p,q r+1Z^{r+1}_{p,q} if even cF pr1C p+q1F prC p+q1\partial c \in F_{p-r-1} C_{p+q-1} \hookrightarrow F_{p-r} C_{p+q-1}.

The second condition is equivalent to c\partial c representing the 0-element in the quotient F prC p+q1/F pr1C p+q1F_{p-r}C_{p+q-1}/ F_{p-r-1}C_{p+q-1}. But this is in turn equivalent to c\partial c being 0 in Z pr,q+r1 rF prC p+q1/F pr1C p+q1Z^r_{p-r,q+r-1} \subset F_{p-r} C_{p+q-1} / F_{p-r-1} C_{p+q-1}.

rr-Almost homology groups: the spectral sequence

Let F C F_\bullet C_\bullet be a filtered chain complex as above.

Definition

For r,p,qr, p, q \in \mathbb{Z} define the rr-almost (p,q)(p,q)-chain homology of the filtered complex to be the quotient of the rr-almost (p,q)(p,q)-cycles by the rr-almost (p,q)(p,q)-boundaries, def. 1:

E p,q r Z p,q rB p,q r ={xF pC p+qxF prC p+q1}(F p+r1C p+q+1)F p1C p+q \begin{aligned} E^r_{p,q} & \coloneqq \frac{Z^r_{p,q}}{B^r_{p,q}} \\ & = \frac{ \left\{ x \in F_p C_{p+q} \,|\, \partial x \in F_{p-r} C_{p+q-1} \right\} } { \partial( F_{p+r-1} C_{p+q+1} ) \oplus F_{p-1} C_{p+q} } \end{aligned}

By prop. 1 the differentials of C C_\bullet restrict on the rr-almost homology groups to maps

r:E p,q rE pr,q+r1 r. \partial^r : E^r_{p,q} \to E^r_{p-r, q + r - 1} \,.
Proposition

Definition 2 indeed gives a spectral sequence in that E , r+1E^{r+1}_{\bullet, \bullet} is indeed the r\partial^r-chain homology of E , rE^r_{\bullet, \bullet}, i.e.

E p,q r+1=ker( r:E p,q rE pr,q+r1 r)im( r:E p+r,qr+1 rE p,q r). E^{r+1}_{p,q} = \frac{ ker(\partial_r : E^r_{p,q} \to E^r_{p-r, q+r-1}) }{ im( \partial_r : E^r_{p+r, q-r+1} \to E^r_{p,q} ) } \,.
Proof

By prop. 3.

Via exact couples

The whole of the spectral sequence can be defined as the spectral sequence of the exact couple

DφDED D\overset{\varphi}{\to} D \to E \to D

where

  • D iH (F i)D \coloneqq \bigoplus_i H^{\bullet}(F_i)

  • and where φ\varphi is the cohomology morphism induced by the inclusion of chain complexes F iF i+1F_i\to F_{i+1}

  • and E iH (F i/F i1)E \coloneqq \bigoplus_i H^\bullet(F_i/F_{i-1}) is the total cohomology of the associated bigraded complex.

At every stage we have a new family of long exact sequences

Properties

In low-degree pages

We characterize the value of the spectral sequence E p,q rE^r_{p,q}, def. 2 for low values of rr and, below in prop. \ref{ConvergingToGenuineHomology}, for rr \to \infty.

Proposition

We have

  • E p,q 0=G pC p+q=F pC p+q/F p1C p+qE^0_{p,q} = G_p C_{p+q} = F_p C_{p+q} / F_{p-1} C_{p+q}

    is the associated p-graded piece of C p+qC_{p+q};

  • E p,q 1=H p+q(G pC )E^1_{p,q} = H_{p+q}(G_p C_\bullet)

    is the chain homology of the associated p-graded complex G pC G_p C_\bullet.

Proof

For r=0r = 0 def. 2 restricts to

E p,q 0=F pC p+qF p1C p+q=G pC p+q E^0_{p,q} = \frac{ F_p C_{p+q}}{F_{p-1} C_{p+q}} = G_p C_{p+q}

because for cF pC p+qc \in F_p C_{p+q} we automatically also have cF pC p+q\partial c \in F_p C_{p+q} since the differential respects the filtering degree by assumption.

For r=1r = 1 def. 2 gives

E p,q 1={cG pC p+qc=0G pC p+q}(F pC p+q)=H p+q(G pC ). E^1_{p,q} = \frac{\{c \in G_p C_{p+q} | \partial c = 0 \in G_p C_{p+q}\} }{\partial(F_p C_{p+q})} = H_{p+q} (G_p C_\bullet) \,.
Remark

There is, in general, a decisive difference between the homology of the associated graded complex H p+q(G pC )H_{p+q}(G_p C_\bullet) and the associated graded piece of the genuine homology G pH p+q(C )G_p H_{p+q}(C_\bullet): in the former the differentials of cycles are required to vanish only up to terms in lower degree, but in the latter they are required to vanish genuinely. The latter expression is instead the value of the spectral sequence for rr \to \infty, see prop. \ref{ConvergingToGenuineHomology} below.

Convergence

General definitions

Definition

Let {E p,q r} r,p,q\{E^r_{p,q}\}_{r,p,q} be a spectral sequence such that for each p,qp,q there is r(p,q)r(p,q) such that for all rr(p,q)r \geq r(p,q) we have

E p,q rr(p,q)E p,q r(p,q). E^{r \geq r(p,q)}_{p,q} \simeq E^{r(p,q)}_{p,q} \,.

Then one says that

  1. the bigraded object

    E {E p,q } p,q{E p,q r(p,q)} p,q E^\infty \coloneqq \{E^\infty_{p,q}\}_{p,q} \coloneqq \{ E^{r(p,q)}_{p,q} \}_{p,q}

    is the limit term of the spectral sequence;

  • the spectral sequence abuts to E E^\infty.
Example

If for a spectral sequence there is r sr_s such that all differentials on pages after r sr_s vanish, rr s=0\partial^{r \geq r_s} = 0, then {E r s} p,q\{E^{r_s}\}_{p,q} is limit term for the spectral sequence. One says in this cases that the spectral sequence collapses at r sr_s.

Proof

By the defining relation

E p,q r+1ker( pr,q+r1 r)/im( p,q r)=E pq r E^{r+1}_{p,q} \simeq ker(\partial^r_{p-r,q+r-1})/im(\partial^r_{p,q}) = E^r_{pq}

the spectral sequence becomes constant in rr from r sr_s on if all the differentials vanish, so that ker( p,q r)=E p,q rker(\partial^r_{p,q}) = E^r_{p,q} for all p,qp,q.

Example

If for a spectral sequence {E p,q r} r,p,q\{E^r_{p,q}\}_{r,p,q} there is r s2r_s \geq 2 such that the r sr_sth page is concentrated in a single row or a single column, then the the spectral sequence degenerates on this pages, example 1, hence this page is a limit term, def. 3. One says in this case that the spectral sequence collapses on this page.

Proof

For r2r \geq 2 the differentials of the spectral sequence

r:E p,q rE pr,q+r1 r \partial^r \colon E^r_{p,q} \to E^r_{p-r, q+r-1}

have domain and codomain necessarily in different rows an columns (while for r=1r = 1 both are in the same row and for r=0r = 0 both coincide). Therefore if all but one row or column vanish, then all these differentials vanish.

Definition

A spectral sequence {E p,q r} r,p,q\{E^r_{p,q}\}_{r,p,q} is said to converge to a graded object H H_\bullet with filtering F H F_\bullet H_\bullet, traditionally denoted

E p,q rH , E^r_{p,q} \Rightarrow H_\bullet \,,

if the associated graded complex {G pH p+q} p,q{F pH p+q/F p1H p+q}\{G_p H_{p+q}\}_{p,q} \coloneqq \{F_p H_{p+q} / F_{p-1} H_{p+q}\} of HH is the limit term of EE, def. 3:

E p,q G pH p+q p,q. E^\infty_{p,q} \simeq G_p H_{p+q} \;\;\;\;\;\;\; \forall_{p,q} \,.
Remark

In practice spectral sequences are often referred to via their first non-trivial page, often also the page at which it collapses, def. 2, oftne the second page. Then one often uses notation such as

E p,q 2H E^2_{p,q} \Rightarrow H_\bullet

to be read as “There is a spectral sequence whose second page is as shown on the left and which converges to a filtered object as shown on the right.”

Definition

A spectral sequence {E p,q r}\{E^r_{p,q}\} is called a bounded spectral sequence if for all n,rn,r \in \mathbb{Z} the number of non-vanishing terms of the form E k,nk rE^r_{k,n-k} is finite.

Example

A spectral sequence {E p,q r}\{E^r_{p,q}\} is called

  • a first quadrant spectral sequence if all terms except possibly for p,q0p,q \geq 0 vanish;

  • a third quadrant spectral sequence** if all terms except possibly for p,q0p,q \leq 0 vanish.

Such spectral sequences are bounded, def. 5.

Proposition

A bounded spectral sequence, def. 5, has a limit term, def. 3.

Proof

First notice that if a spectral sequence has at most NN non-vanishing terms of total degree nn on page rr, then all the following pages have at most at these positions non-vanishing terms, too, since these are the homologies of the previous terms.

Therefore for a bounded spectral sequence for each nn there is L(n)L(n) \in \mathbb{Z} such that E p,np r=0E^r_{p,n-p} = 0 for all pL(n)p \leq L(n) and all rr. Similarly there is T(n)T(n) \in \mathbb{Z} such E nq,q r=0E^r_{n-q,q} = 0 for all qT(n)q \leq T(n) and all rr.

We claim then that the limit term of the bounded spectral sequence is in position (p,q)(p,q) given by the value E p,q rE^r_{p,q} for

r>max(pL(p+q1),q+1L(p+q+1)). r \gt max( p-L(p+q-1), q + 1 - L(p+q+1) ) \,.

This is because for such rr we have

  1. E pr,q+r1 r=0E^r_{p-r, q+r-1} = 0 because pr<L(p+q1)p-r \lt L(p+q-1), and hence the kernel ker( pr,q+r1 r)=0ker(\partial^r_{p-r,q+r-1}) = 0 vanishes;

  2. E p+r,qr+1 r=0E^r_{p+r, q-r+1} = 0 because qr+1<T(p+q+1)q-r + 1 \lt T(p+q+1), and hence the image im( p,q r)=0im(\partial^r_{p,q}) = 0 vanishes.

Therefore

E p,q r+1 =ker( pr,q+r1 r)/im( p,q r) E p,q r/0 E p,q r. \begin{aligned} E^{r+1}_{p,q} &= ker(\partial^r_{p-r,q+r-1})/im(\partial^r_{p,q}) \\ & \simeq E^r_{p,q}/0 \\ & \simeq E^r_{p,q} \end{aligned} \,.

Convergence of bounded filtrations

Definition

A filtration F C F_\bullet C_\bullet on a chain complex C C_\bullet is called a bounded filtration if for all nn \in \mathbb{Z} there is L(n),T(n)L(n), T(n) \in \mathbb{Z} such that

F p<S(n)C n=0F p>T(n)C nC n. F_{p \lt S(n)}C_n = 0 \;\;\;\;\; F_{p \gt T(n)}C_n \simeq C_n \,.
Proposition

The spectral sequence of a filtered complex, def. 6, has a limit term, def. 3.

Proof

The spectral sequence of a filtered complex F C F_\bullet C_\bullet is by def. 2 at E p,q rE^r_{p,q} a quotient of a subobject of F pC p+qF_p C_{p+q}. By def. 6 therefore there are for each n,rn,r \in \mathbb{Z} finitely many non-vanishing terms of the form E p,np rE^r_{p,n-p}. Therefore the spectral sequence is bounded, def. 5 and hence has a limit term by prop. 6.

Example

If XX \in Top is a CW complex with cell filtration X 0X 1XX_0 \hookrightarrow X_1 \hookrightarrow \cdots \hookrightarrow X, then the induced filtering

F pC (X)C (X n) F_p C_\bullet(X) \coloneqq C_\bullet(X_n)

on its singular chain complex C (X)C_\bullet(X) yields a first-quadrant spectral sequence, example 3. Therefore it has a limit term.

Before saying what that the spectral sequence of a filtered complex converges to the homology of that complex, we need to be careful about what the filtering is on that homology:

Definition

For F C F_\bullet C_\bullet a filtered complex, write for pp \in \mathbb{Z}

F pH (C)image(H (F pC )H (C )). F_p H_\bullet(C) \coloneqq image( H_\bullet(F_p C_\bullet) \to H_\bullet(C_\bullet) ) \,.

This defines a filtering F H (C)F_\bullet H_\bullet(C) of the homology, regarded as a graded object.

Proposition

If the spectral sequence of a filtered complex F C F_\bullet C_\bullet, def. 6 has a limit term, def. 3, then it converges, def. 4, to the chain homology of this complex:

E p,q rH (C). E^r_{p,q} \Rightarrow H_\bullet(C) \,.

Hence for each p,qp,q there is r(p,q)r(p,q) such that

E p,q rr(p,q)G pH p+q(C), E^{r \geq r(p,q)}_{p,q} \simeq G_p H_{p+q}(C) \,,

with the filtering on the right as in def. 7.

Proof

By assumption, there is for each p,qp,q an r(p,q)r(p,q) such that for all rr(p,q)r \geq r(p,q) the rr-almost cycles and rr-almost boundaries, def. 1, in F pC p+qF_p C_{p+q} are the ordinary cycles and boundaries. Therefore for rr(p,q)r \geq r(p,q) def. 2 gives E p,q rG pH p+q(C)E^r_{p,q} \simeq G_p H_{p+q}(C).

Via exact couples

It is instructive to note that in the nnth derived exact couple φ nDE (n)φ nD\varphi^n D\to E_{(n)} \to \varphi^n D\to{}, the hidden part φ nD\varphi^n D is the submodule D (n)D_{(n)} of iH(F i+n)\bigoplus_{i} H(F_{i+n}), as it meets H(F i+n)H(F_{i+n}) representable by elements of F iF_i; that is, we may sensibly call it

F iD (n)=ker(d)F iF idF i+n.F_{i} D_{(n)} = \frac{\ker(d)\cap F_i}{F_i\cap dF_{i+n}}.

Separating the grades, the exactness of the couple at E (n)E_{(n)} then says

F iH j(F i+n)F i+1H j(F i+n+1)E (n) iF iH j+1(F i+n)F i+1H j+1(F i+n+1) F_{i} H^j(F_{i+n})\to F_{i+1}H^j(F_{i+n+1}) \to E_{(n)}^{i\dots} \to F_i H^{j+1}(F_{i+n}) \to F_{i+1}H^{j+1}(F_{i+n+1})

One can see this as converging (if it sensibly converges) to either a subquotient of F iF_i or to a submodule F iH (C)<H (C)F_i H^\bullet(C) \lt H^\bullet(C). Taking the latter interpretation, we hope to find in the limiting case exact sequences

F iH j(C)F i+1H j(C)E () iF iH j+1(C)F i+1H j+1(C). F_i H^j(C)\to F_{i+1} H^j(C) \to E_{(\infty)}^{i\dots} \to F_i H^{j+1}(C) \to F_{i+1} H^{j+1}(C).

At this stage one can check that the morphisms F iH j(C)F i+1H j(C)F_i H^j(C)\to F_{i+1} H^j(C) are indeed definable, and in fact injective, so that whatever E ()E_{(\infty)} should be, the morphism E () iF iH j+1E_{(\infty)}^{i\dots}\to F_i H^{j+1} is null; that is, our long-exact sequence breaks up into the short exact sequences

0F iH j(C)F i+1H j(C)E () i0. 0\to F_i H^j(C) \to F_{i+1} H^j(C)\to E_{(\infty)}^{i\dots} \to 0 .

In summary, if the spectral sequence E (n)E_{(n)} converges in a sensible way to the correct thing E ()E_{(\infty)}, then that correct thing is also the associated graded module of the filtration of H (C)H^\bullet(C) induced by the filtration of CC.

Examples

2-term filtering

The special case where the filtering has just length one is that where we simply have a sub-complex C (1)C C^{(1)}_\bullet \hookrightarrow C_\bullet and want to compute the homology of C C_\bullet from that of C (1)C^{(1)}_\bullet and C /C (1)C_\bullet/C^{(1)}_\bullet.

This case is easily solved by elementary means and it serves as an instructive blueprint for the general case.

Proposition

Given a sub-chain complex C (1)C C^{(1)}_\bullet \hookrightarrow C_\bullet, consider the following constructions

  1. Consider the short exact sequence

    0C (1)C C /C (1)0 0 \to C^{(1)}_\bullet \to C_\bullet \to C_\bullet/C^{(1)}_\bullet \to 0
  2. Its long exact sequence in homology contains the connecting homomorphism

    δ:H (C /C (1))H 1(C (1)). \delta : H_\bullet(C_\bullet/C^{(1)}_\bullet) \to H_{\bullet-1}(C^{(1)}_\bullet) \,.

    Define

    • G 1H kerδG_1 H_\bullet \coloneqq ker \delta

    • G 2H cokerδG_2 H_\bullet \coloneqq coker \delta.

Then H (C )H_\bullet(C_\bullet) is sits in the short exact sequence

0G 0H H (C )G 1H 0. 0 \to G_0 H_\bullet \to H_\bullet(C_\bullet) \to G_1 H_\bullet \to 0 \,.

(…)

Homology of a tensor product of chain complexes

Consider two chain complexes C ,C C_\bullet, C'_\bullet of vector spaces over a field kk, both in non-negative degree.

Their tensor product of chain complexes is

C C = i+j=kC iC j C_\bullet \otimes C'_\bullet = \oplus_{i+j = k} C_i \otimes C_j

with differential on homogenous elements

(αβ)=(α)β+(1) degααβ. \partial (\alpha \otimes\beta) = (\partial \alpha) \otimes \beta + (-1)^{deg \alpha} \alpha \otimes \partial \beta \,.

We may compute the chain homology of CCC \otimes C' by a filter spectral sequence as follows.

Define a filtration on CCC \otimes C' by

F p(CC) k ipC iC k1. F_p(C \otimes C')_{k} \coloneqq \oplus_{i \leq p} C_i \otimes C_{k-1} \,.

This means that the associated graded object is simply

E p,q 0=G p(CC) p+q=C pC q. E^0_{p,q} = G_p (C \otimes C')_{p+q} = C_p \otimes C'_q \,.

The differential on this is r=0=(1) pid C\partial_{r = 0} = (-1)^p id_{C} \otimes \partial'. Hence the universal coefficient theorem gives

E p,q 1=C pH q(C). E^1_{p,q} = C_p \otimes H_q(C') \,.

The next differential is 1=id C\partial_1 = \partial \otimes id_{C'} Since kk is assumed to be a field we have thus

E p,q 2=H p(C H q(C ))=H p(C)H q(C). E^2_{p,q} = H_p(C_\bullet \otimes H_q(C'_\bullet)) = H_p(C)\otimes H_q(C') \,.

Therefore every element in E p,q 2E^2_{p,q} is represented by a tensor product of a CC-cycle with a CC'-cycle and is hence itself a (CC)(C \otimes C')-cycle. Since the differentials in the spectral sequence all come from the differential on CCC \otimes C', this means that all higher differentials vanish, and so the sequence collapses on the E 2E^2-page.

The convergence of the spectral sequence to the the homology of CCC \otimes C' thus says that this is given by

H k(CC) i+j=kH i(C)H j(C). H_{k}(C \otimes C') \simeq \oplus_{i+j = k} H_i(C) \otimes H_j(C') \,.

Homology of the total complex of a double complex

The total complex of a double complex is naturally filtered either by either row-degree of column-degree. The corresponding filtering spectral sequence converges under good conditions to the homology of the total complex. See at spectral sequence of a double complex.

Singular homology of a CW-complex

Let XTopX \in Top be a CW-complex equipped explicitly with the structure of a filtered topological space X 0X nXX^0 \hookrightarrow \cdots \hookrightarrow X^n \hookrightarrow \cdots X. This induces on the singular homology complex C (X)C_\bullet(X) the structure of a filtered chain complex by

F qC (X)C (X p). F_q C_\bullet(X) \coloneqq C_\bullet(X^p) \,.

We discuss how the corresponding spectral sequence shows that the singular homology of XX coincides with the cellular homology of the filtering.

The associated graded object is

G pC p+q(X)=E p,q 0=C p+q(X p)/C p+q(X p1). G_p C_{p+q}(X) = E^0_{p,q} = C_{p+q}(X^p)/C_{p+q}(X^{p-1}) \,.
Proposition

The chain homology of the associated graded chain complex is the relative homology

E p,q 1=H p+q(X pX p1). E^1_{p,q} = H_{p+q}(X^p | X^{p-1}) \,.

Now by assumption that X X^\bullet is the cell decomposition of a cell complex we have

H p+q(X pX p+1){[pCells(X)] q=0 0 otherwise. H_{p+q}(X^p | X^{p+1}) \simeq \left\{ \array{ \mathbb{Z}[pCells(X)] & q = 0 \\ 0 & otherwise } \right. \,.

The chain homology of

:H p(X p,X p1)H p1(X p1,X p2) \partial \colon H_p(X^p, X^{p-1}) \to H_{p-1}(X^{p-1}, X^{p-2})

is the cellular chain homology H p cell(X)H^{cell}_p(X). One finds that

E p,q 2={H p cell(X) q=0 0 otherwise. E^2_{p,q} = \left\{ \array{ H_p^{cell}(X) & q = 0 \\ 0 & otherwise } \right. \,.

Since this is concentrated in the q=0q = 0 -row all higher-rr differentials vanish.

Hence H p(X)H p cell(X)H_p(X) \simeq H^{cell}_p(X).

Remark

The generalization of this argument from ordinary homology to generalized homology is given by the Atiyah–Hirzebruch spectral sequence.

Serre’s spectral sequence of a fibration

We indicate the Leray-Serre spectral sequence of a Serre fibration as a special case of the filtering spectral sequence. For more discussion see there.

A X * B\begin{matrix} A & \to & X \\ \downarrow & & \downarrow \\ * & \to & B \end{matrix}

be a Serre fibration of pointed topological spaces in which BB is a connected CW-complex. Then the kk-skeleta of BB naturally give filtered-space structures to both BB and XX:

X (k) X (k+1) X B k B k+1 B\begin{matrix} X_{(k)} & \to & X_{(k+1)} & \to & X \\ \downarrow & & \downarrow & &\downarrow \\ B_{k} & \to & B_{k+1} & \to & B \end{matrix}

and in turn induce filtrations of the singular chain complex of XX.

The homology Serre spectral sequence for the fibration is essentially that of this filtered complex.

It is straight-forward to show that the pair (X k+1;X k)(X_{k+1};X_k) is kk-connected, and in particular the relative homology H i(X k+1;X k)H_i(X_{k+1};X_k) vanishes for iki\leq k; this ensures that the spectral sequence is 1st/3rd quadrant (And nota bene: this is also a handy way to remember what the bigrading actually is).

There is also an important result about the second page of this spectral sequence

Theorem

The page E (2)E^{(2)} of the homology Serre spectral sequence is given by

E p,q (2)H p(B, q(A b)) E^{(2)}_{p,q} \simeq H_p(B,\mathcal{H}_q(A|_{b}))

the homology of BB with coefficients in the local system defined by the action of Π 1(B)\Pi_1 (B) on H q(A b)H_q(A|_{b}). In the special case that BB is simply connected, these local systems are canonically equivalent to H q(A)H_q(A), the homology of the fiber over the basepoint.

tower diagram/filteringspectral sequence of a filtered stable homotopy type
filtered chain complexspectral sequence of a filtered complex
Whitehead towerAtiyah-Hirzebruch spectral sequence
chromatic towerchromatic spectral sequence
skeleta of simplicial objectspectral sequence of a simplicial stable homotopy type
skeleta of Cech nerve of E-∞ algebraAdams spectral sequence
filtration by support
slice filtrationslice spectral sequence

References

A detailed account is in

  • Jennifer Orlich, Spectral sequences and an application (pdf)

Lecture notes include

and section 3 of

For further references see those listed at spectral sequence, for instance section 5 of

Revised on December 2, 2013 08:24:28 by Urs Schreiber (145.116.131.63)