spectral sequence

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###Context###
#### Homological algebra
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[[!include homological algebra - contents]]
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#### Algebraic topology
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[[!include algebraic topology - contents]]
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#### Stable Homotopy theory
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[[!include stable homotopy theory - contents]]
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#Contents#
* table of contents
{:toc}

## Idea
 {#Idea}

The notion of _spectral sequence_ is an [[algorithm]] or computational tool in [[homological algebra]] and more generally in [[homotopy theory]] which allows to compute [[chain homology|chain homology groups]]/[[homotopy groups]] of _bi_-[[graded objects]] from the homology/homotopy of the two graded components.

Notably there is a spectral sequence for computing the homology of the [[total complex]] of a [[double complex]] from the homology of its row and column complexes separately.
This in turn allows to compute [[derived functors]] of composite functors $G\circ F$ from the double complex $\mathbb{R}^\bullet G (\mathbb{R}^\bullet F(-))$ obtained by non-totally deriving the two functors separately (called the [Grothendieck spectral sequence](#GrothendieckSpectralSequence)). By choosing various functors $F$ and $G$ here this gives rise to various important classes of examples of spectral sequences, see [below](#SpecialGrothendieckSpectralSequence).

More concretely, a homology spectral sequence is a sequence of graded chain complexes that provides the higher order corrections to the _naïve_ idea of computing the homology of the [[total complex]] $Tot(V)_\bullet$ of a [[double complex]] $V_{\bullet, \bullet}$: by first computing those of the vertical differential, then those of the horizontal differential induced on these vertical homology groups (or the other way around). This simple idea in general does not produce the correct homology groups of $Tot(V)_\bullet$, but it does produce a "first-order approximation" to them, in a useful sense. The spectral sequence is the sequence of higher-order corrections that make this naive idea actually work.

Being, therefore, an iterative perturbative approximation scheme of bigraded differential objects, fully-fledged spectral sequences can look a bit intricate. However, a standard experience in mathematical practice is that for most problems of practical interest the relevant spectral sequence "perturbation series" yields the exact result already at the second stage. This reduces the computational complexity immensely and makes spectral sequences a wide-spread useful computational tool.

Despite their name, there seemed to be nothing specifically "spectral" about spectral sequences, for any of the technical meanings of the word [[spectrum - disambiguation|spectrum]]. Together with the concept, this term was introduced by [[Jean Leray]] and has long become standard, but was never really motivated (see p. 5 of [Chow](#Chow)). But then, by lucky coincidence it turns out in the refined context of [[stable (∞,1)-category]] theory/[[stable homotopy theory]] that spectral sequences frequently arise by considering the [[homotopy groups]] of _[[sequential diagram|sequences]] of [[spectra]]_. This is discussed at _[[spectral sequence of a filtered stable homotopy type]]_.

While therefore spectral sequences are a notion considered in the context of [[homological algebra]] and more generally in [[stable homotopy theory]],
there is also an "unstable" or [[nonabelian cohomology|nonabelian]] variant of the notion in plain [[homotopy theory]], called _[[homotopy spectral sequence]]_.

## Definition

We give the general definition of a (co)homology spectral sequence. For motivation see the example _[Spectral sequence of a filtered complex](#SpectralSequenceOfFilteredComplex)_ below.

Throughout, let $\mathcal{A}$ be an [[abelian category]].

### Spectral sequence

+-- {: .num_defn #CohomologySpectralSequence}
###### Definition

A __cohomology spectral sequence__ in $\mathcal{A}$ is

* a family $(E^{p,q}_r)$ of [[objects]] in $\mathcal{A}$, for all [[integers]] $p,q,r$ with $r\geq 1$

  (for a fixed $r$ these are said to form the __$r$-th page__ of the spectral sequence)

* for each $p,q,r$ as above a [[morphism]] (called the __differential__)

  \[
    \label{FormOfDifferentialInCohomologySpectralSequence}
    d^{p,q}_r
    \;\colon\;
    E^{p,q}_r
      \longrightarrow
    E^{p+r,q-r+1}_r
  \]

  satisfying $d_r^2 = 0$ (more precisely, $d_r^{p+r,q-r+1}\circ d_r^{p,q} = 0$)

* [[isomorphisms]] $\;$ $\alpha_r^{p,q}: H^{p,q}(E_r)\to E^{p,q}_{r+1}$ where
  the [[chain cohomology]] is given by

  $$
    H^{p,q}(E_r) = \mathrm{ker} d^{p,q}_r/ \mathrm{im} d^{p-r,q+r-1}_r
   \,.
  $$

=--

Analogously, a **homology spectral sequence** is collection of objects $(E_{p,q}^r)$ with the differential $d_r$ of degree $(-r,r-1)$.

### Convergence
 {#ConvergenceOfSpectralSequences}

+-- {: .num_defn #LimitTerm}
###### Definition

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

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

Then one says equivalently that

1. the [[bigraded object]]

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

   is the **limit term** of the spectral sequence;

1.  the spectral sequence **abuts** to $E^\infty$.

=--

+-- {: .num_example #Degeneration}
###### Example

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

=--

+-- {: .proof}
###### Proof

By the defining relation

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

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

=--

+-- {: .num_example #Collaps}
###### Example

If for a [[spectral sequence]] $\{E^r_{p,q}\}_{r,p,q}$ there is $r_s \geq 2$ such that the $r_s$th page is concentrated in a single row or a single column, then the spectral sequence degenerates on this pages, example \ref{Degeneration}, hence this page is a limit term, def. \ref{LimitTerm}. One says in this case that the spectral sequence **collapses** on this page.

=--

+-- {: .proof}
###### Proof

For $r \geq 2$ the [[differentials]] of the spectral sequence

$$
  \partial^r \colon E^r_{p,q} \to E^r_{p-r, q+r-1}
$$

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

=--

+-- {: .num_defn #Convergence}
###### Definition

A [[spectral sequence]] $\{E^r_{p,q}\}_{r,p,q}$ **converges weakly** to a [[graded object]] $H_\bullet$ with [[exhaustive filtering]] $F_\bullet H_\bullet$, traditionally denoted

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

if the [[associated graded]] complex $\{G_p H_{p+q}\}_{p,q} \coloneqq \{F_p H_{p+q} / F_{p-1} H_{p+q}\}$ of $H$ is the limit term of $E$, def. \ref{LimitTerm}:

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

Furthermore one says that $E$

* **converges** if $F_\bullet$ is also a [[Hausdorff filtration]];

* **converges strongly** if $F_\bullet$ is also a [[complete Hausdorff filtration]].

=--

([Boardman 99, def. 2.5](#Boardman99)) See also _[[conditional convergence of spectral sequences]]_.

+-- {: .num_remark }
###### Remark

In practice, spectral sequences are often referred to via their first interesting page, usually the first or the second. Then one often uses notation such as

$$
  E^1_{p,q} \;\Rightarrow\; H_\bullet
$$

or

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

to be read as "There is a spectral sequence whose first/second page is as shown on the left and which converges (weakly, strongly, or conditionally) to a filtered object as shown on the right."

=--

+-- {: .num_example}
###### Example

The nature of the convergence condition in def. \ref{Convergence} is well illuminated for instance by the [[Serre spectral sequence|Serre]]-[[Atiyah-Hirzebruch spectral sequence]]: for $E^\bullet$ a [[generalized cohomology theory]] and $X$ a finite [[CW-complex]], then it converges to the $E$-cohomology of $X$, filtered by $E$-cohomology relative to the [[simplicial skeleton|skeleta]] $F^p E^{\bullet}(X) im(  E^\bullet(X,X_{p-1}) \to E^\bullet(X))$. Moreover, the second page is the [[ordinary cohomology]] of $X$ with coefficients in the $E$-ground ring, like so:

$$
  H^p(X,E^q(\ast)) \Rightarrow E^{p+q}(X)
  \,.
$$

Here the elements on the left in bidegree $(p,q)$ are manifestly given by cocycles that trivialize on the $(p-1)$-skeleton (being $p$-cocycles), hence it is natural that these contribute to the filtering stage $F^p E^{p+q}(X) = im(E^{p+q}(X,X_{p-1}) \to E^{p+q}(X))$.

=--

+-- {: .num_remark}
###### Remark

In applications one is interested in computing the $H_n$ and uses spectral sequences converging to this as tools for approximating $H_n$ in terms of the given filtration.

Therefore usually spectral sequences are required to converge in each degree, or even that for each pair $(p,q)$ there exists an $r_0$ such that for all $r\geq r_0$, $d_r^{p-r,q+r-1} = 0$.

=--

+-- {: .num_remark}
###### Remark

If $(E^r)$ collapses at $r$, then it converges to $H_\bullet$ with $H_n$ being the unique entry $E_{p,q}^r$ on the non-vanishing row/column with $p+q = n$.

=--

### Boundedness
 {#Boundedness}

+-- {: .num_defn #BoundedSpectralSequence}
###### Definition

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

=--

+-- {: .num_defn #QuadrantSpectralSequence}
###### Example

A [[spectral sequence]] $\{E^r_{p,q}\}$ is called

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

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

Such spectral sequences are bounded, def. \ref{BoundedSpectralSequence}.

=--

+-- {: .num_prop #BoundedSpectralSequenceHasLimitTerm}
###### Proposition

A bounded spectral sequence, def. \ref{BoundedSpectralSequence}, has a limit term, def. \ref{LimitTerm}.

=--

+-- {: .proof}
###### Proof

First notice that if a spectral sequence has at most $N$ non-vanishing terms of total degree $n$ on page $r$, 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 $n$ there is $L(n) \in \mathbb{Z}$ such that $E^r_{p,n-p} = 0$ for all $p \leq L(n)$ and all $r$. Similarly there is $T(n) \in \mathbb{Z}$ such $E^r_{n-q,q} = 0$ for all $q \leq T(n)$ and all $r$.

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

$$
  r \gt max(  p-L(p+q-1), q + 1 - T(p+q+1) )
  \,.
$$

This is because for such $r$ we have

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

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

Therefore

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

=--

### Extension problem
 {#ExtensionProblem}

Given a spectral sequence, then even if it converges strongly (def. \ref{Convergence}), computing its infinity-page still just gives the [[associated graded]] of the [[filtered object]] that it converges to, not the filtered object itself. The latter is in each filter stage an [[extension]] of the previous stage by the corresponding stage of the infinity-page, but there are in general several possible extensions (the trivial extension or some twisted extensions). The problem of determining these extensions and hence the problem of actually determining the filtered object from a spectral sequence converging to it is often referred to as the _extension problem_.

More in detail, consider, for definiteness, a cohomology spectral sequence (def. \ref{CohomologySpectralSequence}) converging (def. \ref{Convergence}) to some [[filtered object|filtered]] $F^\bullet H^\bullet$

$$
  E^{p,q} \;\Rightarrow\; H^\bullet
  \,.
$$

Then by definition of convergence there are isomorphisms

$$
  E_\infty^{p,\bullet} \simeq F^p H^{p + \bullet} / F^{p+1} H^{p + \bullet}
  \,.
$$

Equivalently this means that there are [[short exact sequences]] of the form

$$
  0 \to F^{p+1}H^{p +\bullet} \hookrightarrow F^p H^{p +\bullet} \longrightarrow E_\infty^{p,\bullet} \to 0
  \,.
$$

for all $p$. The extension problem then is to inductively deduce $F^p H^\bullet$ from knowledge of $F^{p+1}H^\bullet$ and $E_\infty^{p,\bullet}$.

In good cases these short exact sequences happen to be [[split exact sequences]], which means that the extension problem is solved by the [[direct sum]]

$$
  F^p H^{p+\bullet} \simeq F^{p+1} H^{p+\bullet} \oplus E_\infty^{p,\bullet}
  \,.
$$

But in general this need not be the case.

One sufficient condition that these exact sequences split is that they consist of homomorphisms of $R$-[[modules]], for some $R$, and that $E_\infty^{p,\bullet}$ are [[projective modules]] (for instance [[free modules]]) over $R$. Because then the [[Ext]]-group $Ext^1_R(E_\infty^{p,\bullet},-)$ vanishes, and hence all extensions are trivial, hence split.

So for instance for every spectral sequence in [[vector spaces]] the extension problem is trivial (since every vector space is a free module).

## Examples
 {#Examples}

The basic class of examples are

* [Spectral sequences of filtered complexes](#SpectralSequenceOfFilteredComplex)

which compute the cohomology of a [[filtered object|filtered complex]] from the cohomologies of its [[associated graded objects]].

From this one obtains as a special case the class of

* [Spectral sequences of double complexes](#SpectralSequenceOfADoubleComplex)

which compute the cohomology of the [[total complex]] of a [[double complex]] using the two canonical filtrations of this by row- and by column-degree.

From this in turn one obtains as a special case the class of

* [Grothendieck spectral sequences](#GrothendieckSpectralSequence)

which compute the [[derived functor]] $\mathbb{R}^\bullet(G \circ F (-))$ of the composite of two functors from the spectral sequence of the double complex $\mathbb{R}^\bullet (F (\mathbb{R}^\bullet G (-)))$.

Many special cases of this for various choices of $F$ and $G$ go by special names, this we tabulate at

* _[List of examples](#ListOfExamples)_.

### Spectral sequence of a filtered complex
 {#SpectralSequenceOfFilteredComplex}

The fundamental example of a spectral sequence, from which essentially all the other examples arise as special cases, is the _[[spectral sequence of a filtered complex]]_. (See there for details). Or more generally in [[stable homotopy theory]]: the [[spectral sequence of a filtered stable homotopy type]].

If a [[cochain complex]] $C^\bullet$ is equipped with a [[filtered object|filtration]] $F^\bullet C^\bullet$, there is an induced filtration $F^\bullet H(C)$ of its [[cohomology groups]], according to which levels of the filtration contain representatives for the various cohomology classes.

A filtration $F$ also gives rise to an  [[associated graded object]] $Gr(F)$, whose grades are the successive level inclusion [[cokernels]].  Generically, the operations of grading and cohomology do not commute:

$$
  Gr(F^\bullet H^\bullet(C)) \neq H^\bullet (Gr(F^\bullet) C)
  \,.
$$

But the [[spectral sequence of a filtered complex|spectral sequence associated to a filtered complex]] $F^\bullet C^\bullet$, passes through $H^\bullet (Gr(F^\star) C)$ in the page $E_{(1)}$ and in good cases converges to $Gr(F^* H^\bullet(C))$.

### Spectral sequence of a double complex
 {#SpectralSequenceOfADoubleComplex}

The [[total complex]] of a [[double complex]] is naturally filtered in two ways: by columns and by rows. By the above [[spectral sequence of a filtered complex]] this gives two different spectral sequences associated computing the cohomology of a double complex from the cohomologies of its rows and columns. Many other classes of spectral sequences are special cases of this cases, notably the [[Grothendieck spectral sequence]] and _its_ special cases.

This is discussed at _[[spectral sequence of a double complex]]_.

### Spectral sequences for hyper-derived functors
  {#HyperDerivedFunctors}

From the [[spectral sequence of a double complex]] one obtains as a special case a spectral sequence that computes [[hyper-derived functors]].

(...)

### Grothendieck spectral sequence
 {#GrothendieckSpectralSequence}

The _[[Grothendieck spectral sequence]]_ computes the composite of two [[derived functors]] from the two derived functors separately.

Let $\mathcal{A} \stackrel{F}{\to} \mathcal{B} \stackrel{G}{\to} \mathcal{C}$ be two [[left exact functors]] between [[abelian categories]].

Write $R^p F : \mathcal{D} \to Ab$ for the [[cochain cohomology]] of the [[derived functor]] of $F$ in degree $p$ etc. .

+-- {: .num_theorem}
###### Theorem

If $F$ sends [[injective objects]] of $\mathcal{A}$ to $G$-[[acyclic objects]] in $\mathcal{B}$ then for each $A \in \mathcal{A}$ there is a [first quadrant](#FirstQuadrant) cohomology spectral sequence

$$
  E_r^{p,q} := (R^p G \circ R^q F)(A)
$$

that converges to the right [[derived functor]] of the composite functor

$$
  E_r^{p,q} \Rightarrow R^{p+q} (G \circ F)(A).
$$

Moreover

1. the [[edge maps]] in this spectral sequence are the canonical morphisms

   $$
     R^p G (F A) \to R^p (G \circ F)(A)
   $$

   induced from applying $F$ to an injective resolution
   $A \to \hat A$ and the morphism

   $$
     R^q (G \circ F)(A) \to G(R^q F (A))
     \,.
   $$

1. the [[exact sequence]] of low degree terms is

   $$
     0 \to (R^1 G)(F(A)) \to R^1(G \circ F)(A)
      \to G(R^1(F(A)))
      \to (R^2 G)(F(A))
      \to R^2(G \circ F)(A)
   $$

=--

This is called the _[[Grothendieck spectral sequence]]_.

+-- {: .proof}
###### Proof

Since for $A \to \hat A$ an injective [[resolution]] of $A$ the complex $F(\hat A)$ is a chain complex not concentrated in a single degree, we have that $R^p (G \circ F)(A)$ is equivalently the [[hyper-derived functor]] evaluation $\mathbb{R}^p(G) (F(A))$.

Therefore the second spectral sequence discussed at [hyper-derived functor spectral sequences](#HyperDerivedFunctors) converges as

$$
  (R^p G)H^q(F(\hat A)) \Rightarrow R^p (G \circ F)(A)
  \,.
$$

Now since by construction $H^q(F(\hat A)) = R^q F(A)$ this is a spectral sequence

$$
  (R^p G)(R^q F) A) \Rightarrow R^p (G \circ F)(A)
  \,.
$$

This is the Grothendieck spectral sequence.

=--

### Special Grothendieck spectral sequences
 {#SpecialGrothendieckSpectralSequence}

* [Leray spectral sequence](#LeraySpectralSequence)

* [Hochschild-Serre spectral sequence](#HochschildSerreSpectralSequence)

* [Base-change spectral sequence](#BaseChangeSpectralSequence)

#### Leray spectral sequence
 {#LeraySpectralSequence}

The _[[Leray spectral sequence]]_ is the special case of the
[[Grothendieck spectral sequence]] for the case where the two functors being composed are a [[direct image|push-forward]] of [[sheaves]] of [[abelian groups]] along a [[continuous map]] $f : X \to Y$ followed by the push-forward $X \to *$ to the point. This yields a spectral sequence that computes the [[abelian sheaf cohomology]] on $X$ in terms of the abelian sheaf cohomology on $Y$.

+-- {: .num_theorem}
###### Theorem

Let $X, Y$ be suitable [[sites]] and $f : X \to Y$ be a morphism of sites. () Let $\mathcal{C} = Ch_\bullet(Sh(X,Ab))$ and $\mathcal{D} = Ch_\bullet(Sh(Y,Ab))$ be the [[model structure on chain complexes|model categories of complexes of sheaves of abelian groups]]. The [[direct image]] $f_*$ and [[global section]] functor $\Gamma_Y$ compose to $\Gamma_X$:

$$
  \Gamma_X : \mathcal{C} \stackrel{f_*}{\to} \mathcal{D} \stackrel{\Gamma_Y}{\to} Ch_\bullet(Ab)
  \,.
$$

Then for $A \in Sh(X,Ab)$ a sheaf of abelian groups on $X$ there is a cohomology spectral sequence

$$
  E_r^{p,q} := H^p(Y, R^q f_* A)
$$

that converges as

$$
  E_r^{p,q} \Rightarrow H^{p+q}(X, A)
$$

and hence computes the cohomology of $X$ with coefficients in $A$ in terms of the cohomology of $Y$ with coefficients in the push-forward of $A$.

=--

#### Base change spectral sequence for $Tor$ and $Ext$
 {#BaseChangeSpectralSequence}

For $R$ a [[ring]] write $R$[[Mod]] for its category of [[modules]].
Given a [[homomorphism]] of [[ring]] $f : R_1 \to R_2$ and an $R_2$-[[module]] $N$ there are composites of [[base change]] along $f$ with the [[hom-functor]] and the [[tensor product]] functor

$$
  R_1 Mod \stackrel{\otimes_{R_1} R_2}{\to} R_2 Mod \stackrel{\otimes_{R_2} N}{\to} Ab
$$

$$
  R_1 Mod \stackrel{Hom_{R_1 Mod}(-,R_2)}{\to}
  R_2 Mod
  \stackrel{Hom_{R_2}(-,N)}{\to}
  Ab
  \,.
$$

The [[derived functors]] of $Hom_{R_2}(-,N)$ and $\otimes_{R_2} N$ are the [[Ext]]- and the [[Tor]]-functors, respectively, so the [[Grothendieck spectral sequence]] applied to these composites yields [[base change spectral sequence]] for these.

#### Hochschild-Serre spectral sequence
 {#HochschildSerreSpectralSequence}

* [[Hochschild-Serre spectral sequence]]

### Exact couples
 {: #exact_couples}

The above examples are all built on the [spectral sequence of a filtered complex](#SpectralSequenceOfFilteredComplex). An alternatively universal construction builds spectral sequences from _exact couples_.

An **[[exact couple]]** is an exact sequence of three arrows among two objects

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

These creatures construct spectral sequences by a two-step process:

*  first, the composite $ d \coloneqq k j \colon E\to E$ is nilpotent, in that $d^2=0$
*  second, the homology $E'$ of $(E,d)$ supports a map $j':E'\to \varphi D$, and receives a map $k':\varphi D\to E'$.  Setting $D'=\varphi D$, by general reasoning

$$
  E' \overset{j'}{\to} D' \overset{\varphi}{\to} D' \overset{k'}{\to} E' \overset{j'}{\to}
  \,.
  $$
is again an exact couple.

The sequence of complexes $(E,d),(E',d'),\dots$ is a spectral sequence, by construction. For more see at _[exact couple -- Spectral sequences from exact couples](exact+couple#SpectralSequencesFromExactCouples)_

Examples of exact couples can be constructed in a number of ways. Notably there are naturally [exact couples of towers of (co-)fibrations](exact+couple#ExactCoupleOfATowerOfFibrations). For instance [[Adams spectral sequences]] are usually produced this way (from towers which are [[Adams resolutions]]). For a list of examples in this class see [below](#SpectralSequencesInducedFromTowersOfSpectra).

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 the filtered complex described (???) somewhere else on this page.  For another example, choosing a chain complex of flat modules $(C^\dot,d)$, tensoring with the short exact sequence
$$ \mathbb{Z}/p\mathbb{Z} \to \mathbb{Z}/p^2\mathbb{Z} \to \mathbb{Z}/p\mathbb{Z} $$
gives the exact couple

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

in which $\beta$ is the _mod-$p$ Bockstein_ homomorphism.

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 $\coker \varphi\to E\to \ker\varphi$, 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 $j,k,\varphi$, leading to the conventions mentioned earlier.

### List of examples
 {#ListOfExamples}

[[!include Lurie spectral sequences -- table]]

The following list of examples orders the various classes of spectral sequences by special cases: items further to the right are special cases of items further to the left.

* ***[[spectral sequence of a filtered complex]]*** (approximate [[chain homology]] by higher-order [[relative homology]] in the presence of a [[filtered object|filtering]])

  * [[Atiyah-Hirzebruch spectral sequence]] (compute [[generalized homology]] of a [[Serre fibration]] in terms of that of the base with [[coefficients]] in that of the [[fibers]])

    * [[Serre spectral sequence]] (compute [[ordinary homology]] of a [[Serre fibration]] in terms of that of the base with [[coefficients]] in that of the [[fibers]])

  * [[Eilenberg-Moore spectral sequence]] (compute homology of [[homotopy fiber products]])

  * **[[spectral sequence of a double complex]]** (compute homology of [[total complex]] by filtering by row/column degree)

    * **[[Grothendieck spectral sequence]]** (compute [[composition]] of two [[derived functors in homological algebra|derived functors]])

      * [[base change spectral sequence]] (compute [[base change]]/[[extension of scalars]] in two stages)

      * [[Leray spectral sequence]] (compute [[global sections]] in two stages)

      * [[Hochschild-Serre spectral sequence]] (compute [[group cohomology]] in two stages)

Here is a more random list (using material from [Wikipedia](#Wikipedia)). Eventually to be merged with the above.

* [[Adams spectral sequence]] in [[stable homotopy theory]]

* [[Adams-Novikov spectral sequence]], converging to [[homotopy groups]] of [[connective spectra]]

*  [[chromatic spectral sequence]] for calculating the initial terms of the [[Adams-Novikov spectral sequence]]

* [[Atiyah-Hirzebruch spectral sequence]] of an [[extraordinary cohomology theory]];

* [[descent spectral sequence]]

* [[Bar spectral sequence]] for the [[homology]] of the [[classifying space]] of a [[group]]

* [[Barratt spectral sequence]] converging to the [[homotopy]] of the initial space of a [[cofibration]]

* [[Bloch-Lichtenbaum spectral sequence]] converging to the [[algebraic K-theory]] of a [[field]]

* [[Bockstein spectral sequence]] relating the [[homology]] with $mod p$ coefficients and the homology reduced $mod p$

* [[Bousfield-Kan spectral sequence]] converging to the [[homotopy colimit]] of a [[functor]]

* [[Cartan-Leray spectral sequence]] converging to the [[homology]] of a [[quotient space]]

* [[Čech-to-derived functor spectral sequence]] from [[Čech cohomology]] to [[abelian sheaf cohomology]]

* [[change of rings spectral sequences]] for calculating [[Tor]] and [[Ext]] groups of modules

* [[Connes spectral sequences]] converging to the [[cyclic homology]] of an algebra

* [[EHP spectral sequence]] converging to [[stable homotopy groups of spheres]]

* [[Eilenberg-Moore spectral sequence]] for the [[singular cohomology]] of the [[pullback]] of a [[fibration]]

* [[Federer spectral sequence]] converging to [[homotopy groups]] of a function space

* [[Frölicher spectral sequence]] starting from the [[Dolbeault cohomology]] and converging to the algebraic [[de Rham cohomology]] of a [[variety]]

* [[Green's spectral sequence]] for [[Koszul cohomology]]

* [[Grothendieck spectral sequence]] for composing [[derived functors]]

* [[Hodge-de Rham spectral sequence]] converging to the algebraic [[de Rham cohomology]] of a [[variety]]

* [[Hurewicz spectral sequence]] for calculating the [[homology]] of a space from its [[homotopy groups]]

* [[hyperhomology spectral sequence]] for calculating [[hyperhomology]]

* [[Künneth spectral sequence]] for calculating the homology of a [[tensor product]] of [[differential graded algebras]]

* [[Leray spectral sequence]] converging to [[abelian sheaf cohomology]]

* [[Leray-Serre spectral sequence]] of a [[fibration]]

* [[Lyndon-Hochschild?Serre spectral sequence]] in [[group cohomology]]

* [[May spectral sequence]] for calculating the [[Tor]] or [[Ext]] groups of an algebra

* [[Miller spectral sequence]] converging to the $mod p$ [[stable homology]] of a space

* [[Quillen spectral sequence]] for calculating the homotopy of a [[simplicial group]]

* [[spectral sequence of an exact couple]]

* [[van Est spectral sequence]] converging to relative [[Lie algebra cohomology]]

* [[van Kampen spectral sequence]] for calculating the homotopy of a wedge of spaces.

* [[slice spectral sequence]]

## Properties

### Basic lemmas

+-- {: .num_lemma}
###### Lemma
**(mapping lemma)**

If $f : (E_r^{p,q} \to (F_r^{p,q}))$ is a morphism of spectral sequences such that for some $r$ we have that $f_r : E_r^{p,q} \toF_r^{p,q}$ is an isomorphism, then also $f_s$ is an isomorphism for all $s \geq r$.

=--

+-- {: .num_lemma}
###### Lemma
**(classical convergence theorem)**

(...)

=--

This is recalled in ([Weibel, theorem 5.51](#Weibel)).

### First quadrant spectral sequence
 {#FirstQuadrant}

+-- {: .num_defn}
###### Definition

A **first quadrant spectral sequence** is one for wich all pages are concentrated in the first quadrant of the $(p,q)$-plane, in that

$$
  ((p \lt 0) or (q \lt 0))
  \;\;
  \Rightarrow
  E_r^{p,q} = 0
  \,.
$$

=--

+-- {: .num_prop}
###### Proposition

If the $r$th page is concentrated in the first quadrant, then so the $(r+1)st$ page. So if the first one is, then all are.

=--

+-- {: .num_prop}
###### Proposition

Every first quadrant spectral sequence converges at $(p,q)$ from $r \gt max(p,q+1)$ on

$$
  E_{max(p,q+1)+1}^{p,q} = E_\infty^{p,q}
  \,.
$$

=--

+-- {: .num_prop}
###### Proposition

If a first quadrant spectral sequence converges

$$
  E_r^{p,q} \Rightarrow H^{p+q}
$$

then each $H^n$ has a filtration of length $n+1$

$$
  0 = F^{n+1}H^n \subset F^n H^n \subset \cdots \subset F^1 H^n \subset F^0 H^n = H^n
$$

and we have

* $F^n H^n \simeq E_\infty^{n,0}$

* $H^n/F^1 H^n \simeq E_\infty^{0,n}$.

=--

### Cup product structure
 {#PropertiesCupProductStructure}

Cohomological spectral sequences are compatible with [[cup product]] structure on the $E_2$-page. (e.g. [Hutchings 11, sections 5 and 6](#Hutchings11))

## Related concepts

* [[edge morphism]]

* [[multiplicative spectral sequence]]

* [[spectral sequences in homotopy type theory]]

## References

Spectral sequences were originally introduced in 1946 by [[Jean Leray]] in the paper

* [[Jean Leray]], _Structure de l'anneau d'homologie d'une représentation_, Comptes rendus hebdomadaires des séances de l’Académie des Sciences 222 (1946) 1419–1422.

### General exposition

* [[Frank Adams]], part III section 7 of _[[Stable homotopy and generalised homology]]_, 1974

### Abelian/stable theory

An elementary pedagogical introduction is in

* [[Timothy Chow]], _You could have invented spectral sequences_, Notices of the AMS (2006) ([pdf](http://www.ams.org/notices/200601/fea-chow.pdf))
 {#Chow}

Textbook accounts include

* [[John McCleary]], _A User's Guide to Spectral Sequences_, Cambridge University Press

* {#Weibel} [[Charles Weibel]], chapter 5, _An introduction to homological algebra_ Cambridge studies in advanced mathematics 38 (1994)

* [[Raoul Bott]], [[Loring Tu]], section 14 of _Differential forms in algebraic topology_, Graduate Texts in Mathematics __82__, Springer 1982. xiv+331 pp.

* [[Hal Schenck]], _Chapter 9: Cohomology and spectral sequences_ ([pdf](http://www.math.uiuc.edu/~schenck/tapp.pdf)) .

* {#Hatcher} [[Alan Hatcher]], _Spectral sequences in algebraic topology_ ([web](http://www.math.cornell.edu/~hatcher/SSAT/SSATpage.html))

* {#Kochmann96} [[Stanley Kochmann]], _[[Bordism, Stable Homotopy and Adams Spectral Sequences]]_, AMS 1996

* [[Dai Tamaki]], [[Akira Kono]], Chapter 5 in: _Generalized Cohomology_, Translations of Mathematical Monographs, American Mathematical Society, 2006 ([ISBN: 978-0-8218-3514-2](https://bookstore.ams.org/mmono-230))

The general discussion in the context of [[stable (∞,1)-category]] theory (the [[spectral sequence of a filtered stable homotopy type]]) is in section 1.2.2 of

* [[Jacob Lurie]], _[[Higher Algebra]]_

A review Master thesis is

* Jennifer Orlich, _Spectral sequences and an application_ ([pdf](http://www.math.osu.edu/~flicker.1/orlich.pdf))

Reviews of and lecture notes on standard definitions and facts about spectral sequences include

* Matthew Greenberg, _Spectral sequences_ ([pdf](http://www.math.mcgill.ca/goren/SeminarOnCohomology/specseq.pdf))

* {#Hutchings11} [[Michael Hutchings]], _Introduction to spectral sequences_ ([pdf](http://math.berkeley.edu/~hutching/teach/215b-2011/ss.pdf))

* Daniel Murfet, _Spectral sequences_ ([pdf](http://therisingsea.org/notes/SpectralSequences.pdf))

* [[Neil Strickland]], _Spectral sequences_ ([pdf](http://neil-strickland.staff.shef.ac.uk/courses/bestiary/ss.pdf))

* Ravi Vakil, _Spectral Sequences: Friend or Foe?_ ([pdf](http://math.stanford.edu/~vakil/0708-216/216ss.pdf))

* Brandon Williams, _Spectral sequences_ ([pdf](http://www.math.sunysb.edu/~mbw/notes/orals/Spectral%20Sequences.pdf))

* {#Boardman99} [[Michael Boardman]], _Conditionally convergent spectral sequences_, 1999 ([pdf](http://www.uio.no/studier/emner/matnat/math/MAT9580/v12/undervisningsmateriale/boardman-conditionally-1999.pdf))

See also

* Wikipedia, _[Spectral sequence](http://en.wikipedia.org/wiki/Spectral_sequence)_
 {#Wikipedia}

* A. Romero, J. Rubio, F. Sergeraert, _Computing spectral sequences_ ([pdf](http://www-fourier.ujf-grenoble.fr/~sergerar/Papers/Ana-JSC.pdf))

* Eric Peterson, _[Ext chart](http://ext-chart.org)_ software for computing spectral sequences

### Nonabelian / unstable theory {#ReferencesNonabelian}

Homotopy spectral sequences in model categories are discussed in

* [[Aldridge Bousfield]], _Cosimplicial resolutions and homotopy spectral sequences in model categories_ ([arXiv:math/0312531](http://arxiv.org/abs/math/0312531)).

Spectral sequences in general categories with [[zero morphisms]] are discussed in

* [[Marco Grandis]], _Homotopy spectral sequences_ ([arXiv:1007.0632](http://arxiv.org/abs/1007.0632))

Discussion from a perspective of [[homotopy type theory]] (see also *[[spectral sequences in homotopy type theory]]*):

* {#Shulman13} [[Mike Shulman]], _[[homotopytypetheory:spectral sequences|Spectral sequences]]_ 2013 ([part I](https://golem.ph.utexas.edu/category/2013/08/what_is_a_spectral_sequence.html), [part II](http://homotopytypetheory.org/2013/08/08/spectral-sequences/))

* [[Floris van Doorn]], §5 in: *On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory* (2018) [[arXiv:1808.10690](https://arxiv.org/abs/1808.10690)]

and implementation in [[Lean]]-[[HoTT]] is in

* {#vanDoornSpectral} [[Floris van Doorn]], [[Egbert Rijke]], [[Ulrik Buchholtz]], [[Favonia]], [[Steve Awodey]], [[Jeremy Avigad]], [[Mike Shulman]], [[Jonas Frey]],  _Spectral_ [[github.com/cmu-phil/Spectral](https://github.com/cmu-phil/Spectral)]

### History

* [[John McCleary]], _A history of spectral sequences: Origins to 1953_, in _History of Topology_, edited by Ioan M. James, North Holland (1999) 631–663

[[!redirects spectral sequences]]


spectral sequence of a double complex

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Homological algebra
+--{: .hide}
[[!include homological algebra - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

Given a [[double complex]] $C_{\bullet, \bullet}$ (in some [[abelian category]]), there is a [[spectral sequence]] whose second page is the "naïve double cohomology"

$$
  E^2_{p,q} = H_p^{hor}(H_q^{vert}(C))
$$

and which converges, under mild conditions, to the correct [[chain homology]] of the [[total complex]] $Tot(C_{\bullet, \bullet})$.

This is the special case of the [[spectral sequence of a filtered complex]] with $Tot(C)_\bullet$ [[filtered chain complex|filtered]] by row-degree (or dually, by column-degree).

## Definition

Let $C_{\bullet,\bullet}$ be a [[double complex]]. Its [[total complex]] $Tot C$ is given in degree $n$ by the [[direct sum]]

$$
  (Tot C)_n = \oplus_{p+q = n} C_{p,q}
$$

and the [[differential]] acts on elements $c \in C_{p,q}$ as

$$
  \partial^{Tot} \colon c \mapsto \partial^{hor} c + (-1)^{p} \partial^{ver} c
  \,.
$$

+-- {: .num_defn #HorVertFiltration}
###### Definition

The **horizontal filtration** on $Tot C$ is the [[filtered object|filtration]] $F_\bullet Tot C$ given in degree $n$ by the [[direct sum]] expression

$$
  F^{hor}_p (Tot C)_n \coloneqq
  \oplus_{{n_1+n_2 = n} \atop {n_1 \leq p} } C_{n_1,n_2}
  \,.
$$

Similarly the **vertical filtration** is given by

$$
  F^{vert}_p (Tot C)_n \coloneqq
  \oplus_{{n_1+n_2 = n} \atop {n_2 \leq p} } C_{n_1,n_2}
  \,.
$$

=--

+-- {: .num_defn #SpectralSequenceOfDoubleComplex}
###### Definition

The (vertical/horizontal) _spectral sequence of the double complex_ $C_{\bullet,\bullet}$ is the [[spectral sequence of a filtered complex]] for the filtered total complex from def. \ref{HorVertFiltration}.

=--

## Properties
 {#Properties}

### Low-degree pages
 {#LowDegreePages}

+-- {: .num_prop}
###### Proposition

Let $\{E^r_{p,q}\}_{r,p,q}$ be the spectral sequence of a [[double complex]] $C_{\bullet, \bullet}$, according to def. \ref{SpectralSequenceOfDoubleComplex}, with respect to the horizontal filtration. Then the first few pages are for all $p,q \in \mathbb{Z}$ given by

* $E^0_{p,q} \simeq C_{p,q}$;

* $E^1_{p,q} \simeq H_q(C_{p, \bullet})$;

* $E^2_{p,q} \simeq H_p(H^{vert}_q(C))$.

Moreover, if $C_{\bullet, \bullet}$ is concentrated in the first quadrant ($0 \leq p,q$), then the spectral sequence [converges](spectral+sequence#ConvergenceOfSpectralSequences) to the chain homology of the total complex:

$$
  E^\infty_{p,q} \simeq G_p H_{p+q}(Tot(C)_\bullet)
  \,.
$$

=--

+-- {: .proof}
###### Proof

This is a matter of unwinding the definition, using the general properties of
[spectral sequences of a filtered complex -- in low degree pages](spectral+sequence+of+a+filtered+complex#InLowDegreePages). We display equations for the horizontal filtering, the other case works analogously.

The 0th page is by definition the [[associated graded]] piece

$$
  \begin{aligned}
    E^0_{p,q}
      & \coloneqq
       G_p Tot(C)_{p+q}
      \\
      & \coloneqq
    F_p Tot(C)_{p+q} / F_{p-1} Tot(C)_{p+q}
    \\
    & \coloneqq
    \frac{
      \underset{ {n_1 + n_2 = p+q} \atop {n_1 \leq p} }{\oplus} C_{n_1, n_2}
    }
    {
      \underset{ {n_1 + n_2 = p+q} \atop {n_1 \lt p} }{\oplus} C_{n_1, n_2}
    }
    \\
    & \simeq
    C_{p,q}
    \,.
  \end{aligned}
$$

The first page is the chain homology of the [[associated graded]] chain complex:

$$
  \begin{aligned}
    E^1_{p,q}
    & \simeq
    H_{p+q}(G_p Tot(C)_\bullet)
    \\
    & \simeq
    H_{p+q}( C_{p,\bullet} )
    \\
    & \simeq H_q(C_{p, \bullet})
  \end{aligned}
  \,.
$$

In particular this means that representatives of $[c] \in E^1_{p,q}$ are given by $c \in C_{p,q}$ such that $\partial^{vert} c = 0$. It follows that $\partial^1 \colon E^1_{p,q} \to E^1_{p-1, q}$, which by the definition of a [[total complex]] acts as $\partial^{hor} \pm \partial^{vert}$, acts on these representatives just as $\partial^{hor}$ and this gives the second page

$$
  E^2_{p,q} \simeq ker(\partial^1_{p-1,q})/im(\partial^1_{p,q})
  \simeq
  H_p(H_q^{vert}(C_{\bullet, \bullet}))
  \,.
$$

Finally, for $C_{\bullet, \bullet}$ concentrated in $0 \leq p,q$ the corresponding [[filtered chain complex]] $F_p Tot(C)_\bullet$ is a non-negatively graded chain complex with filtration bounded below. Therefore the spectral sequence converges as claimed by the general discussion at _[spectral sequence of a filtered complex - convergence](spectral+sequence+of+a+filtered+complex#ConvergenceGeneral)_.

=--

## Examples

A plethora of types of spectral sequences are special cases of the spectral sequence of a double complex, for instance

* [[Frölicher spectral sequence]].

For the moment see at _[[spectral sequence]]_ for a list.

## References

Dedicated discussion of the case of spectral sequences of double complexes is for instance in

* Ravi Vakil, _Spectral Sequences: Friend or Foe?_ ([pdf](http://math.stanford.edu/~vakil/0708-216/216ss.pdf))

or in section 25, lecture 9 of

* [[Rick Jardine]], _Lectures on Homotopy Theory_ ([pdf](http://www.math.uwo.ca/~jardine/papers/HomTh/lecture009.pdf))

Details are usually discussed for the more general case of a _[[spectral sequence of a filtered complex]]_.

[[!redirects spectral sequences of a double complex]]
[[!redirects spectral sequences of double complexes]]


spectral sequence of a filtered complex

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Homological algebra
+--{: .hide}
[[!include homological algebra - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

The [[spectral sequence]] of a [[filtered object|filtered]] [[chain complex]]

$$
  \cdots
    \hookrightarrow
   F_{p-1}C_\bullet
     \hookrightarrow
   F_p C_\bullet
     \hookrightarrow
   \cdots
     \hookrightarrow
  C_\bullet
$$

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

$$
  G_p C \coloneqq F_p C_\bullet/F_{p-1}C_\bullet
  \,;
$$

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

If the filtering is bounded then $r$-almost homology for sufficiently large $r$ ("$\infty$-almost homology") is 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 $r$ (it "collapses"), and so allows one to obtain the genuine homology from some finite $r$-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]]

$$
  \cdots
    \to
  H_q (F_{p-1} C_\bullet)
    \to
  H_q (F_p C_\bullet)
    \to
  H_q(F_p C_\bullet / F_{q-1} C_\bullet)
    \to
  H_{q-1}(F_{p-1} C_\bullet)
   \to
 \cdots
$$

which are induced by the short exact sequences

$$
  0 \to F_{p-1}C_q \hookrightarrow F_p C_q \to F_p C_q / F_{p-1} C_q \to 0
$$

coming from the filtering.

## Definition

We give the definition

* [via relative homology](#ExplicitDefinition)

and

* [via exact couples](#ViaExactCouples).

### Via relative homology
 {#ExplicitDefinition}

Let $R$ be a [[ring]] and write $\mathcal{A} = R$[[Mod]] for its [[category]] of [[modules]].

Let

$$
  \cdots
  \hookrightarrow
  F_{p-1}C_\bullet \hookrightarrow F_p C_\bullet \hookrightarrow
  \cdots
  \hookrightarrow
  C_\bullet
$$

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

+-- {: .num_remark}
###### Remark

In more detail this means that

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

1. For each $n \in \mathbb{Z}$ there is a [[filtered object|filtering]] $F_\bullet C_n$ on $C_n$ and all these filterings are compatible with the [[differentials]] in that

   $$
     \partial(F_p C_n) \subset F_p C_{n-1}
   $$

1. The grading associated to the filtering is such that the $p$-graded elements are those in the [[quotient]]

   $$
     G_p C_n \coloneqq \frac{F_p C_n}{ F_{p-1} C_n}
     \,.
   $$

   Since the differentials respect the grading we have chain complexes $G_p C_\bullet$ in each filtering degree $p$.

The base category $\mathcal{A}$ can be any abelian category. In this case, we still use element-notation as if $\mathcal{A}$ were a category of [[modules]].

=--

#### $r$-Almost cycles and boundaries
 {#InterpretationOfTerms}

+-- {: .num_defn #AlmostChainsCyclesBoundaries}
###### Definition

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

1. $G_p C_{p+q}$ is the module of ***$(p,q)$-[[chains]]*** or of ***$(p+q)$-chains in filtering degree $p$***;

1. $\begin{aligned} Z^r_{p,q}  & \coloneqq \left\{ c \in G_p C_{p+q} | \partial c = 0 \, mod\, F_{p-r} C_{\bullet} \right\} \\ & =  \left\{ c \in F_p C_{p+q} | \partial(c) \in F_{p-r} C_{p+q-1} \right\}/ F_{p-1}C_{p+q} \end{aligned}$

   is the module of ***$r$-almost $(p,q)$-[[cycles]]*** (the $(p+q)$-chains whose differentials vanish modulo terms of filtering degree $p-r$);

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

   is the module of **$r$-almost $(p,q)$-[[boundaries]]**.

=--

Similarly we set

$$
  Z^\infty_{p,q} \coloneqq \{c \in F_p C_{p + q} | \partial c = 0 \}/F_{p-1}C_{p+q}
  =
  Z(G_p C_{p+q})
$$

$$
  B^\infty_{p,q} = \cdots
  \,.
$$

> fix

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

+-- {: .num_prop #DifferentialsOnAlmostChains}
###### Proposition

The [[differentials]] of $C_\bullet$ restrict on $r$-almost cycles to morphisms of the form

$$
  \partial^r
   \colon
  Z^r_{p,q}
  \to
  Z^r_{p-r, q+r-1}
  \,.
$$

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

=--

+-- {: .proof}
###### Proof

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

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

=--

+-- {: .num_prop }
###### Proposition

We have a sequence of canonical inclusions

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

=--

+-- {: .num_prop #KernelsInsideAlmostCycles}
###### Proposition

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

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

=--

+-- {: .proof}
###### Proof

An element $c \in F_p C_{p+q}$ represents

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

1. an element in $Z^{r+1}_{p,q}$ if even $\partial c \in F_{p-r-1} C_{p+q-1} \hookrightarrow F_{p-r} C_{p+q-1}$.

The second condition is equivalent to $\partial c$ representing the 0-element in the quotient $F_{p-r}C_{p+q-1}/ F_{p-r-1}C_{p+q-1}$. But this is in turn equivalent to $\partial c$ being 0 in $Z^r_{p-r,q+r-1} \subset F_{p-r} C_{p+q-1} / F_{p-r-1} C_{p+q-1}$.

=--

#### $r$-Almost homology groups: the spectral sequence

Let $F_\bullet C_\bullet$ be a [[filtered chain complex]] as above.

+-- {: .num_defn #ExplicitForm}
###### Definition

For $r, p, q \in \mathbb{Z}$ define the **$r$-almost $(p,q)$-[[chain homology]]** of the filtered complex to be the [[quotient]] of the $r$-almost $(p,q)$-cycles by the $r$-almost $(p,q)$-boundaries, def. \ref{AlmostChainsCyclesBoundaries}:

$$
  \begin{aligned}
    E^r_{p,q}
    & \coloneqq
    \frac{Z^r_{p,q}}{B^r_{p,q}}
    \\
    & =
  \frac{
    \left\{
     x \in F_p C_{p+q} \,|\, \partial x \in F_{p-r} C_{p+q-1}
    \right\} + F_{p-1} C_{p+q}
  }
  {
    \partial( F_{p+r-1} C_{p+q+1} ) + F_{p-1} C_{p+q}
  }
  \end{aligned}
$$

By prop. \ref{DifferentialsOnAlmostChains} the differentials of $C_\bullet$ restrict on the $r$-almost homology groups to maps

$$
  \partial^r : E^r_{p,q} \to E^r_{p-r, q + r - 1}
  \,.
$$

=--

+-- {: .num_prop #ExplicitDefIsIndeedSpectralSequ}
###### Proposition

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

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

=--

+-- {: .proof}
###### Proof

By prop. \ref{KernelsInsideAlmostCycles}.

=--

### Via exact couples
 {#ViaExactCouples}

The spectral sequence may alternatively be obtained as the [spectral sequence of an exact couple](exact+couple#SpectralSequencesFromExactCouples)

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

where

* $D \coloneqq \bigoplus_i H^{\bullet}(F_i) $

* and where $\varphi$ is the cohomology morphism induced by the inclusion of chain complexes $F_i\to F_{i+1}$

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

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

## Properties

### Low-degree pages
 {#InLowDegreePages}

We characterize the value of the spectral sequence $E^r_{p,q}$, def. \ref{ExplicitForm} for low values of $r$ and, below in prop. \ref{AbuttingFiltrationConvergesToHomology}, for $r \to \infty$.

+-- {: .num_prop #FirstPages}
###### Proposition

For the spectral sequence of a filtered complex $G_\bullet C_\bullet$ from def. \ref{ExplicitForm}, the first pages have the following form:

* $E^0_{p,q} = G_p C_{p+q} = F_p C_{p+q} / F_{p-1} C_{p+q}$

  is the [[associated graded|associated p-graded]] piece of $C_{p+q}$;

* $E^1_{p,q} = H_{p+q}(G_p C_\bullet)$

  is the [[chain homology]] of the [[associated graded|associated p-graded]] complex $G_p C_\bullet$.

=--

+-- {: .proof}
###### Proof

For $r = 0$ def. \ref{ExplicitForm} restricts to

$$
  E^0_{p,q} = \frac{ F_p C_{p+q}}{F_{p-1} C_{p+q}} = G_p C_{p+q}
$$

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

For $r = 1$ def. \ref{ExplicitForm} gives

$$
  E^1_{p,q} = \frac{\{c \in G_p C_{p+q} | \partial c = 0 \in G_p C_{p+q}\} }{\partial(F_p C_{p+q})} = H_{p+q} (G_p C_\bullet)
  \,.
$$

=--

+-- {: .num_remark}
###### Remark

There is, in general, a decisive difference between the homology of the associated graded complex $H_{p+q}(G_p C_\bullet)$ and the associated graded piece of the genuine homology $G_p H_{p+q}(C_\bullet)$: 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 $r \to \infty$.

=--

### Convergence
 {#Convergence}

#### General definitions

+-- {: .num_defn #LimitTerm}
###### Definition

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

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

Then one says that

1. the [[bigraded object]]

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

   is the **limit term** of the spectral sequence;

*  the spectral sequence **abuts** to $E^\infty$.

=--

+-- {: .num_example #Degeneration}
###### Example

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

=--

+-- {: .proof}
###### Proof

By the defining relation

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

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

=--

+-- {: .num_example #Collaps}
###### Example

If for a [[spectral sequence]] $\{E^r_{p,q}\}_{r,p,q}$ there is $r_s \geq 2$ such that the $r_s$th page is concentrated in a single row or a single column, then the the spectral sequence degenerates on this page, example \ref{Degeneration}, hence this page is a limit term, def. \ref{LimitTerm}. One says in this case that the spectral sequence **collapses** on this page.

=--

+-- {: .proof}
###### Proof

For $r \geq 2$ the [[differentials]] of the spectral sequence

$$
  \partial^r \colon E^r_{p,q} \to E^r_{p-r, q+r-1}
$$

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

=--

+-- {: .num_defn #Convergence}
###### Definition

A [[spectral sequence]] $\{E^r_{p,q}\}_{r,p,q}$ is said to **converge** to a [[graded object]] $H_\bullet$ with [[filtered chain complex|filtering]] $F_\bullet H_\bullet$, traditionally denoted

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

if the [[associated graded]] complex $\{G_p H_{p+q}\}_{p,q} \coloneqq \{F_p H_{p+q} / F_{p-1} H_{p+q}\}$ of $H$ is the limit term of $E$, def. \ref{LimitTerm}:

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

=--

+-- {: .num_remark }
###### Remark

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

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

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."

=--

+-- {: .num_defn #BoundedSpectralSequence}
###### Definition

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

=--

+-- {: .num_example #QuadrantSpectralSequence}
###### Example

A [[spectral sequence]] $\{E^r_{p,q}\}$ is called

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

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

Such spectral sequences are bounded, def. \ref{BoundedSpectralSequence}.

=--

+-- {: .num_prop #BoundedSpectralSequenceHasLimitTerm}
###### Proposition

A bounded spectral sequence, def. \ref{BoundedSpectralSequence}, has a limit term, def. \ref{LimitTerm}.

=--

+-- {: .proof}
###### Proof

First notice that if a spectral sequence has at most $N$ non-vanishing terms of total degree $n$ on page $r$, 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 $n$ there is $L(n) \in \mathbb{Z}$ such that $E^r_{p,n-p} = 0$ for all $p \leq L(n)$ and all $r$. Similarly there is $T(n) \in \mathbb{Z}$ such $E^r_{n-q,q} = 0$ for all $q \leq T(n)$ and all $r$.

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

$$
  r \gt max(  p-L(p+q-1), q + 1 - L(p+q+1) )
  \,.
$$

This is because for such $r$ we have

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

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

Therefore

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

=--

#### Convergence of bounded filtrations
 {#ConvergenceGeneral}

+-- {: .num_defn #BoundedFiltration}
###### Definition

A [[filtered chain complex|filtration]] $F_\bullet C_\bullet$ on a [[chain complex]] $C_\bullet$ is called a **bounded filtration** if for all $n \in \mathbb{Z}$ there is $L(n), T(n) \in \mathbb{Z}$ such that

$$
  F_{p \lt S(n)}C_n = 0
  \;\;\;\;\;
  F_{p \gt T(n)}C_n \simeq C_n
  \,.
$$

=--

+-- {: .num_prop }
###### Proposition

The spectral sequence of a complex with bounded filtration, def. \ref{BoundedFiltration}, has a limit term, def. \ref{LimitTerm}.

=--

+-- {: .proof}
###### Proof

The spectral sequence of a filtered complex $F_\bullet C_\bullet$ is by def. \ref{ExplicitForm} at $E^r_{p,q}$ a [[quotient]] of a [[subobject]] of $F_p C_{p+q}$. By def. \ref{BoundedFiltration} therefore there are for each $n,r \in \mathbb{Z}$ finitely many non-vanishing terms of the form $E^r_{p,n-p}$. Therefore the spectral sequence is bounded, def. \ref{BoundedSpectralSequence} and hence has a limit term by prop. \ref{BoundedSpectralSequenceHasLimitTerm}.

=--

+-- {: .num_example }
###### Example

If $X \in$ [[Top]] is a [[CW complex]] with cell filtration $X_0 \hookrightarrow X_1 \hookrightarrow \cdots \hookrightarrow X$, then the induced filtering

$$
  F_p C_\bullet(X) \coloneqq C_\bullet(X_n)
$$

on its [[singular chain complex]] $C_\bullet(X)$ yields a first-quadrant spectral sequence, example \ref{QuadrantSpectralSequence}. Therefore it has a limit term.

=--

Before saying 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:

+-- {: .num_defn #FilteringOnHomology}
###### Definition

For $F_\bullet C_\bullet$ a [[filtered complex]], write for $p \in \mathbb{Z}$

$$
  F_p H_\bullet(C)
   \coloneqq
  image( H_\bullet(F_p C_\bullet) \to H_\bullet(C_\bullet) )
  \,.
$$

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

=--

+-- {: .num_prop #AbuttingFiltrationConvergesToHomology}
###### Proposition

If the spectral sequence of a filtered complex $F_\bullet C_\bullet$, def. \ref{BoundedFiltration} has a limit term, def. \ref{LimitTerm}, then it converges, def. \ref{Convergence}, to the [[chain homology]] of the complex:

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

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

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

with the filtering on the right as in def. \ref{FilteringOnHomology}.

=--

+-- {: .proof}
###### Proof

By assumption, there is for each $p,q$ an $r(p,q)$ such that for all $r \geq r(p,q)$ the $r$-almost cycles and $r$-almost boundaries, def. \ref{AlmostChainsCyclesBoundaries}, in $F_p C_{p+q}$ are the ordinary [[cycles]] and [[boundaries]]. Therefore for $r \geq r(p,q)$ def. \ref{ExplicitForm} gives $E^r_{p,q} \simeq G_p H_{p+q}(C)$.

=--

#### Via exact couples

It is instructive to note that in the $n$th derived [[exact couple]] $\varphi^n D\to E_{(n)} \to \varphi^n D\to{}$, the hidden part $\varphi^n D$ is the [[module|submodule]] $D_{(n)}$ of $\bigoplus_{i} H(F_{i+n})$, as it meets $H(F_{i+n})$ representable by elements of $F_i$; that is, we may sensibly call it
$$F_{i} D_{(n)} = \frac{\ker(d)\cap F_i}{F_i\cap dF_{i+n}}.$$
Separating the grades, the exactness of the couple at $E_{(n)}$ then says
$$ F_{i} H^j(F_{i+n})\to F_{i+1}H^j(F_{i+n+1}) \to E_{(n)}^{i\dots} \to F_i H^{j+1}(F_{i+n}) \to F_{i+1}H^{j+1}(F_{i+n+1}) $$

One can see this as converging (if it sensibly converges) to either a subquotient of $F_i$ _or_ to a [[module|submodule]] $F_i H^\bullet(C) \lt H^\bullet(C)$.  Taking the latter interpretation, we hope to find in the limiting case exact sequences
$$ F_i H^j(C)\to F_{i+1} H^j(C) \to E_{(\infty)}^{i\dots} \to F_i H^{j+1}(C) \to F_{i+1} H^{j+1}(C).$$
At this stage one can check that the morphisms $F_i H^j(C)\to F_{i+1} H^j(C)$ are indeed definable, and in fact injective, so that whatever $E_{(\infty)}$ should be, the morphism $E_{(\infty)}^{i\dots}\to F_i H^{j+1}$ is null; that is, our long-exact sequence breaks up into the
[[short exact sequence]]s
$$ 0\to F_i H^j(C) \to F_{i+1} H^j(C)\to E_{(\infty)}^{i\dots} \to 0 .$$

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

## Examples

### 2-term filtering

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

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

+-- {: .num_prop}
###### Proposition

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

1. Consider the [[short exact sequence]]

   $$
     0 \to C^{(1)}_\bullet \to C_\bullet \to C_\bullet/C^{(1)}_\bullet \to 0
   $$

1. Its [[long exact sequence in homology]] contains the [[connecting homomorphism]]

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

   Define

   * $G_1 H_\bullet \coloneqq ker \delta$

   * $G_2 H_\bullet \coloneqq coker \delta$.

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

$$
  0 \to G_0 H_\bullet \to H_\bullet(C_\bullet) \to G_1 H_\bullet \to 0
  \,.
$$

=--

(...)

### Homology of a tensor product of chain complexes
 {#ExampleHomologyOfTensorProduct}

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

Their [[tensor product of chain complexes]] is

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

with [[differential]] on homogenous elements

$$
  \partial (\alpha \otimes\beta) =
  (\partial \alpha)
  \otimes \beta
  +
  (-1)^{deg \alpha} \alpha \otimes \partial \beta
  \,.
$$

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

Define a filtration on $C \otimes C'$ by

$$
  F_p(C \otimes C')_{k}
  \coloneqq
  \oplus_{i \leq p} C_i \otimes C_{k-i}
  \,.
$$

This means that the [[associated graded object]] is simply

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

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

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

The next differential is $\partial_1 = \partial \otimes id_{C'}$.
Since $k$ is assumed to be a field we have thus

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

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

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

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

### Homology of the total complex of a double complex
 {#ExampleHomologyDoubleComplex}

The [[total complex]] of a [[double complex]] is naturally
[[filtered object|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 $X \in Top$ be a [[CW-complex]] equipped explicitly with the structure of a [[filtered topological space]] $X^0 \hookrightarrow \cdots \hookrightarrow X^n \hookrightarrow \cdots X$.
This induces on the [[singular homology]] complex $C_\bullet(X)$ the structure of a [[filtered chain complex]] by

$$
  F_q C_\bullet(X) \coloneqq C_\bullet(X^p)
  \,.
$$

We discuss how the corresponding spectral sequence shows that the [[singular homology]] of $X$ coincides with the [[cellular homology]] of the filtering.

The [[associated graded object]] is

$$
  G_p C_{p+q}(X) = E^0_{p,q} = C_{p+q}(X^p)/C_{p+q}(X^{p-1})
  \,.
$$

+-- {: .num_prop}
###### Proposition

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

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

=--

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

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

The chain homology of

$$
  \partial \colon
   H_p(X^p, X^{p-1})
  \to
  H_{p-1}(X^{p-1}, X^{p-2})
$$

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

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

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

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

+-- {: .num_remark}
###### 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. Let
$$\begin{matrix} A & \to & X \\
 \downarrow & & \downarrow \\
 * & \to & B \end{matrix} $$
be a [[Serre fibration]] of [[pointed object|pointed]] [[topological spaces]] in which $B$ is a [[connected topological space|connected]] [[CW-complex]].  Then the $k$-[[simplicial skeleton|skeleta]] of $B$ naturally give filtered-space structures to both $B$ and $X$:
$$\begin{matrix} X_{(k)} & \to & X_{(k+1)} & \to & X \\
 \downarrow & & \downarrow & &\downarrow \\
 B_{k} & \to & B_{k+1} & \to & B \end{matrix} $$
and in turn induce filtrations of the singular chain complex of $X$.

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

It is straight-forward to show that the pair $(X_{k+1};X_k)$ is $k$-connected, and in particular the relative homology $H_i(X_{k+1};X_k)$ vanishes for $i\leq k$; 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

+-- {: .num_theorem }
###### Theorem

*The page $E^{(2)}$ of the homology Serre spectral sequence is given by*
$$ E^{(2)}_{p,q} \simeq H_p(B,\mathcal{H}_q(A|_{b})) $$
*the homology of $B$ with coefficients in the [[local system]] defined by the action of $\Pi_1 (B)$ on $H_q(A|_{b})$.  In the special case that $B$ is simply connected, these local systems are canonically equivalent to $H_q(A)$, the homology of the fiber over the basepoint.

=--

## Related concepts

* [[Cartan-Eilenberg spectral sequence]]

\linebreak

[[!include Lurie spectral sequences -- table]]

\linebreak

[[!include filtered objects -- contents]]

## References

* {#CartanEilenberg56} [[Henri Cartan]], [[Samuel Eilenberg]], chapter XV of _Homological algebra_, Princeton Univ. Press (1956)

* {#Kochmann96} [[Stanley Kochman]], section 5.3 of  of _[[Bordism, Stable Homotopy and Adams Spectral Sequences]]_, AMS 1996

* {#Orlich} Jennifer Orlich, _Spectral sequences and an application_, 1998 ([pdf](http://www.math.osu.edu/~flicker.1/orlich.pdf))

Lecture notes include

* {#Hutchings} [[Michael Hutchings]], _Introduction to spectral sequences_ (2011) ([pdf](http://math.berkeley.edu/~hutching/teach/215b-2011/ss.pdf))

and section 3 of

* [[Brandon Williams]], _Spectral sequences_ ([pdf](http://www.math.sunysb.edu/~mbw/notes/orals/Spectral%20Sequences.pdf))

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

* [[Charles Weibel]], _[[An Introduction to Homological Algebra]]_

[[!redirects spectral sequence of a filtration]]

[[!redirects spectral sequence of a filtered chain complex]]

[[!redirects spectral sequences of filtered complexes]]