spectral sequence#Terms of convergence

Motivated by problems in algebraic topology, Jean Leray introduced the notion of a sheaf and found himself faced with the problem of computing sheaf cohomology. To compute sheaf cohomology, Leray introduced a computational technique now known as the Leray spectral sequence. This gave a relation between cohomology groups of a sheaf and cohomology groups of the pushforward of the sheaf. The relation involved an infinite process. Leray found that the cohomology groups of the pushforward formed a natural chain complex, so that he could take the cohomology of the cohomology. This was still not the cohomology of the original sheaf, but it was one step closer in a sense. The cohomology of the cohomology again formed a chain complex, and its cohomology formed a chain complex, and so on. The limit of this infinite process was essentially the same as the cohomology groups of the original sheaf.

It was soon realized that Leray's computational technique was an example of a more general phenomenon. Spectral sequences were found in diverse situations, and they gave intricate relationships among homology and cohomology groups coming from geometric situations such as fibrations and from algebraic situations involving derived functors. While their theoretical importance has decreased since the introduction of derived categories, they are still the most effective computational tool available. This is true even when many of the terms of the spectral sequence are incalculable.

Unfortunately, because of the large amount of information carried in spectral sequences, they are difficult to grasp. This information is usually contained in a rank three lattice of abelian groups or modules. The easiest cases to deal with are those in which the spectral sequence eventually collapses, meaning that going out further in the sequence produces no new information. Even when this does not happen, it is often possible to get useful information from a spectral sequence by various tricks.

Fix an abelian category, such as a category of modules over a ring, and a nonnegative integer $r\_0$. A cohomological spectral sequence is a sequence $\backslash \{E\_r,\; d\_r\backslash \}\_\{r\backslash geq\; r\_0\}$ of objects $E\_r$ and endomorphisms $d\_r\; :\; E\_r\; \backslash to\; E\_r$, such that for every $r\backslash geq\; r\_0$

1. $d\_r\; \backslash circ\; d\_r\; =\; 0$,
2. $E\_\{r+1\}\; \backslash cong\; H\_\{*\}(E\_r,\; d\_r)$, the homology of $E\_r$ with respect to $d\_r$.

Usually the isomorphisms are suppressed and we write $E\_\{r+1\}\; =\; H\_\{*\}(E\_r,\; d\_r)$ instead. An object $E\_r$ is called sheet (as in a sheet of paper), or sometimes a page or a term; an endomorphism $d\_r$ is called boundary map or differential. Sometimes $E\_\{r+1\}$ is called the derived object of $E\_r$.{{citation needed|date=June 2015}}

=== Bigraded spectral sequence ===

In reality spectral sequences mostly occur in the category of doubly graded modules over a ring R (or doubly graded sheaves of modules over a sheaf of rings), i.e. every sheet is a bigraded R-module $E\_r\; =\; \backslash bigoplus\_\{p,q\; \backslash in\; \backslash mathbb\{Z\}^2\}\; E\_r^\{p,q\}.$

So in this case a cohomological spectral sequence is a sequence $\backslash \{E\_r,\; d\_r\backslash \}\_\{r\backslash geq\; r\_0\}$ of bigraded R-modules $\backslash \{E\_r^\{p,q\}\backslash \}\_\{p,q\}$ and for every module the direct sum of endomorphisms $d\_r\; =\; (d\_r^\{p,q\}\; :\; E\_r^\{p,q\}\; \backslash to\; E\_r^\{p+r,q-r+1\})\_\{p,q\; \backslash in\; \backslash mathbb\{Z\}^2\}$ of bidegree $(r,1-r)$, such that for every $r\backslash geq\; r\_0$ it holds that:

1. $d\_r^\{p+r,q-r+1\}\; \backslash circ\; d\_r^\{p,q\}\; =\; 0$,
2. $E\_\{r+1\}\; \backslash cong\; H\_\{*\}(E\_r,\; d\_r)$.

The notation used here is called complementary degree. Some authors write $E\_r^\{d,q\}$ instead, where $d\; =\; p\; +\; q$ is the total degree. Depending upon the spectral sequence, the boundary map on the first sheet can have a degree which corresponds to r = 0, r = 1, or r = 2. For example, for the spectral sequence of a filtered complex, described below, r₀ = 0, but for the Grothendieck spectral sequence, r₀ = 2. Usually r₀ is zero, one, or two. In the ungraded situation described above, r₀ is irrelevant.

Mostly the objects we are talking about are chain complexes, that occur with descending (like above) or ascending order. In the latter case, by replacing $E\_r^\{p,q\}$ with $E^r\_\{p,q\}$ and $d\_r^\{p,q\}\; :\; E\_r^\{p,q\}\; \backslash to\; E\_r^\{p+r,q-r+1\}$ with $d^r\_\{p,q\}\; :\; E^r\_\{p,q\}\; \backslash to\; E^r\_\{p-r,q+r-1\}$ (bidegree $(-r,r-1)$), one receives the definition of a homological spectral sequence analogously to the cohomological case.

The most elementary example in the ungraded situation is a chain complex C_•. An object C_• in an abelian category of chain complexes naturally comes with a differential d. Let r₀ = 0, and let E₀ be C_•. This forces E₁ to be the complex H(C_•): At the {{prime|i}}th location this is the {{prime|i}}th homology group of C_•. The only natural differential on this new complex is the zero map, so we let d₁ = 0. This forces $E\_2$ to equal $E\_1$, and again our only natural differential is the zero map. Putting the zero differential on all the rest of our sheets gives a spectral sequence whose terms are:

• E₀ = C_•
• E_r = H(C_•) for all r ≥ 1.

The terms of this spectral sequence stabilize at the first sheet because its only nontrivial differential was on the zeroth sheet. Consequently, we can get no more information at later steps. Usually, to get useful information from later sheets, we need extra structure on the $E\_r$.

Image:SpectralSequence.png

A doubly graded spectral sequence has a tremendous amount of data to keep track of, but there is a common visualization technique which makes the structure of the spectral sequence clearer. We have three indices, r, p, and q. An object $E\_r$ can be seen as the {{mvar|r}}{{sup|th}} checkered page of a book. On these sheets, we will take p to be the horizontal direction and q to be the vertical direction. At each lattice point we have the object $E\_r^\{p,q\}$. Now turning to the next page means taking homology, that is the $(r+1)${{sup|th}} page is a subquotient of the {{mvar|r}}{{sup|th}} page. The total degree n = p + q runs diagonally, northwest to southeast, across each sheet. In the homological case, the differentials have bidegree {{math|(−r, r − 1)}}, so they decrease n by one. In the cohomological case, n is increased by one. The differentials change their direction with each turn with respect to r.

File:Spectral Sequence Visualization.jpg

The red arrows demonstrate the case of a first quadrant sequence (see example below), where only the objects of the first quadrant are non-zero. While turning pages, either the domain or the codomain of all the differentials become zero.

The set of cohomological spectral sequences form a category: a morphism of spectral sequences $f\; :\; E\; \backslash to\; E\text{'}$ is by definition a collection of maps $f\_r\; :\; E\_r\; \backslash to\; E\text{'}\_r$ which are compatible with the differentials, i.e. $f\_r\; \backslash circ\; d\_r\; =\; d\text{'}\_r\; \backslash circ\; f\_r$, and with the given isomorphisms between the cohomology of the r{{sup|th}} step and the {{tmath|(r+1)}}{{sup|th}} sheets of E and {{prime|E}}, respectively: $f\_\{r+1\}(E\_\{r+1\})\; \backslash ,=\backslash ,\; f\_\{r+1\}(H(E\_r))\; \backslash ,=\backslash ,\; H(f\_r(E\_r))$. In the bigraded case, they should also respect the graduation: $f\_r(E\_r^\{p,q\})\; \backslash subset\; \{E\text{'}\_r\}^\{p,q\}.$

A cup product gives a ring structure to a cohomology group, turning it into a cohomology ring. Thus, it is natural to consider a spectral sequence with a ring structure as well. Let $E^\{p,\; q\}\_r$ be a spectral sequence of cohomological type. We say it has multiplicative structure if (i) $E\_r$ are (doubly graded) differential graded algebras and (ii) the multiplication on $E\_\{r+1\}$ is induced by that on $E\_r$ via passage to cohomology.

A typical example is the cohomological Serre spectral sequence for a fibration $F\; \backslash to\; E\; \backslash to\; B$, when the coefficient group is a ring R. It has the multiplicative structure induced by the cup products of fibre and base on the $E\_\{2\}$-page.{{sfn|McCleary|2001|p={{pn|date=August 2021}}}} However, in general the limiting term $E\_\{\backslash infty\}$ is not isomorphic as a graded algebra to H(E; R).{{sfn|Hatcher|loc=Example 1.17}} The multiplicative structure can be very useful for calculating differentials on the sequence.{{sfn|Hatcher|loc=Example 1.18}}

Spectral sequences can be constructed by various ways. In algebraic topology, an exact couple is perhaps the most common tool for the construction. In algebraic geometry, spectral sequences are usually constructed from filtrations of cochain complexes.

{{main|Exact couple}}

right

Another technique for constructing spectral sequences is William Massey's method of exact couples. Exact couples are particularly common in algebraic topology. Despite this they are unpopular in abstract algebra, where most spectral sequences come from filtered complexes.

To define exact couples, we begin again with an abelian category. As before, in practice this is usually the category of doubly graded modules over a ring. An exact couple is a pair of objects (A, C), together with three homomorphisms between these objects: f : A → A, g : A → C and h : C → A subject to certain exactness conditions:

• Image f = Kernel g
• Image g = Kernel h
• Image h = Kernel f

We will abbreviate this data by (A, C, f, g, h). Exact couples are usually depicted as triangles. We will see that C corresponds to the E₀ term of the spectral sequence and that A is some auxiliary data.

To pass to the next sheet of the spectral sequence, we will form the derived couple. We set:

• d = g o h
• A' = f(A)
• C' = Ker d / Im d
• f ' = f|_A', the restriction of f to A'
• h' : C' → A' is induced by h. It is straightforward to see that h induces such a map.
• g' : A' → C' is defined on elements as follows: For each a in A', write a as f(b) for some b in A. g'(a) is defined to be the image of g(b) in C'. In general, g' can be constructed using one of the embedding theorems for abelian categories.

From here it is straightforward to check that (A', C', f ', g', h') is an exact couple. C' corresponds to the E₁ term of the spectral sequence. We can iterate this procedure to get exact couples (A⁽ⁿ⁾, C⁽ⁿ⁾, f⁽ⁿ⁾, g⁽ⁿ⁾, h⁽ⁿ⁾).

In order to construct a spectral sequence, let E_n be C⁽ⁿ⁾ and d_n be g⁽ⁿ⁾ o h⁽ⁿ⁾.

• Serre spectral sequence{{sfn|May}} - used to compute (co)homology of a fibration
• Atiyah–Hirzebruch spectral sequence - used to compute (co)homology of extraordinary cohomology theories, such as K-theory
• Bockstein spectral sequence.
• Spectral sequences of filtered complexes

A very common type of spectral sequence comes from a filtered cochain complex, as it naturally induces a bigraded object. Consider a cochain complex $(C^\{\backslash bullet\},\; d)$ together with a descending filtration, $...\; \backslash supset\backslash ,\; F^\{-2\}C^\{\backslash bullet\}\; \backslash ,\backslash supset\backslash ,\; F^\{-1\}C^\{\backslash bullet\}\; \backslash supset\; F^\{0\}C^\{\backslash bullet\}\; \backslash ,\backslash supset\backslash ,\; F^\{1\}C^\{\backslash bullet\}\; \backslash ,\backslash supset\backslash ,\; F^\{2\}C^\{\backslash bullet\}\; \backslash ,\backslash supset\backslash ,\; F^\{3\}C^\{\backslash bullet\}\; \backslash ,\backslash supset...\; \backslash ,$ . We require that the boundary map is compatible with the filtration, i.e. $d(F^pC^n)\; \backslash subset\; F^pC^\{n+1\}$, and that the filtration is exhaustive, that is, the union of the set of all $F^pC^\{\backslash bullet\}$ is the entire chain complex $C^\{\backslash bullet\}$. Then there exists a spectral sequence with $E\_0^\{p,q\}\; =\; F^\{p\}C^\{p+q\}/F^\{p+1\}C^\{p+q\}$ and $E\_1^\{p,q\}\; =\; H^\{p+q\}(F^\{p\}C^\{\backslash bullet\}/F^\{p+1\}C^\{\backslash bullet\})$.{{citation|surname1=Serge Lang|title=Algebra|edition=Überarbeitete 3.|series=Graduate Texts in Mathematics 211|publisher=Springer-Verlag|publication-place=New York|isbn=038795385X|date=2002|language=German

}} Later, we will also assume that the filtration is Hausdorff or separated, that is, the intersection of the set of all $F^pC^\{\backslash bullet\}$ is zero.

The filtration is useful because it gives a measure of nearness to zero: As p increases, $F^pC^\{\backslash bullet\}$ gets closer and closer to zero. We will construct a spectral sequence from this filtration where coboundaries and cocycles in later sheets get closer and closer to coboundaries and cocycles in the original complex. This spectral sequence is doubly graded by the filtration degree p and the complementary degree {{math|1=q = n − p}}.

$C^\{\backslash bullet\}$ has only a single grading and a filtration, so we first construct a doubly graded object for the first page of the spectral sequence. To get the second grading, we will take the associated graded object with respect to the filtration. We will write it in an unusual way which will be justified at the $E\_1$ step:

:$Z\_\{-1\}^\{p,q\}\; =\; Z\_0^\{p,q\}\; =\; F^p\; C^\{p+q\}$

:$B\_0^\{p,q\}\; =\; 0$

:$E\_0^\{p,q\}\; =\; \backslash frac\{Z\_0^\{p,q\}\}\{B\_0^\{p,q\}\; +\; Z\_\{-1\}^\{p+1,q-1\}\}\; =\; \backslash frac\{F^p\; C^\{p+q\}\}\{F^\{p+1\}\; C^\{p+q\}\}$

:$E\_0\; =\; \backslash bigoplus\_\{p,q\backslash in\backslash mathbf\{Z\}\}\; E\_0^\{p,q\}$

Since we assumed that the boundary map was compatible with the filtration, $E\_0$ is a doubly graded object and there is a natural doubly graded boundary map $d\_0$ on $E\_0$. To get $E\_1$, we take the homology of $E\_0$.

:$\backslash bar\{Z\}\_1^\{p,q\}\; =\; \backslash ker\; d\_0^\{p,q\}\; :\; E\_0^\{p,q\}\; \backslash rightarrow\; E\_0^\{p,q+1\}\; =\; \backslash ker\; d\_0^\{p,q\}\; :\; F^p\; C^\{p+q\}/F^\{p+1\}\; C^\{p+q\}\; \backslash rightarrow\; F^p\; C^\{p+q+1\}/F^\{p+1\}\; C^\{p+q+1\}$

:$\backslash bar\{B\}\_1^\{p,q\}\; =\; \backslash mbox\{im\; \}\; d\_0^\{p,q-1\}\; :\; E\_0^\{p,q-1\}\; \backslash rightarrow\; E\_0^\{p,q\}\; =\; \backslash mbox\{im\; \}\; d\_0^\{p,q-1\}\; :\; F^p\; C^\{p+q-1\}/F^\{p+1\}\; C^\{p+q-1\}\; \backslash rightarrow\; F^p\; C^\{p+q\}/F^\{p+1\}\; C^\{p+q\}$

:$E\_1^\{p,q\}\; =\; \backslash frac\{\backslash bar\{Z\}\_1^\{p,q\}\}\{\backslash bar\{B\}\_1^\{p,q\}\}\; =\; \backslash frac\{\backslash ker\; d\_0^\{p,q\}\; :\; E\_0^\{p,q\}\; \backslash rightarrow\; E\_0^\{p,q+1\}\}\{\backslash mbox\{im\; \}\; d\_0^\{p,q-1\}\; :\; E\_0^\{p,q-1\}\; \backslash rightarrow\; E\_0^\{p,q\}\}$

:$E\_1\; =\; \backslash bigoplus\_\{p,q\backslash in\backslash mathbf\{Z\}\}\; E\_1^\{p,q\}\; =\; \backslash bigoplus\_\{p,q\backslash in\backslash mathbf\{Z\}\}\; \backslash frac\{\backslash bar\{Z\}\_1^\{p,q\}\}\{\backslash bar\{B\}\_1^\{p,q\}\}$

Notice that $\backslash bar\{Z\}\_1^\{p,q\}$ and $\backslash bar\{B\}\_1^\{p,q\}$ can be written as the images in $E\_0^\{p,q\}$ of

:$Z\_1^\{p,q\}\; =\; \backslash ker\; d\_0^\{p,q\}\; :\; F^p\; C^\{p+q\}\; \backslash rightarrow\; C^\{p+q+1\}/F^\{p+1\}\; C^\{p+q+1\}$

:$B\_1^\{p,q\}\; =\; (\backslash mbox\{im\; \}\; d\_0^\{p,q-1\}\; :\; F^p\; C^\{p+q-1\}\; \backslash rightarrow\; C^\{p+q\})\; \backslash cap\; F^p\; C^\{p+q\}$

and that we then have

:$E\_1^\{p,q\}\; =\; \backslash frac\{Z\_1^\{p,q\}\}\{B\_1^\{p,q\}\; +\; Z\_0^\{p+1,q-1\}\}.$

$Z\_1^\{p,q\}$ are exactly the elements which the differential pushes up one level in the filtration, and $B\_1^\{p,q\}$ are exactly the image of the elements which the differential pushes up zero levels in the filtration. This suggests that we should choose $Z\_r^\{p,q\}$ to be the elements which the differential pushes up r levels in the filtration and $B\_r^\{p,q\}$ to be image of the elements which the differential pushes up r-1 levels in the filtration. In other words, the spectral sequence should satisfy

:$Z\_r^\{p,q\}\; =\; \backslash ker\; d\_0^\{p,q\}\; :\; F^p\; C^\{p+q\}\; \backslash rightarrow\; C^\{p+q+1\}/F^\{p+r\}\; C^\{p+q+1\}$

:$B\_r^\{p,q\}\; =\; (\backslash mbox\{im\; \}\; d\_0^\{p-r+1,q+r-2\}\; :\; F^\{p-r+1\}\; C^\{p+q-1\}\; \backslash rightarrow\; C^\{p+q\})\; \backslash cap\; F^p\; C^\{p+q\}$

:$E\_r^\{p,q\}\; =\; \backslash frac\{Z\_r^\{p,q\}\}\{B\_r^\{p,q\}\; +\; Z\_\{r-1\}^\{p+1,q-1\}\}$

and we should have the relationship

:$B\_r^\{p,q\}\; =\; d\_0^\{p,q\}(Z\_\{r-1\}^\{p-r+1,q+r-2\}).$

For this to make sense, we must find a differential $d\_r$ on each $E\_r$ and verify that it leads to homology isomorphic to $E\_\{r+1\}$. The differential

:$d\_r^\{p,q\}\; :\; E\_r^\{p,q\}\; \backslash rightarrow\; E\_r^\{p+r,q-r+1\}$

is defined by restricting the original differential $d$ defined on $C^\{p+q\}$ to the subobject $Z\_r^\{p,q\}$. It is straightforward to check that the homology of $E\_r$ with respect to this differential is $E\_\{r+1\}$, so this gives a spectral sequence. Unfortunately, the differential is not very explicit. Determining differentials or finding ways to work around them is one of the main challenges to successfully applying a spectral sequence.

• Hodge–de Rham spectral sequence
• Spectral sequence of a double complex
• Can be used to construct Mixed Hodge structures{{cite arXiv|last1=Elzein|first1=Fouad|last2=Trang|first2=Lê Dung|date=2013-02-23|title=Mixed Hodge Structures|eprint=1302.5811|pages=40, 4.0.2|class=math.AG}}

Another common spectral sequence is the spectral sequence of a double complex. A double complex is a collection of objects C_i,j for all integers i and j together with two differentials, {{mvar|d}}{{i sup|{{rn|I}}}} and {{mvar|d}}{{i sup|{{rn|II}}}}. {{mvar|d}}{{i sup|{{rn|I}}}} is assumed to decrease i, and {{mvar|d}}{{i sup|{{rn|II}}}} is assumed to decrease j. Furthermore, we assume that the differentials anticommute, so that d{{i sup|{{rn|I}}}} d{{i sup|{{rn|II}}}} + d{{i sup|{{rn|II}}}} d{{i sup|{{rn|I}}}} = 0. Our goal is to compare the iterated homologies $H^\backslash textrm\{I\}\_i(H^\backslash textrm\{II\}\_j(C\_\{\backslash bullet,\backslash bullet\}))$ and $H^\backslash textrm\{II\}\_j(H^\backslash textrm\{I\}\_i(C\_\{\backslash bullet,\backslash bullet\}))$. We will do this by filtering our double complex in two different ways. Here are our filtrations:

:$(C\_\{i,j\}^\backslash textrm\{I\})\_p\; =\; \backslash begin\{cases\}$

0 & \text{if } i < p \\

C_{i,j} & \text{if } i \ge p \end{cases}

:$(C\_\{i,j\}^\backslash textrm\{II\})\_p\; =\; \backslash begin\{cases\}$

0 & \text{if } j < p \\

C_{i,j} & \text{if } j \ge p \end{cases}

To get a spectral sequence, we will reduce to the previous example. We define the total complex T(C_•,•) to be the complex whose {{prime|n}}th term is $\backslash bigoplus\_\{i+j=n\}\; C\_\{i,j\}$ and whose differential is {{mvar|d}}{{i sup|{{rn|I}}}} + {{mvar|d}}{{i sup|{{rn|II}}}}. This is a complex because {{mvar|d}}{{i sup|{{rn|I}}}} and {{mvar|d}}{{i sup|{{rn|II}}}} are anticommuting differentials. The two filtrations on C_i,j give two filtrations on the total complex:

:$T\_n(C\_\{\backslash bullet,\backslash bullet\})^\backslash textrm\{I\}\_p\; =\; \backslash bigoplus\_\{i+j=n\; \backslash atop\; i\; >\; p-1\}\; C\_\{i,j\}$

:$T\_n(C\_\{\backslash bullet,\backslash bullet\})^\backslash textrm\{II\}\_p\; =\; \backslash bigoplus\_\{i+j=n\; \backslash atop\; j\; >\; p-1\}\; C\_\{i,j\}$

To show that these spectral sequences give information about the iterated homologies, we will work out the E{{i sup|0}}, E{{i sup|1}}, and E{{i sup|2}} terms of the {{rn|I}} filtration on T(C_•,•). The E{{i sup|0}} term is clear:

:$\{\}^\backslash textrm\{I\}E^0\_\{p,q\}\; =$

T_n(C_{\bullet,\bullet})^\textrm{I}_p / T_n(C_{\bullet,\bullet})^\textrm{I}_{p+1} =

\bigoplus_{i+j=n \atop i > p-1} C_{i,j} \Big/

\bigoplus_{i+j=n \atop i > p} C_{i,j} =

C_{p,q},

where {{nowrap|n {{=}} p + q}}.

To find the E{{i sup|1}} term, we need to determine {{mvar|d}}{{i sup|{{rn|I}}}} + {{mvar|d}}{{i sup|{{rn|II}}}} on E{{i sup|0}}. Notice that the differential must have degree −1 with respect to n, so we get a map

:$d^\backslash textrm\{I\}\_\{p,q\}\; +\; d^\backslash textrm\{II\}\_\{p,q\}\; :$

T_n(C_{\bullet,\bullet})^\textrm{I}_p / T_n(C_{\bullet,\bullet})^\textrm{I}_{p+1} =

C_{p,q} \rightarrow

T_{n-1}(C_{\bullet,\bullet})^\textrm{I}_p / T_{n-1}(C_{\bullet,\bullet})^\textrm{I}_{p+1} =

C_{p,q-1}

Consequently, the differential on E{{i sup|0}} is the map C_p,q → C_p,q−1 induced by {{mvar|d}}{{i sup|{{rn|I}}}} + {{mvar|d}}{{i sup|{{rn|II}}}}. But {{mvar|d}}{{i sup|{{rn|I}}}} has the wrong degree to induce such a map, so {{mvar|d}}{{i sup|{{rn|I}}}} must be zero on E{{i sup|0}}. That means the differential is exactly {{mvar|d}}{{i sup|{{rn|II}}}}, so we get

:$\{\}^\backslash textrm\{I\}E^1\_\{p,q\}\; =\; H^\backslash textrm\{II\}\_q(C\_\{p,\backslash bullet\}).$

To find E{{i sup|2}}, we need to determine

:$d^\backslash textrm\{I\}\_\{p,q\}\; +\; d^\backslash textrm\{II\}\_\{p,q\}\; :$

H^\textrm{II}_q(C_{p,\bullet}) \rightarrow

H^\textrm{II}_q(C_{p+1,\bullet})

Because E{{i sup|1}} was exactly the homology with respect to {{mvar|d}}{{i sup|{{rn|II}}}}, {{mvar|d}}{{i sup|{{rn|II}}}} is zero on E{{i sup|1}}. Consequently, we get

:$\{\}^\backslash textrm\{I\}E^2\_\{p,q\}\; =\; H^\backslash textrm\{I\}\_p(H^\backslash textrm\{II\}\_q(C\_\{\backslash bullet,\backslash bullet\})).$

Using the other filtration gives us a different spectral sequence with a similar E{{i sup|2}} term:

:$\{\}^\backslash textrm\{II\}E^2\_\{p,q\}\; =\; H^\backslash textrm\{II\}\_q(H^\{I\}\_p(C\_\{\backslash bullet,\backslash bullet\})).$

What remains is to find a relationship between these two spectral sequences. It will turn out that as r increases, the two sequences will become similar enough to allow useful comparisons.

Let E_r be a spectral sequence, starting with say r = 1. Then there is a sequence of subobjects

:$0\; =\; B\_0\; \backslash subset\; B\_1\; \backslash subset\; B\_\{2\}\; \backslash subset\; \backslash dots\; \backslash subset\; B\_r\; \backslash subset\; \backslash dots\; \backslash subset\; Z\_r\; \backslash subset\; \backslash dots\; \backslash subset\; Z\_2\; \backslash subset\; Z\_1\; \backslash subset\; Z\_0\; =\; E\_1$

such that $E\_r\; \backslash simeq\; Z\_\{r-1\}/B\_\{r-1\}$; indeed, recursively we let $Z\_0\; =\; E\_1,\; B\_0\; =\; 0$ and let $Z\_r,\; B\_r$ be so that $Z\_r/B\_\{r-1\},\; B\_r/B\_\{r-1\}$ are the kernel and the image of $E\_r\; \backslash overset\{d\_r\}\backslash to\; E\_r.$

We then let $Z\_\{\backslash infty\}\; =\; \backslash cap\_r\; Z\_r,\; B\_\{\backslash infty\}\; =\; \backslash cup\_r\; B\_r$ and

:$E\_\{\backslash infty\}\; =\; Z\_\{\backslash infty\}/B\_\{\backslash infty\}$;

it is called the limiting term. (Of course, such $E\_\{\backslash infty\}$ need not exist in the category, but this is usually a non-issue since for example in the category of modules such limits exist or since in practice a spectral sequence one works with tends to degenerate; there are only finitely many inclusions in the sequence above.)

We say a spectral sequence converges weakly if there is a graded object $H^\{\backslash bullet\}$ with a filtration $F^\{\backslash bullet\}\; H^\{n\}$ for every $n$, and for every $p$ there exists an isomorphism $E\_\{\backslash infty\}^\{p,q\}\; \backslash cong\; F^pH^\{p+q\}/F^\{p+1\}H^\{p+q\}$. It converges to $H^\{\backslash bullet\}$ if the filtration $F^\{\backslash bullet\}\; H^\{n\}$ is Hausdorff, i.e. $\backslash cap\_\{p\}F^pH^\{\backslash bullet\}=0$. We write

:$E\_r^\{p,q\}\; \backslash Rightarrow\_p\; E\_\backslash infty^n$

to mean that whenever p + q = n, $E\_r^\{p,q\}$ converges to $E\_\backslash infty^\{p,q\}$.

We say that a spectral sequence $E\_r^\{p,q\}$ abuts to $E\_\backslash infty^\{p,q\}$ (the spectral sequence abutment{{Anchor|Abutment}}) if for every $p,q$ there is $r(p,q)$ such that for all $r\; \backslash geq\; r(p,q)$, $E\_r^\{p,q\}\; =\; E\_\{r(p,q)\}^\{p,q\}$. Then $E\_\{r(p,q)\}^\{p,q\}\; =\; E\_\backslash infty^\{p,q\}$ is the limiting term. The spectral sequence is regular or degenerates at $r\_0$ if the differentials $d\_r^\{p,q\}$ are zero for all $r\; \backslash geq\; r\_0$. If in particular there is $r\_0\; \backslash geq\; 2$, such that the $r\_0^\{th\}$ sheet is concentrated on a single row or a single column, then we say it collapses. In symbols, we write:

:$E\_r^\{p,q\}\; \backslash Rightarrow\_p\; E\_\backslash infty^\{p,q\}$

The p indicates the filtration index. It is very common to write the $E\_2^\{p,q\}$ term on the left-hand side of the abutment, because this is the most useful term of most spectral sequences. The spectral sequence of an unfiltered chain complex degenerates at the first sheet (see first example): since nothing happens after the zeroth sheet, the limiting sheet $E\_\{\backslash infty\}$ is the same as $E\_1$.

The five-term exact sequence of a spectral sequence relates certain low-degree terms and E_∞ terms.

Notice that we have a chain of inclusions:

:$Z\_0^\{p,q\}\; \backslash supe\; Z\_1^\{p,q\}\; \backslash supe\; Z\_2^\{p,q\}\backslash supe\backslash cdots\backslash supe\; B\_2^\{p,q\}\; \backslash supe\; B\_1^\{p,q\}\; \backslash supe\; B\_0^\{p,q\}$

We can ask what happens if we define

:$Z\_\backslash infty^\{p,q\}\; =\; \backslash bigcap\_\{r=0\}^\backslash infty\; Z\_r^\{p,q\},$

:$B\_\backslash infty^\{p,q\}\; =\; \backslash bigcup\_\{r=0\}^\backslash infty\; B\_r^\{p,q\},$

:$E\_\backslash infty^\{p,q\}\; =\; \backslash frac\{Z\_\backslash infty^\{p,q\}\}\{B\_\backslash infty^\{p,q\}+Z\_\backslash infty^\{p+1,q-1\}\}.$

$E\_\backslash infty^\{p,q\}$ is a natural candidate for the abutment of this spectral sequence. Convergence is not automatic, but happens in many cases. In particular, if the filtration is finite and consists of exactly r nontrivial steps, then the spectral sequence degenerates after the rth sheet. Convergence also occurs if the complex and the filtration are both bounded below or both bounded above.

To describe the abutment of our spectral sequence in more detail, notice that we have the formulas:

:$Z\_\backslash infty^\{p,q\}\; =\; \backslash bigcap\_\{r=0\}^\backslash infty\; Z\_r^\{p,q\}\; =\; \backslash bigcap\_\{r=0\}^\backslash infty\; \backslash ker(F^p\; C^\{p+q\}\; \backslash rightarrow\; C^\{p+q+1\}/F^\{p+r\}\; C^\{p+q+1\})$

:$B\_\backslash infty^\{p,q\}\; =\; \backslash bigcup\_\{r=0\}^\backslash infty\; B\_r^\{p,q\}\; =\; \backslash bigcup\_\{r=0\}^\backslash infty\; (\backslash mbox\{im\; \}\; d^\{p,q-r\}\; :\; F^\{p-r\}\; C^\{p+q-1\}\; \backslash rightarrow\; C^\{p+q\})\; \backslash cap\; F^p\; C^\{p+q\}$

To see what this implies for $Z\_\backslash infty^\{p,q\}$ recall that we assumed that the filtration was separated. This implies that as r increases, the kernels shrink, until we are left with $Z\_\backslash infty^\{p,q\}\; =\; \backslash ker(F^p\; C^\{p+q\}\; \backslash rightarrow\; C^\{p+q+1\})$. For $B\_\backslash infty^\{p,q\}$, recall that we assumed that the filtration was exhaustive. This implies that as r increases, the images grow until we reach $B\_\backslash infty^\{p,q\}\; =\; \backslash text\{im\; \}(C^\{p+q-1\}\; \backslash rightarrow\; C^\{p+q\})\; \backslash cap\; F^p\; C^\{p+q\}$. We conclude

:$E\_\backslash infty^\{p,q\}\; =\; \backslash mbox\{gr\}\_p\; H^\{p+q\}(C^\backslash bull)$,

that is, the abutment of the spectral sequence is the pth graded part of the (p+q)th homology of C. If our spectral sequence converges, then we conclude that:

:$E\_r^\{p,q\}\; \backslash Rightarrow\_p\; H^\{p+q\}(C^\backslash bull)$

Using the spectral sequence of a filtered complex, we can derive the existence of long exact sequences. Choose a short exact sequence of cochain complexes 0 → {{var|A}}{{i sup|•}} → {{var|B}}{{i sup|•}} → {{var|C}}{{i sup|•}} → 0, and call the first map {{var|f}}{{i sup|•}} : {{var|A}}{{i sup|•}} → {{var|B}}{{i sup|•}}. We get natural maps of homology objects Hⁿ({{var|A}}{{i sup|•}}) → Hⁿ({{var|B}}{{i sup|•}}) → Hⁿ({{var|C}}{{i sup|•}}), and we know that this is exact in the middle. We will use the spectral sequence of a filtered complex to find the connecting homomorphism and to prove that the resulting sequence is exact.To start, we filter {{mvar|B}}{{i sup|•}}:

:$F^0\; B^n\; =\; B^n$

:$F^1\; B^n\; =\; A^n$

:$F^2\; B^n\; =\; 0$

This gives:

:$E^\{p,q\}\_0$

= \frac{F^p B^{p+q}}{F^{p+1} B^{p+q}} = \begin{cases}

0 & \text{if } p < 0 \text{ or } p > 1 \\

C^q & \text{if } p = 0 \\

A^{q+1} & \text{if } p = 1 \end{cases}

:$E^\{p,q\}\_1$

= \begin{cases}

0 & \text{if } p < 0 \text{ or } p > 1 \\

H^q(C^\bull) & \text{if } p = 0 \\

H^{q+1}(A^\bull) & \text{if } p = 1 \end{cases}

The differential has bidegree (1, 0), so d_0,q : H^q({{var|C}}{{i sup|•}}) → H^q+1({{var|A}}{{i sup|•}}). These are the connecting homomorphisms from the snake lemma, and together with the maps {{mvar|A}}{{i sup|•}} → {{mvar|B}}{{i sup|•}} → {{mvar|C}}{{i sup|•}}, they give a sequence:

:$\backslash cdots\backslash rightarrow\; H^q(B^\backslash bull)\; \backslash rightarrow\; H^q(C^\backslash bull)\; \backslash rightarrow\; H^\{q+1\}(A^\backslash bull)\; \backslash rightarrow\; H^\{q+1\}(B^\backslash bull)\; \backslash rightarrow\backslash cdots$

It remains to show that this sequence is exact at the A and C spots. Notice that this spectral sequence degenerates at the E₂ term because the differentials have bidegree (2, −1). Consequently, the E₂ term is the same as the E_∞ term:

:$E^\{p,q\}\_2$

\cong \text{gr}_p H^{p+q}(B^\bull)

= \begin{cases}

0 & \text{if } p < 0 \text{ or } p > 1 \\

H^q(B^\bull)/H^q(A^\bull) & \text{if } p = 0 \\

\text{im } H^{q+1}f^\bull : H^{q+1}(A^\bull) \rightarrow H^{q+1}(B^\bull) &\text{if } p = 1 \end{cases}

But we also have a direct description of the E₂ term as the homology of the E₁ term. These two descriptions must be isomorphic:

:$H^q(B^\backslash bull)/H^q(A^\backslash bull)\; \backslash cong\; \backslash ker\; d^1\_\{0,q\}\; :\; H^q(C^\backslash bull)\; \backslash rightarrow\; H^\{q+1\}(A^\backslash bull)$

:$\backslash text\{im\; \}\; H^\{q+1\}f^\backslash bull\; :\; H^\{q+1\}(A^\backslash bull)\; \backslash rightarrow\; H^\{q+1\}(B^\backslash bull)\; \backslash cong\; H^\{q+1\}(A^\backslash bull)\; /\; (\backslash mbox\{im\; \}\; d^1\_\{0,q\}\; :\; H^q(C^\backslash bull)\; \backslash rightarrow\; H^\{q+1\}(A^\backslash bull))$

The former gives exactness at the C spot, and the latter gives exactness at the A spot.

Using the abutment for a filtered complex, we find that:

:$H^\backslash textrm\{I\}\_p(H^\backslash textrm\{II\}\_q(C\_\{\backslash bull,\backslash bull\}))\; \backslash Rightarrow\_p\; H^\{p+q\}(T(C\_\{\backslash bull,\backslash bull\}))$

:$H^\backslash textrm\{II\}\_q(H^\backslash textrm\{I\}\_p(C\_\{\backslash bull,\backslash bull\}))\; \backslash Rightarrow\_q\; H^\{p+q\}(T(C\_\{\backslash bull,\backslash bull\}))$

In general, the two gradings on {{tmath|H^{p+q}(T(C_{\bull,\bull}))}} are distinct. Despite this, it is still possible to gain useful information from these two spectral sequences.

Let R be a ring, let M be a right R-module and N a left R-module. Recall that the derived functors of the tensor product are denoted Tor. Tor is defined using a projective resolution of its first argument. However, it turns out that $\backslash operatorname\{Tor\}\_i(M,N)\; =\backslash operatorname\{Tor\}\_i(N,M)$. While this can be verified without a spectral sequence, it is very easy with spectral sequences.

Choose projective resolutions $P\_\backslash bull$ and $Q\_\backslash bull$ of M and N, respectively. Consider these as complexes which vanish in negative degree having differentials d and e, respectively. We can construct a double complex whose terms are $C\_\{i,j\}\; =\; P\_i\; \backslash otimes\; Q\_j$ and whose differentials are $d\; \backslash otimes\; 1$ and $(-1)^\backslash textrm\{I\}(1\; \backslash otimes\; e)$. (The factor of −1 is so that the differentials anticommute.) Since projective modules are flat, taking the tensor product with a projective module commutes with taking homology, so we get:

:$H^\backslash textrm\{I\}\_p(H^\backslash textrm\{II\}\_q(P\_\backslash bull\; \backslash otimes\; Q\_\backslash bull))\; =\; H^\backslash textrm\{I\}\_p(P\_\backslash bull\; \backslash otimes\; H^\backslash textrm\{II\}\_q(Q\_\backslash bull))$

:$H^\backslash textrm\{II\}\_q(H^\backslash textrm\{I\}\_p(P\_\backslash bull\; \backslash otimes\; Q\_\backslash bull))\; =\; H^\backslash textrm\{II\}\_q(H^\backslash textrm\{I\}\_p(P\_\backslash bull)\; \backslash otimes\; Q\_\backslash bull)$

Since the two complexes are resolutions, their homology vanishes outside of degree zero. In degree zero, we are left with

:$H^\backslash textrm\{I\}\_p(P\_\backslash bull\; \backslash otimes\; N)\; =\; \backslash operatorname\{Tor\}\_p(M,N)$

:$H^\backslash textrm\{II\}\_q(M\; \backslash otimes\; Q\_\backslash bull)\; =\; \backslash operatorname\{Tor\}\_q(N,M)$

In particular, the $E^2\_\{p,q\}$ terms vanish except along the lines q = 0 (for the {{rn|I}} spectral sequence) and p = 0 (for the {{rn|II}} spectral sequence). This implies that the spectral sequence degenerates at the second sheet, so the E^∞ terms are isomorphic to the E² terms:

:$\backslash operatorname\{Tor\}\_p(M,N)\; \backslash cong\; E^\backslash infty\_p\; =\; H\_p(T(C\_\{\backslash bull,\backslash bull\}))$

:$\backslash operatorname\{Tor\}\_q(N,M)\; \backslash cong\; E^\backslash infty\_q\; =\; H\_q(T(C\_\{\backslash bull,\backslash bull\}))$

Finally, when p and q are equal, the two right-hand sides are equal, and the commutativity of Tor follows.

Consider a spectral sequence where $E\_r^\{p,q\}$ vanishes for all $p$ less than some $p\_0$ and for all $q$ less than some $q\_0$. If $p\_0$ and $q\_0$ can be chosen to be zero, this is called a first-quadrant spectral sequence.

The sequence abuts because $E\_\{r+i\}^\{p,q\}\; =\; E\_r^\{p,q\}$ holds for all $i\backslash geq\; 0$ if $r>p$ and $r>q+1$. To see this, note that either the domain or the codomain of the differential is zero for the considered cases. In visual terms, the sheets stabilize in a growing rectangle (see picture above). The spectral sequence need not degenerate, however, because the differential maps might not all be zero at once. Similarly, the spectral sequence also converges if $E\_r^\{p,q\}$ vanishes for all $p$ greater than some $p\_0$ and for all $q$ greater than some $q\_0$.

Let $E^r\_\{p,\; q\}$ be a homological spectral sequence such that $E^2\_\{p,\; q\}\; =\; 0$ for all p other than 0, 1. Visually, this is the spectral sequence with $E^2$-page

:$\backslash begin\{matrix\}$

& \vdots & \vdots & \vdots & \vdots & \\

\cdots & 0 & E^2_{0,2} & E^2_{1,2} & 0 & \cdots \\

\cdots & 0 & E^2_{0,1} & E^2_{1,1} & 0 & \cdots \\

\cdots & 0 & E^2_{0,0} & E^2_{1,0} & 0 & \cdots \\

\cdots & 0 & E^2_{0,-1} & E^2_{1,-1} & 0 & \cdots \\

& \vdots & \vdots & \vdots & \vdots &

\end{matrix}

The differentials on the second page have degree (-2, 1), so they are of the form

:$d^2\_\{p,q\}:E^2\_\{p,q\}\; \backslash to\; E^2\_\{p-2,q+1\}$

These maps are all zero since they are

:$d^2\_\{0,q\}:E^2\_\{0,q\}\; \backslash to\; 0$, $d^2\_\{1,q\}:E^2\_\{1,q\}\; \backslash to\; 0$

hence the spectral sequence degenerates: $E^\{\backslash infty\}\; =\; E^2$. Say, it converges to $H\_*$ with a filtration

:$0\; =\; F\_\{-1\}\; H\_n\; \backslash subset\; F\_0\; H\_n\; \backslash subset\; \backslash dots\; \backslash subset\; F\_n\; H\_n\; =\; H\_n$

such that $E^\{\backslash infty\}\_\{p,\; q\}\; =\; F\_p\; H\_\{p+q\}/F\_\{p-1\}\; H\_\{p+q\}$. Then $F\_0\; H\_n\; =\; E^2\_\{0,\; n\}$, $F\_1\; H\_n\; /\; F\_0\; H\_n\; =\; E^2\_\{1,\; n\; -1\}$, $F\_2\; H\_n\; /\; F\_1\; H\_n\; =\; 0$, $F\_3\; H\_n\; /\; F\_2\; H\_n\; =\; 0$, etc. Thus, there is the exact sequence:{{harvnb|Weibel|1994|loc=Exercise 5.2.1.}}; there are typos in the exact sequence, at least in the 1994 edition.

:$0\; \backslash to\; E^2\_\{0,\; n\}\; \backslash to\; H\_n\; \backslash to\; E^2\_\{1,\; n\; -\; 1\}\; \backslash to\; 0$.

Next, let $E^r\_\{p,\; q\}$ be a spectral sequence whose second page consists only of two lines q = 0, 1. This need not degenerate at the second page but it still degenerates at the third page as the differentials there have degree (-3, 2). Note $E^3\_\{p,\; 0\}\; =\; \backslash operatorname\{ker\}\; (d:\; E^2\_\{p,\; 0\}\; \backslash to\; E^2\_\{p\; -\; 2,\; 1\})$, as the denominator is zero. Similarly, $E^3\_\{p,\; 1\}\; =\; \backslash operatorname\{coker\}(d:\; E^2\_\{p+2,\; 0\}\; \backslash to\; E^2\_\{p,\; 1\})$. Thus,

:$0\; \backslash to\; E^\{\backslash infty\}\_\{p,\; 0\}\; \backslash to\; E^2\_\{p,\; 0\}\; \backslash overset\{d\}\backslash to\; E^2\_\{p-2,\; 1\}\; \backslash to\; E^\{\backslash infty\}\_\{p-2,\; 1\}\; \backslash to\; 0$.

Now, say, the spectral sequence converges to H with a filtration F as in the previous example. Since $F\_\{p-2\}\; H\_\{p\}\; /\; F\_\{p-3\}\; H\_\{p\}\; =\; E^\{\backslash infty\}\_\{p-2,\; 2\}\; =\; 0$, $F\_\{p-3\}\; H\_p\; /\; F\_\{p-4\}\; H\_p\; =\; 0$, etc., we have: $0\; \backslash to\; E^\{\backslash infty\}\_\{p\; -\; 1,\; 1\}\; \backslash to\; H\_p\; \backslash to\; E^\{\backslash infty\}\_\{p,\; 0\}\; \backslash to\; 0$. Putting everything together, one gets:{{harvnb|Weibel|1994|loc=Exercise 5.2.2.}}

:$\backslash cdots\; \backslash to\; H\_\{p+1\}\; \backslash to\; E^2\_\{p\; +\; 1,\; 0\}\; \backslash overset\{d\}\backslash to\; E^2\_\{p\; -\; 1,\; 1\}\; \backslash to\; H\_p\; \backslash to\; E^2\_\{p,\; 0\}\; \backslash overset\{d\}\backslash to\; E^2\_\{p\; -\; 2,\; 1\}\; \backslash to\; H\_\{p-1\}\; \backslash to\; \backslash dots.$

The computation in the previous section generalizes in a straightforward way. Consider a fibration over a sphere:

:$F\; \backslash overset\{i\}\backslash to\; E\; \backslash overset\{p\}\backslash to\; S^n$

with n at least 2. There is the Serre spectral sequence:

:$E^2\_\{p,\; q\}\; =\; H\_p(S^n;\; H\_q(F))\; \backslash Rightarrow\; H\_\{p+q\}(E)$;

that is to say, $E^\{\backslash infty\}\_\{p,\; q\}\; =\; F\_p\; H\_\{p+q\}(E)/F\_\{p-1\}\; H\_\{p+q\}(E)$ with some filtration $F\_\backslash bullet$.

Since $H\_p(S^n)$ is nonzero only when p is zero or n and equal to Z in that case, we see $E^2\_\{p,\; q\}$ consists of only two lines $p\; =\; 0,n$, hence the $E^2$-page is given by

:$\backslash begin\{matrix\}$

& \vdots & \vdots & \vdots & & \vdots & \vdots & \vdots & \\

\cdots & 0 & E^2_{0,2} & 0 & \cdots & 0 & E^2_{n,2} & 0 & \cdots \\

\cdots & 0 & E^2_{0,1} & 0 & \cdots & 0 & E^2_{n,1} & 0 & \cdots \\

\cdots & 0 & E^2_{0,0} & 0 & \cdots & 0 & E^2_{n,0} & 0 & \cdots \\

\end{matrix}

Moreover, since

:$E^2\_\{p,\; q\}\; =\; H\_p(S^n;H\_q(F))\; =\; H\_q(F)$

for $p\; =\; 0,n$ by the universal coefficient theorem, the $E^2$ page looks like

:$\backslash begin\{matrix\}$

& \vdots & \vdots & \vdots & & \vdots & \vdots & \vdots & \\

\cdots & 0 & H_2(F) & 0 & \cdots & 0 & H_2(F) & 0 & \cdots \\

\cdots & 0 & H_1(F) & 0 & \cdots & 0 & H_1(F) & 0 & \cdots \\

\cdots & 0 & H_0(F) & 0 & \cdots & 0 & H_0(F) & 0 & \cdots \\

\end{matrix}

Since the only non-zero differentials are on the $E^n$-page, given by

:$d^n\_\{n,q\}:E^n\_\{n,q\}\; \backslash to\; E^n\_\{0,q+n-1\}$

which is

:$d^n\_\{n,q\}:H\_q(F)\; \backslash to\; H\_\{q+n-1\}(F)$

the spectral sequence converges on $E^\{n+1\}\; =\; E^\{\backslash infty\}$. By computing $E^\{n+1\}$ we get an exact sequence

:$0\; \backslash to\; E^\{\backslash infty\}\_\{n,\; q-n\}\; \backslash to\; E^n\_\{n,\; q-n\}\; \backslash overset\{d\}\backslash to\; E^n\_\{0,\; q-1\}\; \backslash to\; E^\{\backslash infty\}\_\{0,\; q-1\}\; \backslash to\; 0.$

and written out using the homology groups, this is

:$0\; \backslash to\; E^\{\backslash infty\}\_\{n,\; q-n\}\; \backslash to\; H\_\{q-n\}(F)\; \backslash overset\{d\}\backslash to\; H\_\{q-1\}(F)\; \backslash to\; E^\{\backslash infty\}\_\{0,\; q-1\}\; \backslash to\; 0.$

To establish what the two $E^\backslash infty$-terms are, write $H\; =\; H(E)$, and since $F\_1\; H\_q/F\_0\; H\_q\; =\; E^\{\backslash infty\}\_\{1,\; q\; -\; 1\}\; =\; 0$, etc., we have: $E^\{\backslash infty\}\_\{n,\; q-n\}\; =\; F\_n\; H\_q\; /\; F\_0\; H\_q$ and thus, since $F\_n\; H\_q\; =\; H\_q$,

:$0\; \backslash to\; E^\{\backslash infty\}\_\{0,\; q\}\; \backslash to\; H\_q\; \backslash to\; E^\{\backslash infty\}\_\{n,\; q\; -\; n\}\; \backslash to\; 0.$

This is the exact sequence

:$0\; \backslash to\; H\_q(F)\; \backslash to\; H\_q(E)\; \backslash to\; H\_\{q-n\}(F)\backslash to\; 0.$

Putting all calculations together, one gets:{{harvnb|Weibel|1994|loc=Application 5.3.5.}}

:$\backslash dots\; \backslash to\; H\_q(F)\; \backslash overset\{i\_*\}\backslash to\; H\_q(E)\; \backslash to\; H\_\{q-n\}(F)\; \backslash overset\{d\}\backslash to\; H\_\{q-1\}(F)\; \backslash overset\{i\_*\}\backslash to\; H\_\{q-1\}(E)\; \backslash to\; H\_\{q-n\; -1\}(F)\; \backslash to\; \backslash dots$

(The Gysin sequence is obtained in a similar way.)

With an obvious notational change, the type of the computations in the previous examples can also be carried out for cohomological spectral sequence. Let $E\_r^\{p,\; q\}$ be a first-quadrant spectral sequence converging to H with the decreasing filtration

:$0\; =\; F^\{n+1\}\; H^n\; \backslash subset\; F^n\; H^n\; \backslash subset\; \backslash dots\; \backslash subset\; F^0\; H^n\; =\; H^n$

so that $E\_\{\backslash infty\}^\{p,q\}\; =\; F^p\; H^\{p+q\}/F^\{p+1\}\; H^\{p+q\}.$

Since $E\_2^\{p,\; q\}$ is zero if p or q is negative, we have:

:$0\; \backslash to\; E^\{0,\; 1\}\_\{\backslash infty\}\; \backslash to\; E^\{0,\; 1\}\_2\; \backslash overset\{d\}\backslash to\; E^\{2,\; 0\}\_2\; \backslash to\; E^\{2,\; 0\}\_\{\backslash infty\}\; \backslash to\; 0.$

Since $E\_\{\backslash infty\}^\{1,\; 0\}\; =\; E\_2^\{1,\; 0\}$ for the same reason and since $F^2\; H^1\; =\; 0,$

:$0\; \backslash to\; E\_2^\{1,\; 0\}\; \backslash to\; H^1\; \backslash to\; E^\{0,\; 1\}\_\{\backslash infty\}\; \backslash to\; 0$.

Since $F^3\; H^2\; =\; 0$, $E^\{2,\; 0\}\_\{\backslash infty\}\; \backslash subset\; H^2$. Stacking the sequences together, we get the so-called five-term exact sequence:

:$0\; \backslash to\; E^\{1,\; 0\}\_2\; \backslash to\; H^1\; \backslash to\; E^\{0,\; 1\}\_2\; \backslash overset\{d\}\backslash to\; E^\{2,\; 0\}\_2\; \backslash to\; H^2.$

Let $E^r\_\{p,\; q\}$ be a spectral sequence. If $E^r\_\{p,\; q\}\; =\; 0$ for every q < 0, then it must be that: for r ≥ 2,

:$E^\{r+1\}\_\{p,\; 0\}\; =\; \backslash operatorname\{ker\}(d:\; E^r\_\{p,\; 0\}\; \backslash to\; E^r\_\{p-r,\; r-1\})$

as the denominator is zero. Hence, there is a sequence of monomorphisms:

:$E^\{r\}\_\{p,\; 0\}\; \backslash to\; E^\{r-1\}\_\{p,\; 0\}\; \backslash to\; \backslash dots\; \backslash to\; E^3\_\{p,\; 0\}\; \backslash to\; E^2\_\{p,\; 0\}$.

They are called the edge maps. Similarly, if $E^r\_\{p,\; q\}\; =\; 0$ for every p < 0, then there is a sequence of epimorphisms (also called the edge maps):

:$E^2\_\{0,\; q\}\; \backslash to\; E^3\_\{0,\; q\}\; \backslash to\; \backslash dots\; \backslash to\; E^\{r-1\}\_\{0,\; q\}\; \backslash to\; E^r\_\{0,\; q\}$.

The transgression is a partially-defined map (more precisely, a map from a subobject to a quotient)

:$\backslash tau:\; E^2\_\{p,\; 0\}\; \backslash to\; E^2\_\{0,\; p\; -\; 1\}$

given as a composition $E^2\_\{p,\; 0\}\; \backslash to\; E^p\_\{p,\; 0\}\; \backslash overset\{d\}\backslash to\; E^p\_\{0,\; p-1\}\; \backslash to\; E^2\_\{0,\; p\; -\; 1\}$, the first and last maps being the inverses of the edge maps.{{sfn|May|loc=§ 1}}

For a spectral sequence $E\_r^\{p,\; q\}$ of cohomological type, the analogous statements hold. If $E\_r^\{p,\; q\}\; =\; 0$ for every q < 0, then there is a sequence of epimorphisms

:$E\_\{2\}^\{p,\; 0\}\; \backslash to\; E\_\{3\}^\{p,\; 0\}\; \backslash to\; \backslash dots\; \backslash to\; E\_\{r-1\}^\{p,\; 0\}\; \backslash to\; E\_r^\{p,\; 0\}$.

And if $E\_r^\{p,\; q\}\; =\; 0$ for every p < 0, then there is a sequence of monomorphisms:

:$E\_\{r\}^\{0,\; q\}\; \backslash to\; E\_\{r-1\}^\{0,\; q\}\; \backslash to\; \backslash dots\; \backslash to\; E\_\{3\}^\{0,\; q\}\; \backslash to\; E\_2^\{0,\; q\}$.

The transgression is a not necessarily well-defined map:

:$\backslash tau:\; E\_2^\{0,\; q-1\}\; \backslash to\; E\_2^\{q,\; 0\}$

induced by $d:\; E\_q^\{0,\; q-1\}\; \backslash to\; E\_q^\{q,\; 0\}$.

Determining these maps are fundamental for computing many differentials in the Serre spectral sequence. For instance the transgression map determines the differential{{sfn|Hatcher|pp=540, 564}}

:$d\_n:E\_\{n,0\}^n\; \backslash to\; E\_\{0,n-1\}^n$

for the homological spectral spectral sequence, hence on the Serre spectral sequence for a fibration $F\; \backslash to\; E\; \backslash to\; B$ gives the map

:$d\_n:H\_n(B)\; \backslash to\; H\_\{n-1\}(F)$.

Some notable spectral sequences are:

• Atiyah–Hirzebruch spectral sequence of an extraordinary cohomology theory
• Bar spectral sequence for the homology of the classifying space of a group.
• Bockstein spectral sequence relating the homology with mod p coefficients and the homology reduced mod p.
• Cartan–Leray spectral sequence converging to the homology of a quotient space.
• Eilenberg–Moore spectral sequence for the singular cohomology of the pullback of a fibration
• Serre spectral sequence of a fibration

• Adams spectral sequence in stable homotopy theory
• Adams–Novikov spectral sequence, a generalization to extraordinary cohomology theories.
• Barratt spectral sequence converging to the homotopy of the initial space of a cofibration.
• Bousfield–Kan spectral sequence converging to the homotopy colimit of a functor.
• Chromatic spectral sequence for calculating the initial terms of the Adams–Novikov spectral sequence.
• Cobar spectral sequence
• EHP spectral sequence converging to stable homotopy groups of spheres
• Federer spectral sequence converging to homotopy groups of a function space.
• Homotopy fixed point spectral sequence{{cite journal |arxiv=math/0406081|first1=Robert R. |last1=Bruner |first2=John |last2=Rognes |title=Differentials in the homological homotopy fixed point spectral sequence |journal=Algebr. Geom. Topol. |volume=5 |issue=2 |date=2005 |pages=653–690 |doi=10.2140/agt.2005.5.653 |doi-access=free}}
• Hurewicz spectral sequence for calculating the homology of a space from its homotopy.
• Miller spectral sequence converging to the mod p stable homology of a space.
• Milnor spectral sequence is another name for the bar spectral sequence.
• Moore spectral sequence is another name for the bar spectral sequence.
• Quillen spectral sequence for calculating the homotopy of a simplicial group.
• Rothenberg–Steenrod spectral sequence is another name for the bar spectral sequence.
• van Kampen spectral sequence for calculating the homotopy of a wedge of spaces.

• Čech-to-derived functor spectral sequence from Čech cohomology to 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.
• Gersten–Witt spectral sequence
• Green's spectral sequence for Koszul cohomology
• Grothendieck spectral sequence for composing derived functors
• Hyperhomology spectral sequence for calculating hyperhomology.
• Künneth spectral sequence for calculating the homology of a tensor product of differential algebras.
• Leray spectral sequence converging to the cohomology of a sheaf.
• Local-to-global Ext spectral sequence
• Lyndon–Hochschild–Serre spectral sequence in group (co)homology
• May spectral sequence for calculating the Tor or Ext groups of an algebra.
• Spectral sequence of a differential filtered group: described in this article.
• Spectral sequence of a double complex: described in this article.
• Spectral sequence of an exact couple: described in this article.
• Universal coefficient spectral sequence
• van Est spectral sequence converging to relative Lie algebra cohomology.

• Arnold's spectral sequence in singularity theory.
• Bloch–Lichtenbaum spectral sequence converging to the algebraic K-theory of a field.
• Frölicher spectral sequence starting from the Dolbeault cohomology and converging to the algebraic de Rham cohomology of a variety.
• Hodge–de Rham spectral sequence converging to the algebraic de Rham cohomology of a variety.
• Motivic-to-K-theory spectral sequence

{{reflist|2}}

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• {{cite journal | last = Chow | first = Timothy Y. | year =2006 | title = You Could Have Invented Spectral Sequences | url = http://www.ams.org/notices/200601/fea-chow.pdf | journal = Notices of the American Mathematical Society | volume = 53 | pages = 15–19 }}

• {{cite web|url=https://mathoverflow.net/q/17357|title= What is so "spectral" about spectral sequences?| publisher= MathOverflow}}
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