Spectral sequence

Formal definition

Cohomological spectral sequence

Fix an abelian category, such as a category of modules over a ring, and a nonnegative integer ${\displaystyle r_0}$. A cohomological spectral sequence is a sequence ${\displaystyle \{E_r,d_r{\}}_{r\ge r_0}}$ of objects ${\displaystyle E_r}$ and endomorphisms ${\displaystyle d_r:E_r\to E_r}$, such that for every ${\displaystyle r\ge r_0}$

1. ${\displaystyle d_r\circ d_r=0}$,
2. ${\displaystyle E_{r+1}\cong H_{\ast }(E_r,d_r)}$, the homology of ${\displaystyle E_r}$ with respect to ${\displaystyle d_r}$.

Usually the isomorphisms are suppressed and we write ${\displaystyle E_{r+1}=H_{\ast }(E_r,d_r)}$ instead. An object ${\displaystyle E_r}$ is called sheet (as in a sheet of paper), or sometimes a page or a term; an endomorphism ${\displaystyle d_r}$ is called boundary map or differential. Sometimes ${\displaystyle E_{r+1}}$ is called the derived object of ${\displaystyle E_r}$.

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 $E_r=\underset{p,q\in {\mathbb{Z}}^2}{⨁}{E}_r^{p,q}.$ So in this case a cohomological spectral sequence is a sequence ${\displaystyle \{E_r,d_r{\}}_{r\ge r_0}}$ of bigraded R-modules ${\displaystyle \{E_r^{p,q}{\}}_{p,q}}$ and for every module the direct sum of endomorphisms ${\displaystyle d_r=(d_r^{p,q}:E_r^{p,q}\to E_r^{p+r,q-r+1}{)}_{p,q\in {\mathbb{Z}}^2}}$ of bidegree ${\displaystyle (r,1-r)}$, such that for every ${\displaystyle r\ge r_0}$ it holds that:

1. ${\displaystyle d_r^{p+r,q-r+1}\circ d_r^{p,q}=0}$,
2. ${\displaystyle E_{r+1}\cong H_{\ast }(E_r,d_r)}$.

The notation used here is called complementary degree. Some authors write ${\displaystyle E_r^{d,q}}$ instead, where ${\displaystyle d=p+q}$ 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.

Homological spectral sequence

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 ${\displaystyle E_r^{p,q}}$ with ${\displaystyle E_{p,q}^r}$ and ${\displaystyle d_r^{p,q}:E_r^{p,q}\to E_r^{p+r,q-r+1}}$ with ${\displaystyle d_{p,q}^r:E_{p,q}^r\to E_{p-r,q+r-1}^r}$ (bidegree ${\displaystyle (-r,r-1)}$), one receives the definition of a homological spectral sequence analogously to the cohomological case.

Spectral sequence from a chain complex

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 i′th location this is the 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 ${\displaystyle E_2}$ to equal ${\displaystyle E_1}$, 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 ${\displaystyle E_r}$.

Visualization

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 ${\displaystyle E_r}$ can be seen as the r^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 ${\displaystyle E_r^{p,q}}$. Now turning to the next page means taking homology, that is the ${\displaystyle (r+1)}$^th page is a subquotient of the r^th page. The total degree n = p + q runs diagonally, northwest to southeast, across each sheet. In the homological case, the differentials have bidegree (−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.

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.

Properties

Categorical properties

The set of cohomological spectral sequences form a category: a morphism of spectral sequences ${\displaystyle f:E\to E^{\prime }}$ is by definition a collection of maps ${\displaystyle f_r:E_r\to E_r^{\prime }}$ which are compatible with the differentials, i.e. ${\displaystyle f_r\circ d_r=d_r^{\prime }\circ f_r}$, and with the given isomorphisms between the cohomology of the r^th step and the ⁠${\displaystyle (r+1)}$⁠^th sheets of E and E′, respectively: ${\displaystyle f_{r+1}(E_{r+1})=f_{r+1}(H(E_r))=H(f_r(E_r))}$. In the bigraded case, they should also respect the graduation: ${\displaystyle f_r(E_r^{p,q})\subset {E_r^{\prime }}^{p,q}.}$

Multiplicative structure

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 ${\displaystyle E_r^{p,q}}$ be a spectral sequence of cohomological type. We say it has multiplicative structure if (i) ${\displaystyle E_r}$ are (doubly graded) differential graded algebras and (ii) the multiplication on ${\displaystyle E_{r+1}}$ is induced by that on ${\displaystyle E_r}$ via passage to cohomology.

A typical example is the cohomological Serre spectral sequence for a fibration ${\displaystyle F\to E\to B}$, when the coefficient group is a ring R. It has the multiplicative structure induced by the cup products of fibre and base on the ${\displaystyle E_2}$-page. However, in general the limiting term ${\displaystyle E_{\mathrm{\infty }}}$ is not isomorphic as a graded algebra to H(E; R). The multiplicative structure can be very useful for calculating differentials on the sequence.

Constructions of spectral sequences

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.

Spectral sequence of an exact couple

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⁾.

Spectral sequences constructed with this method

• Serre spectral sequence - 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

The spectral sequence of a filtered complex

A very common type of spectral sequence comes from a filtered cochain complex, as it naturally induces a bigraded object. Consider a cochain complex ${\displaystyle (C^{\bullet },d)}$ together with a descending filtration, $...\supset F^{-2}{C}^{\bullet }\supset F^{-1}{C}^{\bullet }\supset F^0{C}^{\bullet }\supset F^1{C}^{\bullet }\supset F^2{C}^{\bullet }\supset F^3{C}^{\bullet }\supset ...$ . We require that the boundary map is compatible with the filtration, i.e. $d(F^p{C}^n)\subset F^p{C}^{n+1}$, and that the filtration is exhaustive, that is, the union of the set of all $F^p{C}^{\bullet }$ is the entire chain complex $C^{\bullet }$. Then there exists a spectral sequence with $E_0^{p,q}=F^p{C}^{p+q}/F^{p+1}{C}^{p+q}$ and $E_1^{p,q}=H^{p+q}(F^p{C}^{\bullet }/F^{p+1}{C}^{\bullet })$. Later, we will also assume that the filtration is Hausdorff or separated, that is, the intersection of the set of all $F^p{C}^{\bullet }$ is zero.

The filtration is useful because it gives a measure of nearness to zero: As p increases, $F^p{C}^{\bullet }$ 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 q = n − p.

Construction

${\displaystyle C^{\bullet }}$ 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 ${\displaystyle E_1}$ step:

${\displaystyle Z_{-1}^{p,q}=Z_0^{p,q}=F^p{C}^{p+q}}$

${\displaystyle B_0^{p,q}=0}$

${\displaystyle E_0^{p,q}=\frac{Z_0^{p,q}}{B_0^{p,q}+Z_{-1}^{p+1,q-1}}=\frac{F^p{C}^{p+q}}{F^{p+1}{C}^{p+q}}}$

${\displaystyle E_0=\underset{p,q\in \mathbf{Z}}{⨁}{E}_0^{p,q}}$

Since we assumed that the boundary map was compatible with the filtration, ${\displaystyle E_0}$ is a doubly graded object and there is a natural doubly graded boundary map ${\displaystyle d_0}$ on ${\displaystyle E_0}$. To get ${\displaystyle E_1}$, we take the homology of ${\displaystyle E_0}$.

${\displaystyle {\overline{Z}}_1^{p,q}=\ker d_0^{p,q}:E_0^{p,q}\to E_0^{p,q+1}=\ker d_0^{p,q}:F^p{C}^{p+q}/F^{p+1}{C}^{p+q}\to F^p{C}^{p+q+1}/F^{p+1}{C}^{p+q+1}}$

${\displaystyle {\overline{B}}_1^{p,q}=\text{im }{d}_0^{p,q-1}:E_0^{p,q-1}\to E_0^{p,q}=\text{im }{d}_0^{p,q-1}:F^p{C}^{p+q-1}/F^{p+1}{C}^{p+q-1}\to F^p{C}^{p+q}/F^{p+1}{C}^{p+q}}$

${\displaystyle E_1^{p,q}=\frac{{\overline{Z}}_1^{p,q}}{{\overline{B}}_1^{p,q}}=\frac{\ker d_0^{p,q}:E_0^{p,q}\to E_0^{p,q+1}}{\text{im }{d}_0^{p,q-1}:E_0^{p,q-1}\to E_0^{p,q}}}$

${\displaystyle E_1=\underset{p,q\in \mathbf{Z}}{⨁}{E}_1^{p,q}=\underset{p,q\in \mathbf{Z}}{⨁}\frac{{\overline{Z}}_1^{p,q}}{{\overline{B}}_1^{p,q}}}$

Notice that ${\displaystyle {\overline{Z}}_1^{p,q}}$ and ${\displaystyle {\overline{B}}_1^{p,q}}$ can be written as the images in ${\displaystyle E_0^{p,q}}$ of

${\displaystyle Z_1^{p,q}=\ker d_0^{p,q}:F^p{C}^{p+q}\to C^{p+q+1}/F^{p+1}{C}^{p+q+1}}$

${\displaystyle B_1^{p,q}=(\text{im }{d}_0^{p,q-1}:F^p{C}^{p+q-1}\to C^{p+q})\cap F^p{C}^{p+q}}$

and that we then have

${\displaystyle E_1^{p,q}=\frac{Z_1^{p,q}}{B_1^{p,q}+Z_0^{p+1,q-1}}.}$

${\displaystyle Z_1^{p,q}}$ are exactly the elements which the differential pushes up one level in the filtration, and ${\displaystyle B_1^{p,q}}$ are exactly the image of the elements which the differential pushes up zero levels in the filtration. This suggests that we should choose ${\displaystyle Z_r^{p,q}}$ to be the elements which the differential pushes up r levels in the filtration and ${\displaystyle B_r^{p,q}}$ 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

${\displaystyle Z_r^{p,q}=\ker d_0^{p,q}:F^p{C}^{p+q}\to C^{p+q+1}/F^{p+r}{C}^{p+q+1}}$

${\displaystyle B_r^{p,q}=(\text{im }{d}_0^{p-r+1,q+r-2}:F^{p-r+1}{C}^{p+q-1}\to C^{p+q})\cap F^p{C}^{p+q}}$

${\displaystyle E_r^{p,q}=\frac{Z_r^{p,q}}{B_r^{p,q}+Z_{r-1}^{p+1,q-1}}}$

and we should have the relationship

${\displaystyle B_r^{p,q}=d_0^{p,q}(Z_{r-1}^{p-r+1,q+r-2}).}$

For this to make sense, we must find a differential ${\displaystyle d_r}$ on each ${\displaystyle E_r}$ and verify that it leads to homology isomorphic to ${\displaystyle E_{r+1}}$. The differential

${\displaystyle d_r^{p,q}:E_r^{p,q}\to E_r^{p+r,q-r+1}}$

is defined by restricting the original differential ${\displaystyle d}$ defined on ${\displaystyle C^{p+q}}$ to the subobject ${\displaystyle Z_r^{p,q}}$. It is straightforward to check that the homology of ${\displaystyle E_r}$ with respect to this differential is ${\displaystyle E_{r+1}}$, 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.

Spectral sequences constructed with this method

• Hodge–de Rham spectral sequence
• Spectral sequence of a double complex
• Can be used to construct Mixed Hodge structures

The spectral sequence of a double complex

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, d^I and d^II. d^I is assumed to decrease i, and d^II is assumed to decrease j. Furthermore, we assume that the differentials anticommute, so that d^I d^II + d^II d^I = 0. Our goal is to compare the iterated homologies ${\displaystyle H_i^{\text{I}}(H_j^{\text{II}}(C_{\bullet ,\bullet }))}$ and ${\displaystyle H_j^{\text{II}}(H_i^{\text{I}}(C_{\bullet ,\bullet }))}$. We will do this by filtering our double complex in two different ways. Here are our filtrations:

${\displaystyle (C_{i,j}^{\text{I}}{)}_p=\{\begin{array}{ll}0& \text{if }i<p\\ C_{i,j}& \text{if }i\ge p\end{array}}$

${\displaystyle (C_{i,j}^{\text{II}}{)}_p=\{\begin{array}{ll}0& \text{if }j<p\\ C_{i,j}& \text{if }j\ge p\end{array}}$

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

${\displaystyle T_n(C_{\bullet ,\bullet }{)}_p^{\text{I}}=\underset{\genfrac{}{}{0ex}{}{i+j=n}{i>p-1}}{⨁}{C}_{i,j}}$

${\displaystyle T_n(C_{\bullet ,\bullet }{)}_p^{\text{II}}=\underset{\genfrac{}{}{0ex}{}{i+j=n}{j>p-1}}{⨁}{C}_{i,j}}$

To show that these spectral sequences give information about the iterated homologies, we will work out the E⁰, E¹, and E² terms of the I filtration on T(C_•,•). The E⁰ term is clear:

${\displaystyle {}^{\text{I}}{E}_{p,q}^0=T_n(C_{\bullet ,\bullet }{)}_p^{\text{I}}/T_n(C_{\bullet ,\bullet }{)}_{p+1}^{\text{I}}=\underset{\genfrac{}{}{0ex}{}{i+j=n}{i>p-1}}{⨁}{C}_{i,j}/\underset{\genfrac{}{}{0ex}{}{i+j=n}{i>p}}{⨁}{C}_{i,j}=C_{p,q},}$

where n = p + q.

To find the E¹ term, we need to determine d^I + d^II on E⁰. Notice that the differential must have degree −1 with respect to n, so we get a map

${\displaystyle d_{p,q}^{\text{I}}+d_{p,q}^{\text{II}}:T_n(C_{\bullet ,\bullet }{)}_p^{\text{I}}/T_n(C_{\bullet ,\bullet }{)}_{p+1}^{\text{I}}=C_{p,q}\to T_{n-1}(C_{\bullet ,\bullet }{)}_p^{\text{I}}/T_{n-1}(C_{\bullet ,\bullet }{)}_{p+1}^{\text{I}}=C_{p,q-1}}$

Consequently, the differential on E⁰ is the map C_p,q → C_p,q−1 induced by d^I + d^II. But d^I has the wrong degree to induce such a map, so d^I must be zero on E⁰. That means the differential is exactly d^II, so we get

${\displaystyle {}^{\text{I}}{E}_{p,q}^1=H_q^{\text{II}}(C_{p,\bullet }).}$

To find E², we need to determine

${\displaystyle d_{p,q}^{\text{I}}+d_{p,q}^{\text{II}}:H_q^{\text{II}}(C_{p,\bullet })\to H_q^{\text{II}}(C_{p+1,\bullet })}$

Because E¹ was exactly the homology with respect to d^II, d^II is zero on E¹. Consequently, we get

${\displaystyle {}^{\text{I}}{E}_{p,q}^2=H_p^{\text{I}}(H_q^{\text{II}}(C_{\bullet ,\bullet })).}$

Using the other filtration gives us a different spectral sequence with a similar E² term:

${\displaystyle {}^{\text{II}}{E}_{p,q}^2=H_q^{\text{II}}(H_p^I(C_{\bullet ,\bullet })).}$

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.

Convergence, degeneration, and abutment

Interpretation as a filtration of cycles and boundaries

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

${\displaystyle 0=B_0\subset B_1\subset B_2\subset \cdots \subset B_r\subset \cdots \subset Z_r\subset \cdots \subset Z_2\subset Z_1\subset Z_0=E_1}$

such that ${\displaystyle E_r\simeq Z_{r-1}/B_{r-1}}$; indeed, recursively we let ${\displaystyle Z_0=E_1,B_0=0}$ and let ${\displaystyle Z_r,B_r}$ be so that ${\displaystyle Z_r/B_{r-1},B_r/B_{r-1}}$ are the kernel and the image of ${\displaystyle E_r\stackrel{d_r}{\to }{E}_r.}$

We then let ${\displaystyle Z_{\mathrm{\infty }}=\bigcap _r{Z}_r,B_{\mathrm{\infty }}=\bigcup _r{B}_r}$ and

${\displaystyle E_{\mathrm{\infty }}=Z_{\mathrm{\infty }}/B_{\mathrm{\infty }}}$;

it is called the limiting term. (Of course, such ${\displaystyle E_{\mathrm{\infty }}}$ 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.)

Terms of convergence

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

${\displaystyle E_r^{p,q}{\Rightarrow }_p{E}_{\mathrm{\infty }}^n}$

to mean that whenever p + q = n, ${\displaystyle E_r^{p,q}}$ converges to ${\displaystyle E_{\mathrm{\infty }}^{p,q}}$. We say that a spectral sequence ${\displaystyle E_r^{p,q}}$ abuts to ${\displaystyle E_{\mathrm{\infty }}^{p,q}}$ (the spectral sequence abutment) if for every ${\displaystyle p,q}$ there is ${\displaystyle r(p,q)}$ such that for all ${\displaystyle r\ge r(p,q)}$, ${\displaystyle E_r^{p,q}=E_{r(p,q)}^{p,q}}$. Then ${\displaystyle E_{r(p,q)}^{p,q}=E_{\mathrm{\infty }}^{p,q}}$ is the limiting term. The spectral sequence is regular or degenerates at ${\displaystyle r_0}$ if the differentials ${\displaystyle d_r^{p,q}}$ are zero for all ${\displaystyle r\ge r_0}$. If in particular there is ${\displaystyle r_0\ge 2}$, such that the ${\displaystyle r_0^{th}}$ sheet is concentrated on a single row or a single column, then we say it collapses. In symbols, we write:

${\displaystyle E_r^{p,q}{\Rightarrow }_p{E}_{\mathrm{\infty }}^{p,q}}$

The p indicates the filtration index. It is very common to write the ${\displaystyle E_2^{p,q}}$ 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 ${\displaystyle E_{\mathrm{\infty }}}$ is the same as ${\displaystyle E_1}$.

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

Examples of degeneration

The spectral sequence of a filtered complex, continued

Notice that we have a chain of inclusions:

${\displaystyle Z_0^{p,q}\supseteq Z_1^{p,q}\supseteq Z_2^{p,q}\supseteq \cdots \supseteq B_2^{p,q}\supseteq B_1^{p,q}\supseteq B_0^{p,q}}$

We can ask what happens if we define

${\displaystyle Z_{\mathrm{\infty }}^{p,q}=\bigcap _{r=0}^{\mathrm{\infty }}{Z}_r^{p,q},}$

${\displaystyle B_{\mathrm{\infty }}^{p,q}=\bigcup _{r=0}^{\mathrm{\infty }}{B}_r^{p,q},}$

${\displaystyle E_{\mathrm{\infty }}^{p,q}=\frac{Z_{\mathrm{\infty }}^{p,q}}{B_{\mathrm{\infty }}^{p,q}+Z_{\mathrm{\infty }}^{p+1,q-1}}.}$

${\displaystyle E_{\mathrm{\infty }}^{p,q}}$ 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:

${\displaystyle Z_{\mathrm{\infty }}^{p,q}=\bigcap _{r=0}^{\mathrm{\infty }}{Z}_r^{p,q}=\bigcap _{r=0}^{\mathrm{\infty }}\ker (F^p{C}^{p+q}\to C^{p+q+1}/F^{p+r}{C}^{p+q+1})}$

${\displaystyle B_{\mathrm{\infty }}^{p,q}=\bigcup _{r=0}^{\mathrm{\infty }}{B}_r^{p,q}=\bigcup _{r=0}^{\mathrm{\infty }}(\text{im }{d}^{p,q-r}:F^{p-r}{C}^{p+q-1}\to C^{p+q})\cap F^p{C}^{p+q}}$

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

${\displaystyle E_{\mathrm{\infty }}^{p,q}={\text{gr}}_p{H}^{p+q}(C^{\bullet })}$,

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:

${\displaystyle E_r^{p,q}{\Rightarrow }_p{H}^{p+q}(C^{\bullet })}$

Long exact sequences

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 → A^• → B^• → C^• → 0, and call the first map f^• : A^• → B^•. We get natural maps of homology objects Hⁿ(A^•) → Hⁿ(B^•) → Hⁿ(C^•), 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 B^•:

${\displaystyle F^0{B}^n=B^n}$

${\displaystyle F^1{B}^n=A^n}$

${\displaystyle F^2{B}^n=0}$

This gives:

${\displaystyle E_0^{p,q}=\frac{F^p{B}^{p+q}}{F^{p+1}{B}^{p+q}}=\{\begin{array}{ll}0& \text{if }p<0\text{ or }p>1\\ C^q& \text{if }p=0\\ A^{q+1}& \text{if }p=1\end{array}}$

${\displaystyle E_1^{p,q}=\{\begin{array}{ll}0& \text{if }p<0\text{ or }p>1\\ H^q(C^{\bullet })& \text{if }p=0\\ H^{q+1}(A^{\bullet })& \text{if }p=1\end{array}}$

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

${\displaystyle \cdots \to H^q(B^{\bullet })\to H^q(C^{\bullet })\to H^{q+1}(A^{\bullet })\to H^{q+1}(B^{\bullet })\to \cdots }$

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:

${\displaystyle E_2^{p,q}\cong {\text{gr}}_p{H}^{p+q}(B^{\bullet })=\{\begin{array}{ll}0& \text{if }p<0\text{ or }p>1\\ H^q(B^{\bullet })/H^q(A^{\bullet })& \text{if }p=0\\ \text{im }{H}^{q+1}{f}^{\bullet }:H^{q+1}(A^{\bullet })\to H^{q+1}(B^{\bullet })& \text{if }p=1\end{array}}$

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

${\displaystyle H^q(B^{\bullet })/H^q(A^{\bullet })\cong \ker d_{0,q}^1:H^q(C^{\bullet })\to H^{q+1}(A^{\bullet })}$

${\displaystyle \text{im }{H}^{q+1}{f}^{\bullet }:H^{q+1}(A^{\bullet })\to H^{q+1}(B^{\bullet })\cong H^{q+1}(A^{\bullet })/(\text{im }{d}_{0,q}^1:H^q(C^{\bullet })\to H^{q+1}(A^{\bullet }))}$

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

The spectral sequence of a double complex, continued

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

${\displaystyle H_p^{\text{I}}(H_q^{\text{II}}(C_{\bullet ,\bullet })){\Rightarrow }_p{H}^{p+q}(T(C_{\bullet ,\bullet }))}$

${\displaystyle H_q^{\text{II}}(H_p^{\text{I}}(C_{\bullet ,\bullet })){\Rightarrow }_q{H}^{p+q}(T(C_{\bullet ,\bullet }))}$

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

Commutativity of Tor

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 ${\displaystyle {\mathrm{Tor}}_i(M,N)={\mathrm{Tor}}_i(N,M)}$. While this can be verified without a spectral sequence, it is very easy with spectral sequences.

Choose projective resolutions ${\displaystyle P_{\bullet }}$ and ${\displaystyle Q_{\bullet }}$ 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 ${\displaystyle C_{i,j}=P_i\otimes Q_j}$ and whose differentials are ${\displaystyle d\otimes 1}$ and ${\displaystyle (-1{)}^{\text{I}}(1\otimes e)}$. (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:

${\displaystyle H_p^{\text{I}}(H_q^{\text{II}}(P_{\bullet }\otimes Q_{\bullet }))=H_p^{\text{I}}(P_{\bullet }\otimes H_q^{\text{II}}(Q_{\bullet }))}$

${\displaystyle H_q^{\text{II}}(H_p^{\text{I}}(P_{\bullet }\otimes Q_{\bullet }))=H_q^{\text{II}}(H_p^{\text{I}}(P_{\bullet })\otimes Q_{\bullet })}$

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

${\displaystyle H_p^{\text{I}}(P_{\bullet }\otimes N)={\mathrm{Tor}}_p(M,N)}$

${\displaystyle H_q^{\text{II}}(M\otimes Q_{\bullet })={\mathrm{Tor}}_q(N,M)}$

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

${\displaystyle {\mathrm{Tor}}_p(M,N)\cong E_p^{\mathrm{\infty }}=H_p(T(C_{\bullet ,\bullet }))}$

${\displaystyle {\mathrm{Tor}}_q(N,M)\cong E_q^{\mathrm{\infty }}=H_q(T(C_{\bullet ,\bullet }))}$

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

Worked-out examples

First-quadrant sheet

Consider a spectral sequence where ${\displaystyle E_r^{p,q}}$ vanishes for all ${\displaystyle p}$ less than some ${\displaystyle p_0}$ and for all ${\displaystyle q}$ less than some ${\displaystyle q_0}$. If ${\displaystyle p_0}$ and ${\displaystyle q_0}$ can be chosen to be zero, this is called a first-quadrant spectral sequence. The sequence abuts because ${\displaystyle E_{r+i}^{p,q}=E_r^{p,q}}$ holds for all ${\displaystyle i\ge 0}$ if ${\displaystyle r>p}$ and ${\displaystyle r>q+1}$. 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 ${\displaystyle E_r^{p,q}}$ vanishes for all ${\displaystyle p}$ greater than some ${\displaystyle p_0}$ and for all ${\displaystyle q}$ greater than some ${\displaystyle q_0}$.

2 non-zero adjacent columns

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

${\displaystyle \begin{array}{cccccc}& ⋮& ⋮& ⋮& ⋮& \\ \cdots & 0& E_{0,2}^2& E_{1,2}^2& 0& \cdots \\ \cdots & 0& E_{0,1}^2& E_{1,1}^2& 0& \cdots \\ \cdots & 0& E_{0,0}^2& E_{1,0}^2& 0& \cdots \\ \cdots & 0& E_{0,-1}^2& E_{1,-1}^2& 0& \cdots \\ & ⋮& ⋮& ⋮& ⋮& \end{array}}$

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

${\displaystyle d_{p,q}^2:E_{p,q}^2\to E_{p-2,q+1}^2}$

These maps are all zero since they are

${\displaystyle d_{0,q}^2:E_{0,q}^2\to 0}$, ${\displaystyle d_{1,q}^2:E_{1,q}^2\to 0}$

hence the spectral sequence degenerates: ${\displaystyle E^{\mathrm{\infty }}=E^2}$. Say, it converges to ${\displaystyle H_{\ast }}$ with a filtration

${\displaystyle 0=F_{-1}{H}_n\subset F_0{H}_n\subset \cdots \subset F_n{H}_n=H_n}$

such that ${\displaystyle E_{p,q}^{\mathrm{\infty }}=F_p{H}_{p+q}/F_{p-1}{H}_{p+q}}$. Then ${\displaystyle F_0{H}_n=E_{0,n}^2}$, ${\displaystyle F_1{H}_n/F_0{H}_n=E_{1,n-1}^2}$, ${\displaystyle F_2{H}_n/F_1{H}_n=0}$, ${\displaystyle F_3{H}_n/F_2{H}_n=0}$, etc. Thus, there is the exact sequence:

${\displaystyle 0\to E_{0,n}^2\to H_n\to E_{1,n-1}^2\to 0}$.

Next, let ${\displaystyle E_{p,q}^r}$ 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 ${\displaystyle E_{p,0}^3=\ker (d:E_{p,0}^2\to E_{p-2,1}^2)}$, as the denominator is zero. Similarly, ${\displaystyle E_{p,1}^3=\mathrm{coker}(d:E_{p+2,0}^2\to E_{p,1}^2)}$. Thus,

${\displaystyle 0\to E_{p,0}^{\mathrm{\infty }}\to E_{p,0}^2\stackrel{d}{\to }{E}_{p-2,1}^2\to E_{p-2,1}^{\mathrm{\infty }}\to 0}$.

Now, say, the spectral sequence converges to H with a filtration F as in the previous example. Since ${\displaystyle F_{p-2}{H}_p/F_{p-3}{H}_p=E_{p-2,2}^{\mathrm{\infty }}=0}$, ${\displaystyle F_{p-3}{H}_p/F_{p-4}{H}_p=0}$, etc., we have: ${\displaystyle 0\to E_{p-1,1}^{\mathrm{\infty }}\to H_p\to E_{p,0}^{\mathrm{\infty }}\to 0}$. Putting everything together, one gets:

${\displaystyle \cdots \to H_{p+1}\to E_{p+1,0}^2\stackrel{d}{\to }{E}_{p-1,1}^2\to H_p\to E_{p,0}^2\stackrel{d}{\to }{E}_{p-2,1}^2\to H_{p-1}\to \dots .}$

Wang sequence

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

${\displaystyle F\stackrel{i}{\to }E\stackrel{p}{\to }{S}^n}$

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

${\displaystyle E_{p,q}^2=H_p(S^n;H_q(F))\Rightarrow H_{p+q}(E)}$;

that is to say, ${\displaystyle E_{p,q}^{\mathrm{\infty }}=F_p{H}_{p+q}(E)/F_{p-1}{H}_{p+q}(E)}$ with some filtration ${\displaystyle F_{\bullet }}$.

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

${\displaystyle \begin{array}{ccccccccc}& ⋮& ⋮& ⋮& & ⋮& ⋮& ⋮& \\ \cdots & 0& E_{0,2}^2& 0& \cdots & 0& E_{n,2}^2& 0& \cdots \\ \cdots & 0& E_{0,1}^2& 0& \cdots & 0& E_{n,1}^2& 0& \cdots \\ \cdots & 0& E_{0,0}^2& 0& \cdots & 0& E_{n,0}^2& 0& \cdots \end{array}}$

Moreover, since

${\displaystyle E_{p,q}^2=H_p(S^n;H_q(F))=H_q(F)}$

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

${\displaystyle \begin{array}{ccccccccc}& ⋮& ⋮& ⋮& & ⋮& ⋮& ⋮& \\ \cdots & 0& H_2(F)& 0& \cdots & 0& H_2(F)& 0& \cdots \\ \cdots & 0& H_1(F)& 0& \cdots & 0& H_1(F)& 0& \cdots \\ \cdots & 0& H_0(F)& 0& \cdots & 0& H_0(F)& 0& \cdots \end{array}}$

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

${\displaystyle d_{n,q}^n:E_{n,q}^n\to E_{0,q+n-1}^n}$

which is

${\displaystyle d_{n,q}^n:H_q(F)\to H_{q+n-1}(F)}$

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

${\displaystyle 0\to E_{n,q-n}^{\mathrm{\infty }}\to E_{n,q-n}^n\stackrel{d}{\to }{E}_{0,q-1}^n\to E_{0,q-1}^{\mathrm{\infty }}\to 0.}$

and written out using the homology groups, this is

${\displaystyle 0\to E_{n,q-n}^{\mathrm{\infty }}\to H_{q-n}(F)\stackrel{d}{\to }{H}_{q-1}(F)\to E_{0,q-1}^{\mathrm{\infty }}\to 0.}$

To establish what the two ${\displaystyle E^{\mathrm{\infty }}}$-terms are, write ${\displaystyle H=H(E)}$, and since ${\displaystyle F_1{H}_q/F_0{H}_q=E_{1,q-1}^{\mathrm{\infty }}=0}$, etc., we have: ${\displaystyle E_{n,q-n}^{\mathrm{\infty }}=F_n{H}_q/F_0{H}_q}$ and thus, since ${\displaystyle F_0{H}_q=E_{0,q}^{\mathrm{\infty }}}$ and ${\displaystyle F_n{H}_q=H_q}$ as the first and the last term of the filtration,

${\displaystyle 0\to E_{0,q}^{\mathrm{\infty }}\to H_q\to E_{n,q-n}^{\mathrm{\infty }}\to 0.}$

This is the exact sequence

${\displaystyle 0\to H_q(F)\to H_q(E)\to H_{q-n}(F)\to 0.}$

Putting all calculations together, one gets:

${\displaystyle \cdots \to H_q(F)\stackrel{i_{\ast }}{\to }{H}_q(E)\to H_{q-n}(F)\stackrel{d}{\to }{H}_{q-1}(F)\stackrel{i_{\ast }}{\to }{H}_{q-1}(E)\to H_{q-n-1}(F)\to \dots }$

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

Low-degree terms

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

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

so that ${\displaystyle E_{\mathrm{\infty }}^{p,q}=F^p{H}^{p+q}/F^{p+1}{H}^{p+q}.}$ Since ${\displaystyle E_2^{p,q}}$ is zero if p or q is negative, we have:

${\displaystyle 0\to E_{\mathrm{\infty }}^{0,1}\to E_2^{0,1}\stackrel{d}{\to }{E}_2^{2,0}\to E_{\mathrm{\infty }}^{2,0}\to 0.}$

Since ${\displaystyle E_{\mathrm{\infty }}^{1,0}=E_2^{1,0}}$ for the same reason and since ${\displaystyle F^2{H}^1=0,}$

${\displaystyle 0\to E_2^{1,0}\to H^1\to E_{\mathrm{\infty }}^{0,1}\to 0}$.

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

${\displaystyle 0\to E_2^{1,0}\to H^1\to E_2^{0,1}\stackrel{d}{\to }{E}_2^{2,0}\to H^2.}$

Edge maps and transgressions

Homological spectral sequences

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

${\displaystyle E_{p,0}^{r+1}=\ker (d:E_{p,0}^r\to E_{p-r,r-1}^r)}$

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

${\displaystyle E_{p,0}^r\to E_{p,0}^{r-1}\to \cdots \to E_{p,0}^3\to E_{p,0}^2}$.

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

${\displaystyle E_{0,q}^2\to E_{0,q}^3\to \cdots \to E_{0,q}^{r-1}\to E_{0,q}^r}$.