Spectral sequence

In homological algebra and algebraic topology, a spectral sequence is a means of computing homology groups by taking successive approximations. Spectral sequences are a generalization of exact sequences, and since their introduction by Jean Leray (1946), they have become important computational tools, particularly in algebraic topology, algebraic geometry and homological algebra.

Discovery and motivation

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.

Formal definition

Fix an abelian category, such as a category of modules over a ring. A spectral sequence is a choice of a nonnegative integer r₀ and a collection of three sequences:

1. For all integers r ≥ r₀, an object E_r , called a sheet (as in a sheet of paper), or sometimes a page or a term,
2. Endomorphisms d_r : E_r → E_r satisfying d_r o d_r = 0, called boundary maps or differentials,
3. Isomorphisms of E_r+1 with H(E_r), the homology of E_r with respect to d_r.

Usually the isomorphisms between E_r+1 and H(E_r) are suppressed, and we write equalities instead. Sometimes E_r+1 is called the derived object of E_r.

The most elementary example is a chain complex C_•. An object C_• in an abelian category of chain complexes 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 E₂ to equal E₁, 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.

In the ungraded situation described above, r₀ is irrelevant, but in practice most spectral sequences occur in the category of doubly graded modules over a ring R (or doubly graded sheaves of modules over a sheaf of rings). In this case, each sheet is a doubly graded module, so it decomposes as a direct sum of terms with one term for each possible bidegree. The boundary map is defined as the direct sum of boundary maps on each of the terms of the sheet. Their degree depends on r and is fixed by convention. For a homological spectral sequence, the terms are written $E^r_{p,q}$ and the differentials have bidegree (− r,r − 1). For a cohomological spectral sequence, the terms are written $E^{p,q}_r$ and the differentials have bidegree (r, 1 − r). (These choices of bidegree occur naturally in practice; see the example of a double complex below.) 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.

A morphism of spectral sequences E → E' is by definition a collection of maps f_r : E_r → E'_r which are compatible with the differentials and with the given isomorphisms between cohomology of the r-th step and the (r + 1)-st sheets of E and E' , respectively.

Let E_r be a spectral sequence, starting with say r = 0. Then there is a sequence of subobjects

$0 = B_0 \subset B_1 \subset B_{2} \subset \dots \subset B_r \subset \dots \subset Z_r \subset \dots \subset Z_2 \subset Z_1 \subset Z_0 = E_0$

such that $E_r \simeq Z_{r-1}/B_{r-1}$; indeed, recursively we let $Z_0 = E_0, B_0 = 0$ 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 $E_r \overset{d_r}\to E_r.$

We then let $Z_{\infty} = \cap_r Z_r, B_{\infty} = \cup_r B_r$ and

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

it is called the limiting term. (Of course, such $E_{\infty}$ 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 degenerates; there are only finitely many inclusions in the sequence above.)

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. For each r, imagine that we have a sheet of graph paper. On this sheet, we will take p to be the horizontal direction and q to be the vertical direction. At each lattice point we have the object $E_r^{p,q}$.

It is very common for n = p + q to be another natural index in the spectral sequence. n 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. When r is zero, the differential moves objects one space down or up. This is similar to the differential on a chain complex. When r is one, the differential moves objects one space to the left or right. When r is two, the differential moves objects just like a knight's move in chess. For higher r, the differential acts like a generalized knight's move.

Worked-out examples

When learning spectral sequences for the first time, it is often helpful to work with simple computational examples. For more formal and complete discussions, see the sections below. For the examples in this section, it suffices to use this definition: one says a spectral sequence converges to H with an increasing filtration F if $E^{\infty}_{p, q} = F_p H_{p+q}/F_{p-1} H_{p+q}$. The examples below illustrate how one relates such filtrations with the E²-term in the forms of exact sequences; many exact sequences in applications (e.g., Gysin sequence) arise in this fashion.

2 columns and 2 rows

Let $E^r_{p, q}$ be a spectral sequence such that $E^2_{p, q} = 0$ for all p other than 0, 1. The differentials on the second page have degree (-2, 1) and therefore they are all zero; i.e., the spectral sequence degenerates: $E^{\infty} = E^2$. Say, it converges to H with a filtration

$0 = F_{-1} H_n \subset F_0 H_n \subset \dots \subset F_n H_n = H_n$

such that $E^{\infty}_{p, q} = F_p H_{p+q}/F_{p-1} H_{p+q}$. 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:^[1]

$0 \to E^2_{0, n} \to H_n \to E^2_{1, n - 1} \to 0$.

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 $E^3_{p, 0} = \operatorname{ker} (d: E^2_{p, 0} \to E^2_{p - 2, 1})$, as the denominator is zero. Similarly, $E^3_{p, 1} = \operatorname{coker}(d: E^2_{p+2, 0} \to E^2_{p, 1})$. Thus,

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

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

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

Wang sequence

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

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

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

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

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

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 and moreover $E^2_{p, q} = H_q(F)$ for p = 0, n, by the universal coefficient theorem. Clearly, $E^{\infty} = E^{n+1}, E^{n} = E^2$ and by computing $E^{n+1}$ we see:

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

Now, writing $H = H(E)$, since $F_1 H_q/F_0 H_q = E^{\infty}_{1, q - 1} = 0$, etc., we have: $E^{\infty}_{n, q-n} = F_n H_q / F_0 H_q$ and thus, since $F_n H_q = H_q$,

$0 \to E^{\infty}_{0, q} \to H_q \to E^{\infty}_{n, q - n} \to 0.$

Putting all calculations together, one gets:^[3]

$\dots \to H_q(F) \overset{i_*}\to H_q(E) \to H_{q-n}(F) \overset{d}\to H_{q-1}(F) \overset{i_*}\to H_{q-1}(E) \to H_{q-n -1}(F) \to \dots$

(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 $E_r^{p, q}$ be a first-quadrant spectral sequence converging to H with the decreasing filtration

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

so that $E_{\infty}^{p,q} = F^p H^{p+q}/F^{p+1} H^{p+q}.$ Since $E^2_{p, q}$ is zero if p or q is negative, we have:

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

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

$0 \to E_2^{1, 0} \to H^1 \to E^{0, 1}_{\infty} \to 0$.

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

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

Edge maps and transgressions

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,

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

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

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

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):

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

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

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

given as a composition $E^2_{p, 0} \to E^p_{p, 0} \overset{d}\to E^p_{0, p-1} \to E^2_{0, p - 1}$, the first and last maps being the inverses of the edge maps.^[4]

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 are a sequence of epimorphisms

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

and if $E^r_{p, q} = 0$ for every p < 0, then there is a sequence of monomorphisms:

$E_{r}^{0, q} \to E_{r-1}^{0, q} \to \dots \to E_{3}^{0, q} \to E_2^{0, q}$.

The transgression is a not necessarily well-defined map:

$\tau: E_2^{0, q-1} \to E_2^{q, 0}$

induced by $d: E_q^{0, q-1} \to E_q^{q, 0}$.

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 $E^{p, q}_r$ 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 $F \to E \to B$, B simply connected, when the coefficient group is a "ring" R. It has the multiplicative structure such that the limiting term $E_{\infty}$ is isomorphic as graded algebra over R to the associated graded algebra of H(E; R), the latter having a ring structure induced by cup product.^[5] The multiplicative structure on a spectral sequence can be very useful in calculating the spectral sequence; for some concrete example, see Serre spectral sequence#The Cohomology Ring of Complex Projective Space.

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.

Exact couples

The most powerful technique for the construction of spectral sequences is William Massey's method of exact couples. Exact couples are particularly common in algebraic topology, where there are many spectral sequences for which no other construction is known. In fact, all known spectral sequences can be constructed using exact couples. 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 and 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⁾). We let E_n be C⁽ⁿ⁾ and d_n be g⁽ⁿ⁾ o h⁽ⁿ⁾. This gives a spectral sequence.

For a simple example, see the Bockstein spectral sequence.

The spectral sequence of a filtered complex

A very common type of spectral sequence comes from a filtered cochain complex. This is a cochain complex C^• together with a set of subcomplexes F^pC^•, where p ranges across all integers. (In practice, p is usually bounded on one side.) We require that the boundary map is compatible with the filtration; this means that d(F^pCⁿ) ⊆ F^pCⁿ⁺¹. We assume that the filtration is descending, i.e., F^pC^• ⊇ F^p+1C^•. We will number the terms of the cochain complex by n. Later, we will also assume that the filtration is Hausdorff or separated, that is, the intersection of the set of all F^pC^• is zero, and that the filtration is exhaustive, that is, the union of the set of all F^pC^• is the entire chain complex C^•.

The filtration is useful because it gives a measure of nearness to zero: As p increases, F^pC^• 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. (The complementary degree is often a more convenient index than the total degree n. For example, this is true of the spectral sequence of a double complex, explained below.)

We will construct this spectral sequence by hand. C^• has only a single grading and a filtration, so we first construct a doubly graded object from C^•. 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 E₁ step:

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

$B_0^{p,q} = 0$

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

$E_0 = \bigoplus_{p,q\in\bold{Z}} E_0^{p,q}$

Since we assumed that the boundary map was compatible with the filtration, E₀ is a doubly graded object and there is a natural doubly graded boundary map d₀ on E₀. To get E₁, we take the homology of E₀.

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

$\bar{B}_1^{p,q} = \mbox{im } d_0^{p,q-1} : E_0^{p,q-1} \rightarrow E_0^{p,q} = \mbox{im } d_0^{p,q-1} : F^p C^{p+q-1}/F^{p+1} C^{p+q-1} \rightarrow F^p C^{p+q}/F^{p+1} C^{p+q}$

$E_1^{p,q} = \frac{\bar{Z}_1^{p,q}}{\bar{B}_1^{p,q}} = \frac{\ker d_0^{p,q} : E_0^{p,q} \rightarrow E_0^{p,q+1}}{\mbox{im } d_0^{p,q-1} : E_0^{p,q-1} \rightarrow E_0^{p,q}}$

$E_1 = \bigoplus_{p,q\in\bold{Z}} E_1^{p,q} = \bigoplus_{p,q\in\bold{Z}} \frac{\bar{Z}_1^{p,q}}{\bar{B}_1^{p,q}}$

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

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

$B_1^{p,q} = (\mbox{im } d_0^{p,q-1} : F^p C^{p+q-1} \rightarrow C^{p+q}) \cap F^p C^{p+q}$

and that we then have

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

$Z_1^{p,q}$ is exactly the stuff which the differential pushes up one level in the filtration, and $B_1^{p,q}$ is exactly the image of the stuff which the differential pushes up zero levels in the filtration. This suggests that we should choose $Z_r^{p,q}$ to be the stuff which the differential pushes up r levels in the filtration and $B_r^{p,q}$ to be image of the stuff which the differential pushes up r-1 levels in the filtration. In other words, the spectral sequence should satisfy

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

$B_r^{p,q} = (\mbox{im } d_0^{p-r+1,q+r-2} : F^{p-r+1} C^{p+q-1} \rightarrow C^{p+q}) \cap F^p C^{p+q}$

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

and we should have the relationship

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

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

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

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.

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 $H^I_i(H^{II}_j(C_{\bullet,\bullet}))$ and $H^{II}_j(H^I_i(C_{\bullet,\bullet}))$. We will do this by filtering our double complex in two different ways. Here are our filtrations:

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

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

To get a spectral sequence, we will reduce to the previous example. We define the total complex T(C_•,•) to be the complex whose n'th term is $\bigoplus_{i+j=n} C_{i,j}$ 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:

$T_n(C_{\bullet,\bullet})^I_p = \bigoplus_{i+j=n \atop i > p-1} C_{i,j}$

$T_n(C_{\bullet,\bullet})^{II}_p = \bigoplus_{i+j=n \atop j > p-1} C_{i,j}$

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:

${}^IE^0_{p,q} = T_n(C_{\bullet,\bullet})^I_p / T_n(C_{\bullet,\bullet})^I_{p+1} = \bigoplus_{i+j=n \atop i > p-1} C_{i,j} \Big/ \bigoplus_{i+j=n \atop i > p} C_{i,j} = C_{p,q},$

where 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

$d^I_{p,q} + d^{II}_{p,q} : T_n(C_{\bullet,\bullet})^I_p / T_n(C_{\bullet,\bullet})^I_{p+1} = C_{p,q} \rightarrow T_{n-1}(C_{\bullet,\bullet})^I_p / T_{n-1}(C_{\bullet,\bullet})^I_{p+1} = C_{p,q-1}$

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

${}^IE^1_{p,q} = H^{II}_q(C_{p,\bullet}).$

To find E², we need to determine

$d^I_{p,q} + d^{II}_{p,q} : H^{II}_q(C_{p,\bullet}) \rightarrow H^{II}_q(C_{p+1,\bullet})$

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

${}^IE^2_{p,q} = H^I_p(H^{II}_q(C_{\bullet,\bullet})).$

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

${}^{II}E^2_{p,q} = H^{II}_q(H^{I}_p(C_{\bullet,\bullet})).$

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

In the elementary example that we began with, the sheets of the spectral sequence were constant once r was at least 1. In that setup it makes sense to take the limit of the sequence of sheets: Since nothing happens after the zeroth sheet, the limiting sheet E_∞ is the same as E₁.

In more general situations, limiting sheets often exist and are always interesting. They are one of the most powerful aspects of spectral sequences. We say that a spectral sequence $E_r^{p,q}$ converges to or abuts to $E_\infty^{p,q}$ if there is an r(p, q) such that for all r ≥ r(p, q), the differentials $d_r^{p-r,q+r-1}$ and $d_r^{p,q}$ are zero. This forces $E_r^{p,q}$ to be isomorphic to $E_\infty^{p,q}$ for large r. In symbols, we write:

$E_r^{p,q} \Rightarrow_p E_\infty^{p,q}$

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.

In most spectral sequences, the $E_\infty$ term is not naturally a doubly graded object. Instead, there are usually $E_\infty^n$ terms which come with a natural filtration $F^\bullet E_\infty^n$. In these cases, we set $E_\infty^{p,q} = \mbox{gr}_p E_\infty^{p+q} = F^pE_\infty^{p+q}/F^{p+1}E_\infty^{p+q}$. We define convergence in the same way as before, but we write

$E_r^{p,q} \Rightarrow_p E_\infty^n$

to mean that whenever p + q = n, $E_r^{p,q}$ converges to $E_\infty^{p,q}$.

The simplest situation in which we can determine convergence is when the spectral sequences degenerates. We say that the spectral sequences degenerates at sheet r if, for any s ≥ r, the differential d_s is zero. This implies that E_r ≅ E_r+1 ≅ E_r+2 ≅ ... In particular, it implies that E_r is isomorphic to E_∞. This is what happened in our first, trivial example of an unfiltered chain complex: The spectral sequence degenerated at the first sheet. In general, if a doubly graded spectral sequence is zero outside of a horizontal or vertical strip, the spectral sequence will degenerate, because later differentials will always go to or from an object not in the strip.

The spectral sequence also converges if $E_r^{p,q}$ vanishes for all p less than some p₀ and for all q less than some q₀. If p₀ and q₀ can be chosen to be zero, this is called a first-quadrant spectral sequence. This sequence converges because each object is a fixed distance away from the edge of the non-zero region. Consequently, for a fixed p and q, the differential on later sheets always maps $E_r^{p,q}$ from or to the zero object; more visually, the differential leaves the quadrant where the terms are nonzero. 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 p₀ and for all q greater than some q₀.

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

See also Boardman, Conditionally Convergent Spectral Sequences.

Examples of degeneration

The spectral sequence of a filtered complex, continued

Notice that we have a chain of inclusions:

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

We can ask what happens if we define

$Z_\infty^{p,q} = \bigcap_{r=0}^\infty Z_r^{p,q},$

$B_\infty^{p,q} = \bigcup_{r=0}^\infty B_r^{p,q},$

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

$E_\infty^{p,q}$ 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 r'th 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:

$Z_\infty^{p,q} = \bigcap_{r=0}^\infty Z_r^{p,q} = \bigcap_{r=0}^\infty \ker(F^p C^{p+q} \rightarrow C^{p+q+1}/F^{p+r} C^{p+q+1})$

$B_\infty^{p,q} = \bigcup_{r=0}^\infty B_r^{p,q} = \bigcup_{r=0}^\infty (\mbox{im } d^{p,q-r} : F^{p-r} C^{p+q-1} \rightarrow C^{p+q}) \cap F^p C^{p+q}$

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

$E_\infty^{p,q} = \mbox{gr}_p H^{p+q}(C^\bull)$,

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

$E_r^{p,q} \Rightarrow_p H^{p+q}(C^\bull)$

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^•:

$F^0 B^n = B^n$

$F^1 B^n = A^n$

$F^2 B^n = 0$

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(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:

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

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:

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

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

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:

$H^I_p(H^{II}_q(C_{\bull,\bull})) \Rightarrow_p H^{p+q}(T(C_{\bull,\bull}))$

$H^{II}_q(H^I_p(C_{\bull,\bull})) \Rightarrow_q H^{p+q}(T(C_{\bull,\bull}))$

In general, the two gradings on H^p+q(T(C_•,•)) 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 Tor_i(M, N) = Tor_i(N, M). While this can be verified without a spectral sequence, it is very easy with spectral sequences.

Choose projective resolutions P_• and Q_• 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 ⊗ Q_j and whose differentials are d ⊗ 1 and (−1)ⁱ(1 ⊗ 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:

$H^I_p(H^{II}_q(P_\bull \otimes Q_\bull)) = H^I_p(P_\bull \otimes H^{II}_q(Q_\bull))$

$H^{II}_q(H^I_p(P_\bull \otimes Q_\bull)) = H^{II}_q(H^I_p(P_\bull) \otimes Q_\bull)$

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

$H^I_p(P_\bull \otimes N) = \mbox{Tor}_p(M,N)$

$H^{II}_q(M \otimes Q_\bull) = \mbox{Tor}_q(N,M)$

In particular, the $E^2_{p,q}$ 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:

$\mbox{Tor}_p(M,N) \cong E^\infty_p = H_p(T(C_{\bull,\bull}))$

$\mbox{Tor}_q(N,M) \cong E^\infty_q = H_q(T(C_{\bull,\bull}))$

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

Further examples

Some notable spectral sequences are:

• Adams spectral sequence in stable homotopy theory
• Adams–Novikov spectral sequence, a generalization to extraordinary cohomology theories.
• Arnold's spectral sequence in singularity theory.
• Atiyah–Hirzebruch spectral sequence of an extraordinary cohomology theory
• 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 sheaf cohomology.
• Change of rings spectral sequences for calculating Tor and Ext groups of modules.
• Chromatic spectral sequence for calculating the initial terms of the Adams–Novikov spectral sequence.
• Cobar spectral sequence
• 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.
• Gersten–Witt spectral sequence
• 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.
• Homotopy fixed point spectral sequence^[6]
• Hurewicz spectral sequence for calculating the homology of a space from its homotopy.
• 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.
• 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.
• Motivic-to-K-theory 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.
• Serre spectral sequence of a fibration
• 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.
• van Kampen spectral sequence for calculating the homotopy of a wedge of spaces.

Notes

1. Weibel, Exercise 5.2.1.; there are typos in the exact sequence, at least in the 1994 edition.
2. Weibel, Exercise 5.2.2.
3. Weibel, Application 5.3.5.
4. May, § 1
5. Hatcher, Section 1.2.
6. http://arxiv.org/abs/math/0406081

References

• Hatcher, Allen, Spectral Sequences in Algebraic Topology (PDF)
• Leray, Jean (1946), "L'anneau d'homologie d'une représentation", Les Comptes rendus de l'Académie des sciences 222: 1366–1368
• Leray, Jean (1946), "Structure de l'anneau d'homologie d'une représentation", Les Comptes rendus de l'Académie des sciences 222: 1419–1422
• Koszul, Jean-Louis (1947), "Sur les opérateurs de dérivation dans un anneau", Les Comptes rendus de l'Académie des sciences 225: 217–219
• Massey, William S. (1952), "Exact couples in algebraic topology. I, II", Annals of Mathematics. Second Series (Annals of Mathematics) 56 (2): 363–396, doi:10.2307/1969805, JSTOR 1969805
• Massey, William S. (1953), "Exact couples in algebraic topology. III, IV, V", Annals of Mathematics. Second Series (Annals of Mathematics) 57 (2): 248–286, doi:10.2307/1969858, JSTOR 1969858
• J. P. May, A primer on spectral sequences
• McCleary, John (2001), A User's Guide to Spectral Sequences, Cambridge Studies in Advanced Mathematics 58 (2nd ed.), Cambridge University Press, doi:10.2277/0521567599, ISBN 978-0-521-56759-6, MR 1793722
• Mosher, Robert; Tangora, Martin (1968), Cohomology Operations and Applications in Homotopy Theory, Harper and Row, ISBN 978-0-06-044627-7
• Weibel, Charles A. (1994), An introduction to homological algebra, Cambridge Studies in Advanced Mathematics 38, Cambridge University Press, ISBN 978-0-521-55987-4, OCLC 36131259, MR 1269324

Further reading

• Chow, T. "You Could Have Invented Spectral Sequences." Notices of the American Mathematical Society 53: 1519.