In homological algebra, especially in algebraic topology or group cohomology, 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 in 1946, they have become an important research tool, particularly in homotopy theory. However, they have a reputation for being abstruse and difficult to comprehend. Homological algebra is the branch of mathematics which studies the methods of homology and cohomology in a general setting. ... Algebraic topology is a branch of mathematics in which tools from abstract algebra are used to study topological spaces. ... In abstract algebra, homological algebra, algebraic topology and algebraic number theory, as well as in applications to group theory proper, group cohomology is a way to study groups using a sequence of functors H n. ... Jean Leray (7 November 1906-10 November 1998) was a French mathematician, who worked on both partial differential equations and algebraic topology. ... 1946 (MCMXLVI) was a common year starting on Tuesday. ... An illustration of a homotopy between the two bold paths In topology, two continuous functions from one topological space to another are called homotopic (Greek homeos = identical and topos = place) if one can be continuously deformed into the other, such a deformation being called a homotopy between the two functions. ...

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 relationship between cohomology groups of a sheaf and cohomology groups of the pushforward of the sheaf. It was not a direct relation. Instead, Leray found that the cohomology groups of the pushforward formed a natural chain complex, so that he could take the cohomology of the cohomology. The result was not the cohomology of the original sheaf, but it was closer, and it too formed a natural chain complex. Leray found that he could repeat this process, and that each step got him closer to the cohomology groups of the original sheaf. Taking the limit of the iterated cohomologies gave him exactly the cohomology groups of the original sheaf. This allowed Leray to compute sheaf cohomology. Algebraic topology is a branch of mathematics in which tools from abstract algebra are used to study topological spaces. ... In mathematics, a sheaf F on a topological space X is something that assigns a structure F(U) (such as a set, group, or ring) to each open set U of X. The structures F(U) are compatible with the operations of restricting the open set to smaller subsets and... In mathematics, sheaf cohomology is the aspect of sheaf theory, concerned with sheaves of abelian groups, that applies homological algebra to make possible effective calculation of the global sections of a sheaf F. This is the main step, in numerous areas, from sheaf theory as a description of a geometric... In mathematics, the Leray spectral sequence was a pioneering example in homological algebra, introduced in 1946 by Jean Leray. ... In mathematics, in the field of sheaf theory and especially in algebraic geometry, the direct image functor generalizes the notion of a section of a sheaf to the relative case. ... In mathematics, a chain complex is a construct originally used in the field of algebraic topology. ...

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. This article may be too technical for most readers to understand. ... In mathematics, certain functors may be derived to obtain other functors closely related to the original ones. ... In mathematics, the derived category D(C) of a category C is a construction of homological algebra introduced to refine and in a certain sense to simplify the theory of derived functors defined on C (which therefore should already be an abelian category). ...

Unfortunately, because of the large amount of information carried in spectral sequences, they are difficult to grasp. This information is usually contained in a triply-indexed maze of abelian groups or modules. In most cases that can be computed, the spectral sequence eventually collapses, meaning that going out further in the sequence produces no new information. This does not always happen, however, and then it becomes necessary to use tricks to extract useful information. Even in these cases, however, it is still possible to get useful information from a spectral sequence. In mathematics, an abelian group, also called a commutative group, is a group such that for all a and b in G. In other words, the order of elements in a product doesnt matter. ... In abstract algebra, the concept of a module over a ring is a generalization of the notion of vector space, where instead of requiring the scalars to lie in a field, the scalars may lie in an arbitrary ring. ...

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: In mathematics, an abelian category is a category in which morphisms and objects can be added and in which kernels and cokernels exist and have nice properties. ...

1. For all integers r ≥ r₀, an object E_r, called a sheet (as in a sheet of paper),
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. E_r+1 is sometimes called the derived object of E_r. A blank sheet of paper Paper is a commodity of thin material produced by the amalgamation of fibers, typically vegetable fibers composed of cellulose, which are subsequently held together by hydrogen bonding. ...

The most elementary example is a chain complex C_•. C_• is an object in an abelian category of chain complexes, and it 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: In mathematics, a chain complex is a construct originally used in the field of algebraic topology. ...

• 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). The degree of the boundary maps depends on r and is fixed by convention. For a homological spectral sequence, the terms are written and the differentials have bidegree (-r,r-1). For a cohomological spectral sequence, the terms are written 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. In abstract algebra, the concept of a module over a ring is a generalization of the notion of vector space, where instead of requiring the scalars to lie in a field, the scalars may lie in an arbitrary ring. ... In mathematics, a ring is an algebraic structure in which addition and multiplication are defined and have properties listed below. ... In mathematics, a sheaf F on a topological space X is something that assigns a structure F(U) (such as a set, group, or ring) to each open set U of X. The structures F(U) are compatible with the operations of restricting the open set to smaller subsets and... In mathematics, in the field of homological algebra, the Grothendieck spectral sequence is a technique that allows one to compute the derived functors of the composition of two functors , from knowledge of the derived functors of F and G. If and are two additive (covariant) functors between abelian categories such...

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

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: William Schumacher Massey (born 1920) is an American mathematician, known for his work in algebraic topology. ...

• A pair of objects A and C.
• Three homomorphisms between these objects:
  • f : A → A
  • g : A → C
  • h : C → A

subject to certain exactness conditions: Image File history File links No higher resolution available. ...

• 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.

Visualization

The E₂ sheet of a spectral sequence

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 . Image File history File links No higher resolution available. ...

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 to the right or left. This is similar to the differential on a chain complex. When r is one, the differential moves objects one space down or up. 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. The knight moves in an L shape. ... Chess is a recreational and competitive game for two players. ...

Examples of spectral sequences

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^•. In mathematics, a filtration is an indexed set Si of subobjects of a given algebraic structure S, with an index set I that is a totally ordered set, subject only to the condition that if i ≤ j in I then Si is contained in Sj. ...

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 boundaries and cycles in later sheets get closer and closer to boundaries and cycles in the original complex. This spectral sequence will be doubly graded. One of the grades will be the filtration degree p. The other is called the complementary degree and is denoted q. The complementary degree satisfies the relation p + q = n. (We use the complementary degree instead of the location in the chain complex because this is more natural in the common case 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:

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.

Notice that and can be written as the images in of

and that we then have

.

is exactly the stuff which the differential pushes up one level in the filtration, and is exactly the image of the stuff which the differential pushes up one level in the filtration. This suggests that we should choose to be the stuff which the differential pushes up r levels in the filtration and to be image of the stuff which the differential pushes up r levels in the filtration. In other words, the spectral sequence should satisfy

and we should have the relationship

.

For this to make sense, we must find a differential on each E_r which gives the E_r+1 we stated above. Since we have written as a subobject of C^{p + q}, we get a differential on by restricting the differential on C^{p + q}:

Now 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 and . We will do this by filtering our double complex in two different ways. Here are our filtrations:

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 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:

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:

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

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

.

To find E², we need to determine

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

.

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

.

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 converges to or abuts to if there is an r(p, q) such that for all r ≥ r(p, q), the differentials and are zero. This forces to be isomorphic to for large r. In symbols, we write:

The p indicates the filtration index. It is very common to write the 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 term is not naturally a doubly graded object. Instead, there are usually terms which come with a natural filtration . In these cases, we set . We define convergence in the same way as before, but we write

to mean that whenever p + q = n, converges to .

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 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 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 vanishes for all p greater than some p₀ and for all q greater than some q₀.

Examples of degeneration

The spectral sequence of a filtered complex, continued

Notice that we have a chain of inclusions:

We can ask what happens if we define

,

,

.

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:

)

To see what this implies for recall that we assumed that the filtration was separated. This implies that as r increases, the kernels shrink, until we are left with . For , recall that we assumed that the filtration was exhaustive. This implies that as r increases, the images grow until we reach . We conclude

,

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:

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^•: In mathematics, especially in homological algebra and other applications of Abelian category theory, as well as in group theory, an exact sequence is a (finite or infinite) sequence of objects and morphisms between them such that the image of one morphism equals the kernel of the next. ...

F⁰Bⁿ = Bⁿ

F¹Bⁿ = Aⁿ

F²Bⁿ = 0

This gives:

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: In mathematics, particularly homological algebra, the snake lemma, a statement valid in every Abelian category, is the crucial tool used to construct the long exact sequences that are ubiquitous in homological algebra and its applications, for instance in algebraic topology. ...

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:

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

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:

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 M and N be R-modules. 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. In mathematics, the Tor functors of homological algebra are the derived functors of the tensor product functor. ...

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)^j(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:

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

In particular, the terms vanish except along the lines p = 0 (for one spectral sequence) and q = 0 (for the other). This implies that the spectral sequence degenerates at the second sheet. We get isomorphisms:

Finally, when p and q are equal, we get isomorphisms of the two right-hand sides, even after accounting for their different gradings, and the commutativity of Tor follows.

Further examples

Some notable spectral sequences are:

• Leray-Serre spectral sequence of a fibration
• Lyndon/Hochschild-Serre spectral sequence in group cohomology
• Adams spectral sequence in stable homotopy theory
• Atiyah-Hirzebruch spectral sequence of an extraordinary cohomology theory
• Adams-Novikov spectral sequence for an extraordinary cohomology theory
• Grothendieck spectral sequence for composing derived functors
• Chromatic spectral sequence for the stable homotopy groups of spheres
• Eilenberg-Moore spectral sequence
• Bockstein spectral sequence
• Hyperhomology spectral sequence
• Green's spectral sequence for Koszul cohomology

In mathematics, the Serre spectral sequence (sometimes Leray-Serre spectral sequence to acknowledge earlier work of Jean Leray) is a basic tool of algebraic topology. ... This article may be too technical for most readers to understand. ... In abstract algebra, homological algebra, algebraic topology and algebraic number theory, as well as in applications to group theory proper, group cohomology is a way to study groups using a sequence of functors H n. ... In mathematics, the Adams spectral sequence is a spectral sequence introduced by Frank Adams, to provide a computational tool in stable homotopy theory. ... In mathematics, stable homotopy theory is a branch of algebraic topology. ... In mathematics, the Atiyah-Hirzebruch spectral sequence is a computational tool from homological algebra, designed to make possible the calculation of an extraordinary cohomology theory. ... In mathematics, homology theory is the axiomatic study of the intuitive geometric idea of homology of cycles on topological spaces. ... In mathematics, homology theory is the axiomatic study of the intuitive geometric idea of homology of cycles on topological spaces. ... In mathematics, in the field of homological algebra, the Grothendieck spectral sequence is a technique that allows one to compute the derived functors of the composition of two functors , from knowledge of the derived functors of F and G. If and are two additive (covariant) functors between abelian categories such... In mathematics, certain functors may be derived to obtain other functors closely related to the original ones. ... In mathematics, the homotopy groups of spheres are the groups πk(Sn) in algebraic topology, more specifically homotopy theory, where πk(.) for k ≥ 1 denotes the homotopy group and Sn the n-sphere. ... The introduction to this article provides insufficient context for those unfamiliar with the subject matter. ... In homological algebra, the hyperhomology or hypercohomology of a complex of objects of an abelian category is an extension of the usual homology of an object to complexes. ...

References

Historical references

• Leray, Jean (1946). "L'anneau d'homologie d'une représentation". C. R. Acad. Sci. Paris 222: 1366--1368. 
• Leray, Jean (1946). "Structure de l'anneau d'homologie d'une représentation". C. R. Acad. Sci. Paris 222: 1419--1422. 
• Koszul, Jean-Louis (1947). "Sur les opérateurs de dérivation dans un anneau". C. R. Acad. Sci. Paris 225: 217--219. 
• Massey, William S. (1952). "Exact couples in algebraic topology. I, II". Ann. of Math. (2nd series) 56: 363--396. 
• Massey, William S. (1953). "Exact couples in algebraic topology. III, IV, V". Ann. of Math. (2nd series) 57: 248--286. 

Jean Leray (7 November 1906-10 November 1998) was a French mathematician, who worked on both partial differential equations and algebraic topology. ... Jean Leray (7 November 1906-10 November 1998) was a French mathematician, who worked on both partial differential equations and algebraic topology. ... Jean-Louis Koszul (born January 3, 1920 in Strasbourg, France) is a mathematician best known for studying geometry and discovering the Koszul complex. ... William Schumacher Massey (born 1920) is an American mathematician, known for his work in algebraic topology. ... William Schumacher Massey (born 1920) is an American mathematician, known for his work in algebraic topology. ...

Modern references

• McCleary, John (February 2001). A User's Guide to Spectral Sequences, 2nd Edition, Cambridge University Press, 560 pp. DOI:10.2277/0521567599. ISBN 0-521-56759-9. 
• Mosher, Robert; Martin Tangora (1968). Cohomology Operations and Applications in Homotopy Theory. Harper and Row. 
• Hatcher, Allen. Spectral Sequences in Algebraic Topology (PDF). 
• Chow, Timothy Y. (January 2006). "You Could Have Invented Spectral Sequences" (PDF). Notices of the American Mathematical Society 53: 15--19.