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, 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

Cohomological spectral sequence

Fix an abelian category, such as a category of modules over a ring, and a nonnegative integer. A cohomological spectral sequence is a sequence of objects and endomorphisms, such that for every

,
, the homology of with respect to.

Usually the isomorphisms are suppressed and we write instead. An object is called sheet, or sometimes a page or a term; an endomorphism is called boundary map or differential. Sometimes is called the derived object of.

Bigraded spectral sequence

In reality spectral sequences mostly occur in the category of doubly graded modules over a ring R, i.e. every sheet is a bigraded R-module
So in this case a cohomological spectral sequence is a sequence of bigraded R-modules and for every module the direct sum of endomorphisms of bidegree, such that for every it holds that:

The notation used here is called complementary degree. Some authors write instead, where 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 or ascending order. In the latter case, by replacing with and with , 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: At the th location this is the th homology group of C_•. The only natural differential on this new complex is the zero map, so we let d₁ = 0. This forces to equal, 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 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.

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 can be seen as the 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. Now turning to the next page means taking homology, that is the page is a subquotient of the page. The total degree n = p + q runs diagonally, northwest to southeast, across each sheet. In the homological case, the differentials have bidegree, 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, 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 is by definition a collection of maps which are compatible with the differentials, i.e., and with the given isomorphisms between the cohomology of the r step and the sheets of E and, respectively:. In the bigraded case, they should also respect the graduation:

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 be a spectral sequence of cohomological type. We say it has multiplicative structure if are differential graded algebras and the multiplication on is induced by that on via passage to cohomology.
A typical example is the cohomological Serre spectral sequence for a fibration, when the coefficient group is a ring R. It has the multiplicative structure induced by the cup products of fibre and base on the -page. However, in general the limiting term is not isomorphic as a graded algebra to H. 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, 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. 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
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 for some b in A. g' is defined to be the image of g 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 is an exact couple. C' corresponds to the E₁ term of the spectral sequence. We can iterate this procedure to get exact couples , C, f, g, h^{.
In order to construct a spectral sequence, let E_n be C and d_n be g o h.}