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 (as in a sheet of paper), 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 (or doubly graded sheaves of modules over a sheaf of rings), 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 (like above) or ascending order. In the latter case, by replacing with and with (bidegree ), 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 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• Er = 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 .
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 (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 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 (i) are (doubly graded) differential graded algebras and (ii) 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(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 En be C(n) and dn 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 together with a descending filtration, . We require that the boundary map is compatible with the filtration, i.e. , and that the filtration is exhaustive, that is, the union of the set of all is the entire chain complex . Then there exists a spectral sequence with and . Later, we will also assume that the filtration is Hausdorff or separated, that is, the intersection of the set of all is zero.
The filtration is useful because it gives a measure of nearness to zero: As p increases, 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 .
Construction 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 step:
Since we assumed that the boundary map was compatible with the filtration, is a doubly graded object and there is a natural doubly graded boundary map on . To get , we take the homology of .
Notice that and can be written as the images in of
and that we then have
are exactly the elements which the differential pushes up one level in the filtration, and are exactly the image of the elements which the differential pushes up zero levels in the filtration. This suggests that we should choose to be the elements which the differential pushes up r levels in the filtration and 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
and we should have the relationship
For this to make sense, we must find a differential on each and verify that it leads to homology isomorphic to . The differential