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.
Cohomological spectral sequenceEdit
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
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 sequenceEdit
In reality spectral sequences mostly occur in the category of doubly graded modules over a ringR (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 sequenceEdit
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 complexEdit
The most elementary example in the ungraded situation is a chain complexC•. 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 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 .
The E2 sheet of a cohomological 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. 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 (−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.
Four pages of a cohomological spectral sequence
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.
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 rth step and the (r+1)th sheets of E and E' , respectively: . In the bigraded case, they should also respect the graduation:
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 sequencesEdit
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 coupleEdit
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, C), together with three homomorphisms between these objects: f : A → A, g : A → C and h : C → A subject to certain exactness conditions:
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 methodEdit
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 q = n − p.
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
is defined by restricting the original differential defined on to the subobject . It is straightforward to check that the homology of with respect to this differential is , 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 methodEdit
Can be used to construct Mixed Hodge structures
The spectral sequence of a double complexEdit
Another common spectral sequence is the spectral sequence of a double complex. A double complex is a collection of objects Ci,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 complexT(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 Ci,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:
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
Consequently, the differential on E0 is the map Cp,q → Cp,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 abutmentEdit
Interpretation as a filtration of cycles and boundariesEdit
Let Er be a spectral sequence, starting with say r = 1. Then there is a sequence of subobjects
such that ; indeed, recursively we let and let be so that are the kernel and the image of
We then let and
it is called the limiting term. (Of course, such 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 convergenceEdit
We say a spectral sequence converges weakly if there is a graded object with a filtration for every , and for every there exists an isomorphism . It converges to if the filtration is Hausdorff, i.e. . We write
to mean that whenever p + q = n, converges to .
We say that a spectral sequence abuts to if for every there is such that for all , . Then is the limiting term. The spectral sequence is regular or degenerates at if the differentials are zero for all . If in particular there is , such that the sheet is concentrated on a single row or a single column, then we say it collapses. 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. 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 is the same as .
The spectral sequence of a filtered complex, continuedEdit
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 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:
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 pth graded part of the (p+q)th homology of C. If our spectral sequence converges, then we conclude that:
Long exact sequencesEdit
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•:
The differential has bidegree (1, 0), so d0,q : Hq(C•) → Hq+1(A•). These are the connecting homomorphisms from the snake lemma, and together with the maps A• → B• → C•, they give a sequence:
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, continuedEdit
Using the abutment for a filtered complex, we find that:
In general, the two gradings on Hp+q(T(C•,•)) are distinct. Despite this, it is still possible to gain useful information from these two spectral sequences.
Commutativity of TorEdit
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 . While this can be verified without a spectral sequence, it is very easy with spectral sequences.
Choose projective resolutions and 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 and whose differentials are and . (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 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:
Finally, when p and q are equal, the two right-hand sides are equal, and the commutativity of Tor follows.
Consider a spectral sequence where vanishes for all less than some and for all less than some . If and can be chosen to be zero, this is called a first-quadrant spectral sequence.
The sequence abuts because holds for all if and . 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 vanishes for all greater than some and for all greater than some .
2 non-zero adjacent columnsEdit
Let be a homological spectral sequence such that for all p other than 0, 1. Visually, this is the spectral sequence with -page
The differentials on the second page have degree (-2, 1), so they are of the form
These maps are all zero since they are
hence the spectral sequence degenerates: . Say, it converges to with a filtration
such that . Then , , , , etc. Thus, there is the exact sequence:
Next, let 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 , as the denominator is zero. Similarly, . Thus,
Now, say, the spectral sequence converges to H with a filtration F as in the previous example. Since , , etc., we have: . Putting everything together, one gets:
The computation in the previous section generalizes in a straightforward way. Consider a fibration over a sphere:
With an obvious notational change, the type of the computations in the previous examples can also be carried out for cohomological spectral sequence. Let be a first-quadrant spectral sequence converging to H with the decreasing filtration
Since is zero if p or q is negative, we have: