In the area of mathematics known as homological algebra, especially in algebraic topology and 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.
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.
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. 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.
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:
Usually the isomorphisms between Er+1 and H(Er) are suppressed, and we write equalities instead. Er+1 is sometimes called the derived object of Er.
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:
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 Er.
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.
A morphism of spectral sequences E → E' is by definition a collection of maps fr : Er → 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.
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:
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:
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 En be C(n) and dn be g(n) o h(n). This gives 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 .
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.
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 FpC•, 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(FpCn) ⊆ FpCn+1. We assume that the filtration is descending, i.e., FpC• ⊇ Fp+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 FpC• is zero, and that the filtration is exhaustive, that is, the union of the set of all FpC• is the entire chain complex C•.
The filtration is useful because it gives a measure of nearness to zero: As p increases, FpC• 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 Er which gives the Er+1 we stated above. Since we have written as a subobject of , we get a differential on by restricting the differential on :
Now it is straightforward to check that the homology of Er with respect to this differential is Er+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.
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 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 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:
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.
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 ds is zero. This implies that Er ≅ Er+1 ≅ Er+2 ≅ ... In particular, it implies that Er 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.
The five-term exact sequence of a spectral sequence relates certain low-degree terms and E∞ terms.
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:
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.
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.
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 Tori(M, N) = Tori(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 Ci,j = Pi ⊗ Qj 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.
Some notable spectral sequences are: