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 Jean Leray (1946), they have become important computational tools, particularly in algebraic topology, algebraic geometry and homological algebra.
Contents
- Discovery and motivation
- Formal definition
- Visualization
- Worked out examples
- 2 columns and 2 rows
- Wang sequence
- Low degree terms
- Edge maps and transgressions
- Multiplicative structure
- Constructions of spectral sequences
- Exact couples
- The spectral sequence of a filtered complex
- The spectral sequence of a double complex
- Convergence degeneration and abutment
- The spectral sequence of a filtered complex continued
- Long exact sequences
- The spectral sequence of a double complex continued
- Commutativity of Tor
- Further examples
- References
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
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:
- For all integers r ≥ r0, an object Er , called a sheet (as in a sheet of paper), or sometimes a page or a term,
- Endomorphisms dr : Er → Er satisfying dr o dr = 0, called boundary maps or differentials,
- Isomorphisms of Er+1 with H(Er), the homology of Er with respect to dr.
Usually the isomorphisms between Er+1 and H(Er) are suppressed, and we write equalities instead. Sometimes Er+1 is called the derived object of Er.
The most elementary example is a chain complex C•. An object C• in an abelian category of chain complexes 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). In this case, each sheet is a doubly graded module, so it decomposes as a direct sum of terms with one term for each possible bidegree. The boundary map is defined as the direct sum of boundary maps on each of the terms of the sheet. Their degree depends on r and is fixed by convention. For a homological spectral sequence, the terms are written
A morphism of spectral sequences E → E' is by definition a collection of maps fr : Er → E'r which are compatible with the differentials and with the given isomorphisms between cohomology of the r-th step and the (r + 1)-st sheets of E and E' , respectively.
Let Er be a spectral sequence, starting with say r = 0. Then there is a sequence of subobjects
such that
We then let
it is called the limiting term. (Of course, such
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. 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 down or up. This is similar to the differential on a chain complex. When r is one, the differential moves objects one space to the left or right. 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.
Worked-out examples
When learning spectral sequences for the first time, it is often helpful to work with simple computational examples. For more formal and complete discussions, see the sections below. For the examples in this section, it suffices to use this definition: one says a spectral sequence converges to H with an increasing filtration F if
2 columns and 2 rows
Let
such that
Next, let
Now, say, the spectral sequence converges to H with a filtration F as in the previous example. Since
Wang sequence
The computation in the previous section generalizes in a straightforward way. Consider a fibration over a sphere:
with n at least 2. There is the Serre spectral sequence:
that is to say,
Since
Now, writing
Putting all calculations together, one gets:
(The Gysin sequence is obtained in a similar way.)
Low-degree terms
With an obvious notational change, the type of the computations in the previous examples can also be carried out for cohomological spectral sequence. Let
so that
Since
Since
Edge maps and transgressions
Let
as the denominator is zero. Hence, there is a sequence of monomorphisms:
They are called the edge maps. Similarly, if
The transgression is a partially-defined map (more precisely, a map from a subobject to a quotient)
given as a composition
For a spectral sequence
And if
The transgression is a not necessarily well-defined map:
induced by
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
A typical example is the cohomological Serre spectral sequence for a fibration
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.
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 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.
For a simple example, see the Bockstein spectral sequence.
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 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 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. (The complementary degree is often a more convenient index than the total degree n. For example, this is true 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 that we then have
and we should have the relationship
For this to make sense, we must find a differential dr on each Er and verify that it leads to homology isomorphic to Er+1. The differential
is defined by restricting the original differential d defined on
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.
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 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
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
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 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
The p indicates the filtration index. It is very common to write the
In most spectral sequences, the
to mean that whenever p + q = n,
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
The five-term exact sequence of a spectral sequence relates certain low-degree terms and E∞ terms.
See also Boardman, Conditionally Convergent Spectral Sequences.
The spectral sequence of a filtered complex, continued
Notice that we have a chain of inclusions:
We can ask what happens if we define
To describe the abutment of our spectral sequence in more detail, notice that we have the formulas:
To see what this implies for
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•:
This gives:
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, continued
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 Tor
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 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)i(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
Finally, when p and q are equal, the two right-hand sides are equal, and the commutativity of Tor follows.
Further examples
Some notable spectral sequences are: