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.
Definition.
Fix an abelian category, such as a category of modules over a ring. A cohomological spectral sequence is a choice of a nonnegative integer formula_1 and a collection of three sequences:
Usually the isomorphisms between formula_6 and formula_7 are suppressed, and we write equalities instead. Sometimes formula_6 is called the derived object of formula_3.
Spectral sequence from a chain complex.
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 "r"0 = 0, and let "E"0 be "C•". This forces "E"1 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 "d"1 = 0. This forces formula_14 to equal formula_15, 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 formula_3.
Types of spectral sequences.
In the ungraded situation described above, "r"0 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 formula_17 and the differentials have bidegree (− "r","r" − 1). For a cohomological spectral sequence, the terms are written formula_18 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, "r"0 = 0, but for the Grothendieck spectral sequence, "r"0 = 2. Usually "r"0 is zero, one, or two.
Categorical properties.
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)"th sheets of "E" and "E' ", respectively.
Interpretation as a filtration of cycles and boundaries.
Let "E""r" be a spectral sequence, starting with say "r" = 1. Then there is a sequence of subobjects
such that formula_20; indeed, recursively we let formula_21 and let formula_22 be so that formula_23 are the kernel and the image of formula_24
We then let formula_25 and
it is called the limiting term. (Of course, such formula_27 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.)
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 formula_28.
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 formula_29. The examples below illustrate how one relates such filtrations with the formula_30-term in the forms of exact sequences; many exact sequences in applications (e.g., Gysin sequence) arise in this fashion.
2 non-zero adjacent columns.
Let formula_31 be a homological spectral sequence such that formula_32 for all "p" other than 0, 1. Visually, this is the spectral sequence with formula_30-pageformula_34The differentials on the second page have degree (-2, 1), so they are of the formformula_35These maps are all zero since they areformula_36, formula_37hence the spectral sequence degenerates: formula_38. Say, it converges to formula_39 with a filtrationformula_40such that formula_29. Then formula_42, formula_43, formula_44, formula_45, etc. Thus, there is the exact sequence:formula_46.Next, let formula_31 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 formula_48, as the denominator is zero. Similarly, formula_49. Thus,formula_50.Now, say, the spectral sequence converges to "H" with a filtration "F" as in the previous example. Since formula_51, formula_52, etc., we have: formula_53. Putting everything together, one gets:formula_54
Wang sequence.
The computation in the previous section generalizes in a straightforward way. Consider a fibration over a sphere:formula_55with "n" at least 2. There is the Serre spectral sequence:formula_56;that is to say, formula_57 with some filtration formula_58.
Since formula_59 is nonzero only when "p" is zero or "n" and equal to Z in that case, we see formula_60 consists of only two lines formula_61, hence the formula_30-page is given byformula_63Moreover, sinceformula_64for formula_61 by the universal coefficient theorem, the formula_30 page looks likeformula_67Since the only non-zero differentials are on the formula_68-page, given byformula_69which isformula_70the spectral sequence converges on formula_71. By computing formula_72 we get an exact sequenceformula_73and written out using the homology groups, this isformula_74To establish what the two formula_75-terms are, write formula_76, and since formula_77, etc., we have: formula_78 and thus, since formula_79,formula_80This is the exact sequenceformula_81Putting all calculations together, one gets:formula_82(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 formula_83 be a first-quadrant spectral sequence converging to "H" with the decreasing filtration
so that formula_85
Since formula_86 is zero if "p" or "q" is negative, we have:
Since formula_88 for the same reason and since formula_89
Since formula_91, formula_92. Stacking the sequences together, we get the so-called five-term exact sequence:
Edge maps and transgressions.
Homological spectral sequences.
Let formula_31 be a spectral sequence. If formula_95 for every "q" < 0, then it must be that: for "r" ≥ 2,
as the denominator is zero. Hence, there is a sequence of monomorphisms:
They are called the edge maps. Similarly, if formula_95 for every "p" < 0, then there is a sequence of epimorphisms (also called the edge maps):
The transgression is a partially-defined map (more precisely, a map from a subobject to a quotient)
given as a composition formula_101, the first and last maps being the inverses of the edge maps.
Cohomological spectral sequences.
For a spectral sequence formula_83 of cohomological type, the analogous statements hold. If formula_103 for every "q" < 0, then there is a sequence of epimorphisms
And if formula_103 for every "p" < 0, then there is a sequence of monomorphisms:
The transgression is a not necessarily well-defined map:
induced by formula_108.
Application.
Determining these maps are fundamental for computing many differentials in the Serre spectral sequence. For instance the transgression map determines the differentialpg 540,564formula_109for the homological spectral spectral sequence, hence on the Serre spectral sequence for a fibration formula_110 gives the mapformula_111
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 formula_112 be a spectral sequence of cohomological type. We say it has multiplicative structure if (i) formula_3 are (doubly graded) differential graded algebras and (ii) the multiplication on formula_6 is induced by that on formula_3 via passage to cohomology.
A typical example is the cohomological Serre spectral sequence for a fibration formula_110, when the coefficient group is a ring "R". It has the multiplicative structure induced by the cup products of fibre and base on the formula_117-page. However, in general the limiting term formula_27 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.
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 "E"0 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") "h"("n"). This gives a 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") ⊆ "FpC""n"+1. We assume that the filtration is "descending", i.e., "FpC•" ⊇ "F""p"+1"C"•. 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" . (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 "E"1 step:
Since we assumed that the boundary map was compatible with the filtration, "E"0 is a doubly graded object and there is a natural doubly graded boundary map "d"0 on "E"0. To get "E"1, we take the homology of "E"0.
Notice that formula_127 and formula_128 can be written as the images in formula_129 of
and that we then have
formula_133 is exactly the stuff which the differential pushes up one level in the filtration, and formula_134 is exactly the image of the stuff which the differential pushes up zero levels in the filtration. This suggests that we should choose formula_135 to be the stuff which the differential pushes up "r" levels in the filtration and formula_136 to be image of the stuff 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 "dr" on each "Er" and verify that it leads to homology isomorphic to "E""r"+1. The differential 
is defined by restricting the original differential "d" defined on formula_142 to the subobject formula_135.
It is straightforward to check that the homology of "Er" with respect to this differential is "E""r"+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 formula_144 and formula_145. 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 formula_148 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 "E"0, "E"1, and "E"2 terms of the "I" filtration on "T"("C"•,•). The "E"0 term is clear:
where .
To find the "E"1 term, we need to determine "d I" + "d II" on "E"0. 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 "C""p","q" → "C""p","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 "E"0. That means the differential is exactly "d II", so we get
To find "E2", we need to determine
Because "E"1 was exactly the homology with respect to "d II", "d II" is zero on "E"1. Consequently, we get
Using the other filtration gives us a different spectral sequence with a similar "E"2 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 "E"1.
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 formula_28 converges to or abuts to formula_158 if there is an "r"("p", "q") such that for all "r" ≥ "r"("p", "q"), the differentials formula_159 and formula_160 are zero. This forces formula_28 to be isomorphic to formula_158 for large "r". In symbols, we write:
The "p" indicates the filtration index. It is very common to write the formula_164 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 formula_165 term is not naturally a doubly graded object. Instead, there are usually formula_166 terms which come with a natural filtration formula_167. In these cases, we set formula_168. We define convergence in the same way as before, but we write
to mean that whenever "p" + "q" = "n", formula_28 converges to formula_158.
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" ≅ "E""r"+1 ≅ "E""r"+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 formula_28 vanishes for all "p" less than some "p"0 and for all "q" less than some "q"0. If "p"0 and "q"0 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 formula_28 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 formula_28 vanishes for all "p" greater than some "p"0 and for all "q" greater than some "q"0.
The five-term exact sequence of a spectral sequence relates certain low-degree terms and "E"∞ terms.
See also Boardman, Conditionally Convergent Spectral Sequences.
Examples of degeneration.
The spectral sequence of a filtered complex, continued.
Notice that we have a chain of inclusions:
We can ask what happens if we define
formula_158 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 formula_182 recall that we assumed that the filtration was separated. This implies that as "r" increases, the kernels shrink, until we are left with formula_183. For formula_184, recall that we assumed that the filtration was exhaustive. This implies that as "r" increases, the images grow until we reach formula_185. 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:
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•") → "H""q"+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 "E"2 term because the differentials have bidegree (2, −1). Consequently, the "E"2 term is the same as the "E"∞ term:
But we also have a direct description of the "E"2 term as the homology of the "E"1 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 formula_199. While this can be verified without a spectral sequence, it is very easy with spectral sequences.
Choose projective resolutions formula_200 and formula_201 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 formula_202 and whose differentials are formula_203 and formula_204. (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 formula_209 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 "E"2 terms:
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: