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, and a nonnegative integer formula_1. A cohomological spectral sequence is a sequence formula_2 of objects formula_3 and endomorphisms formula_4, such  that for every formula_5 Usually the isomorphisms are suppressed and we write formula_10 instead. An object formula_3 is called "sheet" (as in a sheet of paper), or sometimes a "page" or a "term"; an endomorphism formula_12 is called "boundary map" or "differential". Sometimes formula_13 is called the "derived object" of formula_8. 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 formula_15 So in this case a cohomological spectral sequence is  a sequence formula_2 of bigraded R-modules formula_17 and for every module the direct sum of endomorphisms formula_18 of bidegree formula_19, such that for every formula_5 it holds that: The notation used here is called "complementary degree". Some authors write formula_23 instead, where formula_24 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, "r" = 0, but for the Grothendieck spectral sequence, "r" =  2. Usually "r" is zero, one, or two. In the ungraded situation described above, "r" 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 formula_25 with formula_26 and formula_27 with formula_28 (bidegree formula_29), 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 "r" = 0, and let "E" be "C". This forces "E" to be the complex "H"("C"): At the "i"<nowiki>'</nowiki>th location this is the "i"<nowiki>'</nowiki>th homology group of "C". The only natural differential on this new complex is the zero map, so we let "d" = 0. This forces formula_30 to equal formula_31, 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_8. 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 formula_8 can be seen as the formula_34 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 formula_35. Now turning to the next page means taking homology, that is the formula_36 page is a subquotient of the formula_34 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. 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 formula_38 is by definition a collection of maps formula_39 which are compatible with the differentials, i.e. formula_40, and with the given isomorphisms between the cohomology of the "r"th step and the "(r+1)"th sheets of "E" and "E' ", respectively: formula_41. In the bigraded case, they should also respect the graduation: formula_42 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_43 be a spectral sequence of cohomological type. We say it has multiplicative structure if (i) formula_8 are (doubly graded) differential graded algebras and (ii) the multiplication on formula_13 is induced by that on formula_8 via passage to cohomology. A typical example is the cohomological Serre spectral sequence for a fibration formula_47, 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_48-page. However, in general the limiting term formula_49 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. 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 "E" 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<nowiki>'</nowiki>", "C<nowiki>'</nowiki>", "f<nowiki>'</nowiki>", "g<nowiki>'</nowiki>", "h< nowiki>'</nowiki>") is an exact couple. "C<nowiki>'</nowiki>" corresponds to the "E" term of the spectral sequence. We can iterate this procedure to get exact couples ("A", "C", "f", "g", "h"). In order to construct a spectral sequence, let "E" be "C" and "d" be "g" "h". 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 formula_50 together with a descending filtration, formula_51 . We require that the boundary map is compatible with the filtration, i.e. formula_52 , and that the filtration is "exhaustive", that is, the union of the set of all formula_53 is the entire chain complex formula_54. Then there exists a spectral sequence with formula_55 and formula_56. Later, we will also assume that the filtration is "Hausdorff" or "separated", that is, the intersection of the set of all formula_53 is zero. The filtration is useful because it gives a measure of nearness to zero: As "p" increases, formula_53 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. formula_59 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 formula_60 step: Since we assumed that the boundary map was compatible with the filtration, formula_65 is a doubly graded object and  there is a natural doubly graded boundary map formula_66 on formula_65. To get formula_60, we take the homology of formula_65. Notice that formula_74 and formula_75 can be written as the images in formula_76 of and that we then have formula_80 are exactly the elements which the differential pushes up one level in the filtration, and formula_81 are exactly the image of the elements which the differential pushes up zero levels in the filtration. This suggests that we should choose formula_82 to be the elements which the differential pushes up "r" levels in the filtration and formula_83 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 formula_12 on each formula_3 and verify that it leads to homology isomorphic to formula_90. The differential is defined by restricting the original differential formula_92 defined on formula_93 to the subobject formula_82. It is straightforward to check that the homology of formula_3 with respect to this differential is formula_90, 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 "C" for all integers "i" and "j" together with two differentials, "d " and "d ". "d " is assumed to decrease "i", and "d " is assumed to decrease "j". Furthermore, we assume that the differentials "anticommute", so that "d d " + "d d " = 0. Our goal is to compare the iterated homologies formula_97 and formula_98. 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"<nowiki>'</nowiki>th term is formula_101 and whose differential is "d " + "d ". This is a complex because "d " and "d " are anticommuting differentials. The two filtrations on "C" 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", "E", and "E" terms of the "I  filtration on T"("C"). The "E" term is clear: where . To find the "E" term, we need to determine "d " + "d " on "E". Notice that the differential must have degree −1 with respect to "n", so we get a map Consequently, the differential on "E" is the map "C" → "C" induced by "d " + "d ". But "d " has the wrong degree to induce such a map, so "d " must be zero on "E". That means the differential is exactly "d ", so we get To find "E", we need to determine Because "E" was exactly the homology with respect to  d , "d " is zero on "E". Consequently, we get Using the other filtration gives us a different spectral sequence with a similar "E" 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. Interpretation as a filtration of cycles and boundaries. Let "E" be a spectral sequence, starting with say "r" = 1. Then there is a sequence of subobjects such that formula_111; indeed, recursively we let formula_112 and let formula_113 be so that  formula_114 are the kernel and the image of formula_115 We then let formula_116 and it is called the limiting term. (Of course, such formula_49 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 convergence. We say a spectral sequence converges weakly if there is a graduated object formula_119 with a filtration formula_120 for every formula_121, and for every formula_122 there  exists an isomorphism formula_123. It converges to formula_119 if the filtration formula_120 is Hausdorff, i.e. formula_126. We write to mean that whenever "p" + "q" = "n", formula_35 converges to formula_129. We say that a spectral sequence formula_35 abuts to formula_129 if for every formula_132 there is formula_133 such that for all formula_134, formula_135. Then formula_136 is the limiting term. The spectral sequence is regular or degenerates at formula_137 if the differentials formula_138 are zero for all formula_139. If in particular there is formula_140, such that the formula_141 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 formula_143 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 formula_144 is the same as formula_60. The five-term exact sequence of a spectral sequence relates certain low-degree terms and "E" terms. 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_129 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_153 recall that we assumed that the filtration was separated. This implies that as "r" increases, the kernels shrink, until we are left with formula_154. For formula_155, recall that we assumed that the filtration was exhaustive. This implies that as "r" increases, the images grow until we reach formula_156. 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 "H"("A") → "H"("B") → "H"("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 "d" : "H"("C") → "H"("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" term because the differentials have bidegree (2, −1). Consequently, the "E" term is the same as the "E" term: But we also have a direct description of the "E" term as the homology of the "E" 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 H(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_170. While this can be verified without a spectral sequence, it is very  easy with spectral sequences. Choose projective resolutions formula_171 and formula_172 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_173 and whose differentials are formula_174 and formula_175. (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_180 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" terms: Finally, when "p" and "q" are equal, the two right-hand sides are equal, and the commutativity of Tor follows. Worked-out examples. First-quadrant sheet. Consider a spectral sequence where formula_35 vanishes for all formula_122 less than some formula_185 and for all formula_186 less than some formula_187. If formula_185 and formula_187 can be chosen  to be zero, this is called a first-quadrant spectral sequence. The sequence abuts because formula_190 holds for all formula_191 if formula_192 and formula_193. 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 formula_35 vanishes for all formula_122 greater than some formula_185 and for all formula_186 greater than some formula_187. 2  non-zero adjacent columns. Let formula_199 be a homological spectral sequence such that formula_200 for all "p" other than 0, 1. Visually, this is the spectral sequence with formula_201-pageformula_202The differentials on the second page have degree (-2, 1), so they are of the formformula_203These maps are all zero since they areformula_204, formula_205hence the spectral sequence degenerates: formula_206. Say, it converges to formula_207 with a filtrationformula_208such that formula_209. Then formula_210, formula_211, formula_212, formula_213, etc. Thus, there is the exact sequence:formula_214.Next, let formula_199 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_216, as the denominator is zero. Similarly, formula_217. Thus,formula_218.Now, say, the spectral sequence converges to "H" with a filtration "F" as in the previous example. Since formula_219, formula_220, etc., we have: formula_221. Putting everything together, one gets:formula_222 Wang sequence. The computation in the previous section generalizes in a straightforward way. Consider a fibration over a sphere:formula_223with "n" at least 2. There is the Serre spectral sequence:formula_224;that is to say, formula_225 with some  filtration formula_226. Since formula_227 is nonzero only when "p" is zero or "n" and equal to Z in that case, we see formula_228 consists of only two lines formula_229, hence the formula_201-page is given byformula_231Moreover, sinceformula_232for formula_229 by the universal coefficient theorem, the formula_201 page looks likeformula_235Since the only non-zero differentials are on the formula_236-page, given byformula_237which isformula_238the spectral sequence converges on formula_239. By computing formula_240 we get an exact sequenceformula_241and written out using the homology groups, this isformula_242To establish what the two formula_243-terms are, write formula_244, and since formula_245, etc., we have: formula_246 and thus , since formula_247,formula_248This is the exact sequenceformula_249Putting all calculations together, one gets:formula_250(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_251 be a first-quadrant spectral sequence converging to "H" with the decreasing filtration so that formula_253 Since formula_254 is zero if "p" or "q" is negative, we have: Since formula_256 for the same reason and since formula_257 Since formula_259, formula_260. Stacking the sequences together, we get the so-called five-term exact  sequence: Edge maps and transgressions. Homological spectral sequences. Let formula_199 be a spectral sequence. If formula_263 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_263 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_269, the first and last maps being the inverses of the  edge maps. Cohomological spectral sequences. For a spectral sequence formula_251 of cohomological type, the analogous statements hold. If formula_271 for every "q" < 0, then there is a sequence of epimorphisms And if formula_271 for every "p" < 0, then there is a sequence of monomorphisms: The transgression is a not necessarily well-defined map: induced by formula_276. Application. Determining these maps are fundamental for computing many differentials in the Serre spectral sequence. For instance the transgression map determines the differentialformula_277for the homological spectral spectral sequence, hence on the Serre spectral sequence for a fibration formula_47 gives the mapformula_279 Further  examples. Some notable spectral sequences are: