SPECTRAL SEQUENCES: FRIEND OR FOE? RAVI VAKIL Spectal sequences ae a poweful book-keeping tool fo poving things involving complicated commutative diagams. They wee intoduced by Leay in the 94 s at the same time as he intoduced sheaves. They have a eputation fo being abstuse and difficult. It has been suggested that the name spectal was given because, like spectes, spectal sequences ae teifying, evil, and dangeous. I have head no one disagee with this intepetation, which is pehaps not supising since I just made it up. Nonetheless, the goal of this note is to tell you enough that you can use spectal sequences without hesitation o fea, and why you shouldn t be fightened when they come up in a semina. What is diffeent in this pesentation is that we will use spectal sequence to pove things that you may have aleady seen, and that you can pove easily in othe ways. This will allow you to get some hands-on expeience fo how to use them. We will also see them only in a special case of double complexes (which is the vesion by fa the most often used in algebaic geomety), and not in the geneal fom usually pesented (filteed complexes, exact couples, etc.). See chapte 5 of Weibel s mavelous book fo moe detailed infomation if you wish. If you want to become comfotable with spectal sequences, you must ty the execises. Fo conceteness, we wok in the categoy vecto spaces ove a given field. Howeve, eveything we say will apply in any abelian categoy, such as the categoy Mod A of A- modules... Double complexes. A fist-quadant double complex is a collection of vecto spaces E p,q (p, q Z), which ae zeo unless p, q, and ightwad mophisms d p,q > : E p,q E p,q+ and upwad mophisms d p,q : Ep,q E p+,q. In the supescipt, the fist enty denotes the ow numbe, and the second enty denotes the column numbe, in keeping with the convention fo matices, but opposite to how the (x, y)-plane is labeled. The subscipt is meant to suggest the diection of the aows. We will always wite these as d > and d and ignoe the supescipts. We equie that d > and d satisfying (a) d > =, (b) d =, and one moe condition: (c) eithe d > d = d d > (all the squaes commute) o d > d + d d > = (they all anticommute). Both come up in natue, and you can switch fom one to the othe by eplacing d p,q with dp ( )q. So I ll assume that all the squaes anticommute, but that you know how to tun the commuting case into this one. (You will see that thee is no diffeence in the ecipe, basically because the image and kenel of a homomophism f equal the image and kenel espectively of f.) Date: Tuesday, Mach, 8. Updated vesion late Tuesday aftenoon.
E p+,q d p+,q > E p+,q+ d p,q anticommutes d p,q+ E p,q d p,q > E p,q+ Thee ae vaiations on this definition, whee fo example the vetical aows go downwads, o some diffeent subset of the E p,q ae equied to be zeo, but I ll leave these staightfowad vaiations to you. Fom the double complex we constuct a coesponding (single) complex E with E k = i E i,k i, with d = d > + d. In othe wods, when thee is a single supescipt k, we mean a sum of the kth antidiagonal of the double complex. The single complex is somtimes called the total complex. Note that d = (d > + d ) = d > + (d >d + d d > ) + d =, so E is indeed a complex. The cohomology of the single complex is sometimes called the hypecohomology of the double complex. We will instead use the phase cohomology of the double complex. Ou initial goal will be to find the cohomology of the double complex. You will see late that we secetly also have othe goals. A spectal sequence is a ecipe fo computing some infomation about the cohomology of the double complex. I won t yet give the full ecipe. Supisingly, this fagmentay bit of infomation is sufficent to pove lots of things... Appoximate Definition. A spectal sequence with ightwad oientation is a sequence of tables o pages > E p,q, > E p,q, > E p,q,... (p, q Z), whee > E p,q = E p,q, along with a diffeential >d p,q : > E p,q > E p+,q + with > d p,q ke > d p,q > d p,q / im > d p,q+ =, along with an isomophism of the cohomology of > d at > E p,q (i.e. ) with > E p,q +. The oientation indicates that ou th diffeential is the ightwad one: d = d >. The left subscipt > is usually omitted.
The ode of the mophisms is best undestood visually: () d 3 d d d (the mophisms each apply to diffeent pages). Notice that the map always is degee in the gading of the single complex E. The actual definition descibes what E, and d, actually ae, in tems of E,. We will descibe d, d, and d below, and you should fo now take on faith that this sequence continues in some natual way. Note that E p,q is always a subquotient of the coesponding tem on the th page E p,q = fo all, so E p,q = unless p, q Z. Notice E p,q. In paticula, if E p,q =, then E p,q also that fo any fixed p, q, once is sufficiently lage, E p,q + using the complex = is computed fom (E,, d ) d p,q E p,q and thus we have canonical isomophisms We denote this module E p,q. E p,q d p+,q = E p,q + = E p,q + = We now descibe the fist few pages of the spectal sequence explicitly. As stated above, the diffeential d on E, = E, is defined to be d >. The ows ae complexes: The th page E : 3
and so E is just the table of cohomologies of the ows. You should check that thee ae now vetical maps d p,q : E p,q E p+,q of the ow cohomology goups, induced by d, and that these make the columns into complexes. (We have used up the hoizontal mophisms, but the vetical diffeentials live on.) The st page E : We take cohomology of d on E, giving us a new table, E p,q. It tuns out that thee ae natual mophisms fom each enty to the enty two above and one to the left, and that the composition of these two is. (It is a vey wothwhile execise to wok out how this natual mophism d should be defined. You agument may be eminiscent of the connecting homomophism in the Snake Lemma.5 o in the long exact sequence in cohomology aising fom a shot exact sequence of complexes, Execise.D. This is no coincidence.) The nd page E : This is the beginning of a patten. Then it is a theoem that thee is a filtation of H k (E ) by E p,q whee p + q = k. (We can t yet state it as an official Theoem because we haven t pecisely defined the pages and diffeentials in the spectal sequence.) Moe pecisely, thee is a filtation () E,k E,k? E,k E k, H k (E ) whee the quotients ae displayed above each inclusion. (I always foget which way the quotients ae supposed to go, i.e. whethe E k, o E,k is the subobject. One way of emembeing it is by having some idea of how the esult is poved.) We say that the spectal sequence > E, conveges to H (E ). We often say that > E, (o any othe page) abuts to H (E ). Although the filtation gives only patial infomation about H (E ), sometimes one can find H (E ) pecisely. One example is if all E i,k i ae zeo, o if all but one of them ae zeo (e.g. if E i,k i has pecisely one non-zeo ow o column, in which case one says that the spectal sequence collapses at the th step, although we will not use this tem). Anothe example is in the categoy of vecto spaces ove a field, in which case we can find the dimension of H k (E ). Also, in lucky cicumstances, E (o some othe small page) aleady equals E. 4
.A. EXERCISE: INFORMATION FROM THE SECOND PAGE. Show that H (E ) = E, = E, and E, H (E ) E, d, E, H (E )..3. The othe oientation. You may have obseved that we could as well have done eveything in the opposite diection, i.e. evesing the oles of hoizontal and vetical mophisms. Then the sequences of aows giving the spectal sequence would look like this (compae to ()). (3) This spectal sequence is denoted E, ( with the upwads oientation ). Then we would again get pieces of a filtation of H (E ) (whee we have to be a bit caeful with the ode with which E p,q coesponds to the subquotients it in the opposite ode to that of () and E p,q fo > E p,q ). Waning: in geneal thee is no isomophism between >E p,q In fact, this obsevation that we can stat with eithe the hoizontal o vetical maps was ou secet goal all along. Both algoithms compute infomation about the same thing (H (E )), and usually we don t cae about the final answe we often cae about the answe we get in one way, and we get at it by doing the spectal sequence in the othe way...4. Examples. We e now eady to see how this is useful. The moal of these examples is the following. In the past, you may have poved vaious facts involving vaious sots of diagams, which involved chasing elements aound. Now, you ll just plug them into a spectal sequence, and let the spectal sequence machiney do you chasing fo you..5. Example: Poving the Snake Lemma. Conside the diagam D E F α β γ A B 5 C
whee the ows ae exact and the squaes commute. (Nomally the Snake Lemma is descibed with the vetical aows pointing downwads, but I want to fit this into my spectal sequence conventions.) We wish to show that thee is an exact sequence (4) ke α ke β ke γ im α im β im γ. We plug this into ou spectal sequence machiney. We fist compute the cohomology using the ightwads oientation, i.e. using the ode (). Then because the ows ae exact, E p,q =, so the spectal sequence has aleady conveged: E p,q =. We next compute this in anothe way, by computing the spectal sequence using the upwads oientation. Then E, (with its diffeentials) is: im α im β im γ Then E, is of the fom: ke α ke β ke γ.???????? We see that afte E, all the tems will stabilize except fo the double-question-maks all maps to and fom the single question maks ae to and fom -enties. And afte E 3, even these two double-quesion-mak tems will stabilize. But in the end ou complex must be the complex. This means that in E, all the enties must be zeo, except fo the two double-question-maks, and these two must be the isomophic. This means that ke α ke β ke γ and im α im β im γ ae both exact (that comes fom the vanishing of the single-question-maks), and coke(ke β ke γ) = ke(im α im β) is an isomophism (that comes fom the equality of the double-question-maks). Taken togethe, we have poved the exactness of (4), and hence the Snake Lemma! Spectal sequences make it easy to see how to genealize esults futhe. Fo example, if A B is no longe assumed to be injective, how would the conclusion change? 6
.6. Example: the Five Lemma. Suppose (5) F G H I J whee the ows ae exact and the squaes commute. α A β B Suppose α, β, δ, ɛ ae isomophisms. We ll show that γ is an isomophism. γ C We fist compute the cohomology of the total complex using the ightwads oientation (). We choose this because we see that we will get lots of zeos. Then > E, looks like this: δ D?? ɛ E? Then > E looks simila, and the sequence will convege by E, as we will neve get any aows between two non-zeo enties in a table theeafte. We can t conclude that the cohomology of the total complex vanishes, but we can note that it vanishes in all but fou degees and most impotant, it vanishes in the two degees coesponding to the enties C and H (the souce and taget of γ). We next compute this using the upwads oientation (3). Then E looks like this:??? and the spectal sequence conveges at this step. We wish to show that those two question maks ae zeo. But they ae pecisely the cohomology goups of the total complex that we just showed wee zeo so we e done! The best way to become comfotable with this sot of agument is to ty it out youself seveal times, and ealize that it eally is easy. So you should do the following execises!.b. EXERCISE: THE SUBTLE FIVE LEMMA. By looking at the spectal sequence poof of the Five Lemma above, pove a subtle vesion of the Five Lemma, whee one of the isomophisms can instead just be equied to be an injection, and anothe can instead just be equied to be a sujection. (I am delibeately not telling you which ones, so you can see how the spectal sequence is telling you how to impove the esult.).c. EXERCISE. If β and δ (in (5)) ae injective, and α is sujective, show that γ is injective. State the dual statement (whose poof is of couse essentially the same)..d. EXERCISE. Use spectal sequences to show that a shot exact sequence of complexes gives a long exact sequence in cohomology. 7
.E. EXERCISE (THE MAPPING CONE). Suppose µ : A B is a mophism of complexes. Suppose C is the single complex associated to the double complex A B. (C is called the mapping cone of µ.) Show that thee is a long exact sequence of complexes: H i (C ) H i (A ) H i (B ) H i (C ) H i+ (A ). (Thee is a slight notational ambiguity hee; depending on how you index you double complex, you long exact sequence might look slightly diffeent.) In paticula, we will use the fact that µ induces an isomophism on cohomology if and only if the mapping cone is exact. You ae now eady to go out into the wold and use spectal sequences to you heat s content!.7. Complete definition of the spectal sequence, and poof. You should most definitely not ead this section any time soon afte eading the intoduction to spectal sequences above. Instead, flip quickly though it to convince youself that nothing fancy is involved. We conside the ightwads oientation. The upwads oientation is of couse a tivial vaiation of this..8. Goals. We wish to descibe the pages and diffeentials of the spectal sequence explicitly, and pove that they behave the way we said they did. Moe pecisely, we wish to: (a) descibe E p,q, (b) veify that H k (E ) is filteed by E p,k p as in (), (c) descibe d and veify that d =, and (d) veify that E p,q + is given by cohomology using d. Befoe tacking these goals, you can impess you fiends by giving this shot desciption of the pages and diffeentials of the spectal sequence. We say that an element of E, is a (p, q)-stip if it is an element of l E p+l,q l (see Fig. ). Its non-zeo enties lie on a semi-infinite antidiagonal stating with position (p, q). We say that the (p, q)-enty (the pojection to E p,q ) is the leading tem of the (p, q)-stip. Let E, be the submodule of all the (p, q)-stips. Clealy E p+q, and S,k = E k. Note that the diffeential d = d + d > sends a (p, q)-stip x to a (p, q + )-stip dx. If dx is futhemoe a (p +, q + + )-stip ( Z ), we say that x is an -closed (p, q)-stip. We denote the set of such (so fo example =, and S,k = E k ). An element of 8
... p+,q p+,q p,q p,q+ FIGURE. A (p, q)-stip (in E p+q ). Clealy S,k = E k. may be depicted as:...? p+,q p+,q p,q.9. Peliminay definition of E p,q. We ae now eady to give a fist definition of E p,q, which by constuction should be a subquotient of E p,q = E p,q. We descibe it as such by descibing two submodules Y p,q X p,q E p,q, and defining E p,q = X p,q /Y p,q. Let X p,q be those elements of E p,q that ae the leading tems of -closed (p, q)-stips. Note that by definition, d sends ( )-closed S p (),q+() -stips to (p, q)-stips. Let Y p,q be the leading ((p, q))-tems of the diffeential d of ()-closed (p (), q+() )-stips (whee the diffeential is consideed as a (p, q)-stip). We next give the definition of the diffeential d of such an element x X p,q. We take any -closed (p, q)-stip with leading tem x. Its diffeential d is a (p +, q + )-stip, and we take its leading tem. The choice of the -closed (p, q)-stip means that this is not a well-defined element of E p,q. But it is well-defined modulo the ()-closed (p+, +)- stips, and hence gives a map E p,q E p+,q +. This definition is faily shot, but not much fun to wok with, so we will foget it, and instead dive into a snakes nest of subscipts and supescipts. 9
We begin with making some quick but impotant obsevations about (p, q)-stips..f. EXERCISE. Veify the following. (a) = S p+,q E p,q. (b) (Any closed (p, q)-stip is -closed fo all.) Any element x of = that is a cycle (i.e. dx = ) is automatically in fo all. Fo example, this holds when x is a bounday (i.e. of the fom dy). (c) Show that fo fixed p, q, stabilizes fo (i.e. = + = ). Denote the stabilized module Sp,q. Show is the set of closed (p, q)-stips (those (p, q)-stips annihilated by d, i.e. the cycles). In paticula, S,k is the set of cycles in E k... Defining E p,q. Define X p,q Then Y p,q := /S p+,q X p,q (6) E p,q and Y := ds p (),q+() /S p+,q. by Execise.F(b). We define = Xp,q Y p,q We have completed Goal.8(a). = ds p (),q+() You ae welcome to veify that these definitions of X p,q and Y p,q and hence E p,q agee with the ealie ones of.9 (and in paticula X p,q and Y p,q ae both submodules of E p,q ), but we won t need this fact..g. EXERCISE: E p,k p GIVES SUBQUOTIENTS OF H k (E ). By Execise.F(c), E p,q stabilizes as. Fo, intepet /ds p (),q+() as the cycles in E p+q modulo those bounday elements of de p+q contained in. Finally, show that Hk (E ) is indeed filteed as descibed in (). We have completed Goal.8(b)... Definition of d. We shall see that the map d : E p,q E p+,q + is just induced by ou diffeential d. Notice that d sends -closed (p, q)-stips to (p +, q + )-stips S p+,q +, by the definition -closed. By Execise.F(b), the image lies in S p+,q +.H. EXERCISE. Veify that d sends ds p (),q+() ds (p+) (),(q +)+(). + S (p+)+,(q +).
(The fist tem on the left goes to fom d =, and the second tem on the left goes to the fist tem on the ight.) Thus we may define d : E p,q = ds p (),q+() and clealy d twice). ds p+,q S p+,q + + S p++,q = E p+,q + = (as we may intepet it as taking an element of and applying d We have accomplished Goal.8(c)... Veifying that the cohomology of d at E p,q of veifying that the cohomology of is E p,q +. We ae left with the unpleasant job (7) S p,q+ ds p +,q 3 +S p +,q+ is natually identified with d ds p +,q+ +S p+,q ds p,q+ and this will conclude ou final Goal.8(d). + d S p+,q + ds p+,q +S p++,q Let s begin by undestanding the kenel of the ight map of (7). Suppose a is mapped to. This means that da = db+c, whee b S p+,q. If u = a b, then u, while du = c S p++,q S p++,q, fom which u is -closed, i.e. u +. Hence a = b + u + x whee dx =, fom which a x = b + c S p+,q + +. Howeve, x, so x + by Execise.F(b). Thus a Sp+,q + +. Convesely, any a S p+,q + + satisfies da ds p+,q + + d + dsp+,q + + S p++,q (using d + Sp++,q and Execise.F(b)) so any such a is indeed in the kenel of ds p+,q Hence the kenel of the ight map of (7) is ke = S p+,q + S p+,q ds p +,q+ + S p++,q + + Next, the image of the left map of (7) is immediately im = dsp,q+ + ds p +,q+ ds p +,q+.. = dsp,q+ ds p +,q+
(as S p,q + contains S p +,q+ ). Thus the cohomology of (7) is ke / im = S p+,q ds p,q+ + + = + + (dsp,q+ whee the equality on the ight uses the fact that ds p,q++ theoem. We thus must show Howeve, and + Sp+,q + (dsp,q+ + (dsp,q+ ) ) = ds p,q+. + and an isomophism ) = ds p,q+ + + Sp+,q consists of (p, q)-stips whose diffeential vanishes up to ow p +, = as desied. fom which + Sp+,q This completes the explanation of how spectal sequences wok fo a fist-quadant double complex. The agument applies without significant change to moe geneal situations, including filteed complexes. E-mail addess: vakil@math.stanfod.edu