SPECTRAL SEQUENCES. im(er

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1 SPECTRAL SEQUENCES MATTHEW GREENBERG. Intoduction Definition. Let a. An a-th stage spectal (cohomological) sequence consists of the following data: bigaded objects E = p,q Z Ep,q, a diffeentials d : E E such that d (E p,q satisfying H(E ) = E +, i.e., + = ke(ep,q E p+,q + ) im(e p,q+ E p,q ). ) E p+,q + We usually daw the -th stage of a spectal sequence in a tabula fomat with p inceasing hoizontally to the left and q inceasing vetically to the ight: E 0, 0 E, 0 E, 0 E 3, 0 E 0,0 0 E,0 0 E,0 0 E 3,0 0 E 0, E, E, E 3, E 0,0 E,0 E,0 E 3,0 E 0, E, E, E 3, E 0,0 E,0 E,0 E 3,0. The spectal sequence of a filteed complex Let K = F 0 K F K be a filteed complex (i.e., each object K n in the complex K is filteed and the diffeentials of the complex K espect the filtation). We set G p K = F p K /F p+ K. Note that the filtation on K induces a filtation H(K ) = F 0 H(K ) F H(K ) on cohomology in a natual way. Given this setup, one may pove the following. Theoem. Suppose that K is a nonnegative filteed complex (i.e., K n = 0 if n < 0). Then thee exists a spectal sequence satisfying

2 MATTHEW GREENBERG 0 = G p K p+q = H p+q (G p K ) E p,q = G p H p+q (K ) fo = (p, q) 0. Sketch of poof. Set 0 = G p K p+q. As diffeential of the complex K espects the filtation, it induces a map d 0 : To ealize E p,q,, one defines the set Z p,q = x F p K p+q dx F p+ K p+q+ } of cocycles modulo F p+ K p+q+ and sets = Z p,q /(appopiate coboundaies). The diffeential d then induces the map d : E E (it is at least evident fom the above that we get a map Z p,q Z p+,q + ). It emains to check the desied popeties of these object. We omit this; fo details see [3, Ch. XX, Poposition 9.]. The spectal sequence whose existence is asseted in the above theoem is an example of a fist quadant spectal sequance, by definition a spectal sequence such that E p,q is zeo unless p, q 0. It is easy to see that in a fist quadant spectal sequence, E p,q = + = if > max(p, q + ). In the situation of the theoem, this stable value is G p H p+q (K ). 3. Convegence of spectal sequences Let E p,q be a spectal sequence, and suppose that fo evey pai (p, q), the tem E p,q stabilizes as (a fist quadant spectal sequence, fo example). Denote this stable value by E p,q. Let H n be a collection of objects with finite filtations We say that 0 F s H n F t H n = H n. conveges to H, and wite E p,q H p+q, if E p,q = F p H p+q /F p+ H p+q = G p H p+q. Example 3. Suppose K is a nonnegative filteed complex. Then by Theoem, we have a convegent spectal sequence = H p+q (G p K ) H p+q (K ). Given this definition of convegence, one is led immediately to ask to what extent the limit of a spectal sequence is detemined by the sequence itself. The next two lemmas give some basic though useful esults in this diection. Lemma 4. Suppose E p,q H p+q. () If E p,q = 0 unless q = q 0, then H n = E n q0,q0 () If E p,q = 0 unless p = p 0, then H n = E p0,n p0.. Poof. We pove (), the poof of () being simila. Conside the filtation 0 F n q0+ H n F n q0 H n F n q0 H n H n. Since E p,q = 0 unless q = q 0, the only nonzeo quotient of this filtation is F n q0 H n /F n q0+ H n = E n q0,q0. Thus, F n q0+ H n = 0 and F n q0 H n = H n, implying H n = F n q0 H n = H n /F n q0+ H n = E n q0,q0.

3 SPECTRAL SEQUENCES 3 The next esult shows that in cetain cases the fom of the filtation on the limit object is detemined by the spectal sequence. Lemma 5. Suppose E p,q H p+q. () If E p,q is a fist quate spectal sequence, then H n has a filtation of the fom () If E p,q fom 0 = F n+ H n F n H n F H n F 0 H n = H n. is a thid quate spectal sequence, then H n has a filtation of the 0 = F H n F 0 H n F n+ H n F n H n = H n. A thid quate spectal sequence is one in which = 0 unless p, q 0. Poof. Again we only pove (), the poof of () being simila. We must show that F n+k H n = 0 if k > 0 and F k H n = H n if k 0. Conside the left tail of the (finite!) filtation of H n, As 0 F n+ H n F n+ H n F n H n. is a fist quate spectal sequence, F n+k H n /F n+k+ H n E n+k, k = 0 if k > 0 = E n,0 if k = 0. Theefoe, F n+k H n = 0 if k > 0. One shows that F k H n = H n if k 0 in a simila fashion by consideing the ight tail of the filtation. 4. The spectal sequence of a double complex In this section, we teat one of the most common ways spectal sequences aise fom a double complex. Definition 6. A double complex M consists of a bigaded object M = p,q Z M p,q togethe with diffeentials d : M p,q M p+,q and δ : M p,q M p,q+ satisfying d = δ = dδ + δd = 0. Example 7. Let R be a ing (P, d P ) and (Q, d Q ) be complexes of R-modules. Define a double complex M = P R Q by M p,q = P p R Q q, d = d P, and δ = ( ) p d Q : P p R Q q P p R Q q+. To each double complex M, we attach a (single) complex Tot M called its total complex defined by Tot n M = M p,q. p+q=n The diffeential D on this total complex is given by D = d + δ. Notice that D = (d + δ) = d + δ + dδ + δd = 0, i.e., (Tot M, D) is a complex. Thee ae two canonical filtations on the total complex Tot M of a double complex M given by F p Tot n M = M,s and F q Tot n M = M,s. +s=n p +s=n s q By the theoem of Section, the filtations F p Tot n M and F p Tot n M detemine spectal sequcences E p,q and E p,q, espectively. One obseves easily that 0 =

4 4 MATTHEW GREENBERG M p,q and checks (pehaps not so easily) that the diffeential aising fom the constuction of the spectal sequence of a filtation is simply given by δ. Thus, = H q δ (M p, ). The diffeential E p+,q is induced by d, viewed as a homomophism of complexes M p, M p+,, implying that is equal to the p- th homology goup of the complex (H q δ (M p, ), d). We will wite this simply (though slightly ambiguously) as H p d (Hq δ (M)). One may obtain simila expessions fo, = 0,,. We summaize these impotant esults in the following theoem. Theoem 8. Let M be a double complex with total complex Tot M. Then thee exist two spectal sequences E p,q and E p,q (coesponding to the two canonical filtations on Tot M) such that 0 = M p,q, 0 = M q,p, (M p, ), = H q δ = H q d (M,p ), = H p d (Hq δ (M)), = H p δ (Hq d (M)). Futhe, if M is a fist o thid quadant double complex, then both E p,q convege to H p+q (Tot M). and Example 9. Let R be a ing. In what all follows, all tenso poducts ae taken ove R. We shall use the spectal sequences attached to a double complex to show that fo R-modules A and B, we have equality of left deived functos L p ( B)(A) = L p (A )(B). Let d P d P d P 0 δ A and Q δ Q δ Q 0 B be pojective esolutions of A and B, espectively, and conside the double complex P Q. We use negative indexing on these spectal sequences in ode to stay in a cohomological, as opposed to a homological, situation. Then = H q δ (P p Q ). Since P p is a pojective and hence flat R-module, one can show in an elementay fashion that H q δ (P p Q ) = P p H q δ (Q ). Since Q is a pojective esolution of B, we have Theefoe, = In like manne, one computes = = P p B if q = 0, 0 othewise. L p ( B)(A) if q = 0, 0 othewise. L p (A )(B) if q = 0, 0 othewise. Using Lemma 4 and the fact that both of these (thid quadant!) spectal sequences convege to H p+q (Tot P Q ), we have L p ( B)(A) = E p,0 = E p,0 = H p (Tot P Q ) = E p,0 = E p,0 = L p (A )(B). Example 0. Let M be an R-module and P be a nonpositive complex of flat R- modules (i.e., P p = 0 if p > 0). Again, all tenso poducts and To s ae taken ove R. The elation between H (P M) and H (P ) M is encoded by a spectal sequence.

5 SPECTRAL SEQUENCES 5 Poposition. Let M and P be as above. Then thee exists a thid quadant spectal sequence = To p (H q (P ), M) H p+q (P M). δ Poof. Take a pojective esolution Q δ Q δ Q 0 M of M. We conside the double complex P Q and its associated spectal sequences. By the same agument as in the pevious example, H p (P M) if q = 0, = 0 othewise. Hee, we have used the flatness of P. Theefoe, by Lemma 4, H p+q (Tot P Q ) = = E p+q,0 = H p+q (P M), and we have identified the limit tem of the spectal sequence whose existence the poposition assets. Now let us conside the spectal sequence E p,q. We have = H q (P Q p ) = H q (P ) Q p, as Q p is pojective and hence flat. Theefoe, = H p δ (Hq (P ) Q p ) = L p (H q (P ) )(M) = To p (H q (P ), M). Since H p+q (Tot P Q ) = H p+q (P M), we ae done. This spectal sequence contains much useful infomation, the extaction of which is the topic of the next section. 5. Getting infomation out of spectal sequences 5.. A univesal coefficients theoem. Conside again the situation of Example 0 in the special case whee the ing R is a pincipal ideal domain. In this case, = 0 unless p = 0,. In such a situation, one may apply the following lemma to obtain an exact sequence fom the spectal sequence of Poposition. Lemma. Suppose H p+q is a thid quate spectal sequence, and that = 0 unless p = 0,. Then we have an exact sequence fo all n 0. 0 E 0, n H n E, n+ 0, Poof. By Lemma 5, the filtation on H n has the fom 0 = F H n F 0 H n F n+ H n F n H n = H n. Combining this with the fact that F k H n /F k+ = E k, n+k = E k, n+k = 0 fo k n, it follows that F H n = H n and F 0 H n = E 0, n. Now substitute these values in the exact sequence 0 F 0 H n F H n E, n+ 0. Poposition and Lemma yield the univesal coefficients theoem fo homology.

6 6 MATTHEW GREENBERG Coollay 3. Let R be a pincipal ideal domain, and let M, and P be as in poposition. Then fo all n 0, we have an exact sequence 0 H n (P ) R M H n (P R M) To R (H n+ (P ), M) Edge maps and tems of low degee. Let H p+q be a fist quadant spectal sequence. By Lemma 5, F n+ H n = 0, implying E n,0 = F n H n /F n+ H n = F n H n H n. Fo each, the diffeential leaving E n,0 is the zeo map. Theefoe, we have sujections E n,0 E n,0 + En,0. The composition E n,0 E n,0 H n is called an edge map, as the tems E n,0 stage diagam. Similaly, E 0,n lie along the bottom edge of the -th = F 0 H n /F H n = H n /F H n by Lemma 5, giving a ae all zeo, we have E0,n. The composition sujection H n E 0,n. As the diffeentials mapping into E 0,n inclusions E 0,n E 0,n + H n E 0,n E 0,n is also called an edge map. When = and n is small, we can be moe specific about the kenels and images of these edge maps. Poposition 4 (Exact sequence of tems of low degee). Let H p+q be a fist quadant spectal sequence. Then the sequence 0 E,0 e H e E 0, d E,0 e H is exact, whee d is the E -stage diffeential and the aows labelled e ae the edge maps descibed above. Poof. By Lemma 5, F H = 0. Theefoe, E,0 = E,0 = F H /F H = F H H, and the edge map E,0 H is injective with image F H. The edge map H E 0, is the composition H E 0, 3 = E 0, E 0,. Theefoe, ke(h E 0, ) = ke(h E 0, ) = F H, poving exactness at H. Also, the image of H in E 0, is pecisely E 0, 3 = ke(e 0, d E,0 ), poving exactness at E0,. As E,0 3 = F H /F 3 H = F H H, we have ke(e,0 H ) = ke(e,0 E,0 3 ) = im(e0, E,0 ), completing the veification. 6. The Gothendieck spectal sequence Let A, B, and C be abelian categoies with enough injectives and let A G B F C left exact covaiant functos (so we may fom thei ight deived functos). It is natual to ask if thee is a elationship between ight deived functos of F G and those of F and G. Unde a cetain technical hypothesis, this elationship exists and is encoded in the Gothendieck spectal sequence.

7 SPECTRAL SEQUENCES 7 Theoem 5 (The Gothendieck spectal sequence). Suppose that GI is F -acyclic fo each injective object I of A (i.e., R p F (GI) = 0 if p > 0). Then fo each object A of A, thee exists a spectal sequence = (R p F )(R q G)(A) R p+q (F G)(A). Let C be a complex and let C I,0 I, be an injective esolution (i.e., each tem of each complex I,j is injective). I 0, I, I, I 0,0 I,0 I,0 C 0 C C Fom this aay, we can extact complexes Z p (C ) Z p (I,0 ) Z p (I, ), B p (C ) B p (I,0 ) B p (I, ), H p (C ) H p (I,0 ) H p (I, ). and We shall say that I, is a fully injective esolution of C if the above complexes ae injective esolutions of Z p (C ), B p (C ), and H p (C ), espectively. Such things exist: Lemma 6. Suppose A has enough injectives. Then any complex in A has a fully injective esolution. Poof. [3, Ch. 0, Lemma 9.5] Sketch of poof of Theoem 5. Let A be an object of A, and let 0 A C be an injective esolution of A. Let I, be a fully injective esolution of GC. We examine the spectal sequences associated to the double complex F I,. We have = H q (F I p, ) = R q F (GC p ) = as GC p is F -acyclic. Theefoe, = (F G)C p if q = 0, 0 othewise, H p ((F G)C ) = R p (F G)(A) if q = 0, 0 othewise, and consequently, H p+q (Tot F I, ) = R p (F G)(A), by Lemma 4. To complete the poof, it suffices to show that = (R p F )(R q G)(A). Evidently, = H q (F I,p ). Using the fact that eveything in sight is injective, one may show that H q (F I,p ) = F H q (I,p ). Theefoe, = H p (F H q (I,p )). Now H q (I,p ) is an injective esolution of H q (GC ) = R q G(A) by the full injectivity of I,. Theefoe, = H p (F H q (I,p )) = (R p F )(R q G)(A), as desied.

8 8 MATTHEW GREENBERG Example 7 (Base change fo Ext). Let R S be a ing homomophism (endowing S with an R-module stuctue) and let A be an S-module. Let G = Hom R (S, ) and F = Hom S (A, ), and conside the diagam R-modules G S-modules F Abelian Goups. F and G ae left exact, covaiant functos, and F G(B) = Hom S (A, Hom R (S, B)) = Hom R (A S S, B) = Hom R (A, B), i.e., F G = Hom R (A, ). If I be an injective R-module, then Hom S (, Hom R (S, I)) = Hom R ( S S, I) = Hom R (, I). So Hom S (, Hom R (S, I)) is exact and Hom R (S, ) sends injectives to injectives. Theefoe, Theoem 5 implies that thee exists a spectal sequence = Ext p S (A, Extq R (S, B)) Extp+q R (A, B). Now suppose S is a pojective R module. In this case, Hom R (S, ) is exact, implying Ext q R (S, B) = Rq Hom R (S, )(B) = 0 if q > 0. That is, the above spectal sequence collapses at the E -stage. Theefoe, by Lemma 4, we obtain the identity fo any S-module A and R-module B. Ext p S (A, Hom R(S, B)) = Ext p R (A, B) 7. The Leay spectal sequence In this section, we constuct the Leay spectal sequence, an essential tool in moden algebaic geomety. Its constuction is an application of Theoem 5. Let f : X Y be a continuous map of topological spaces. Then we have a commutative diagam f Sheaves X Sheaves Y Γ(X, ) Γ(Y, ) Ab whee Ab, f and Γ denote the categoy of abelian goups, sheaf pushfowad, and the sections functo, espectively. It is well known that the categoy of sheaves of abelian goups on a topological space has enough injectives. Also, f and Γ(Y, ) ae left exact, covaiant functos. We claim that f sends injectives to injectives (and thus, to Γ(Y, )-acyclics). To see this, we shall use the following useful lemma. Lemma 8. Let G : A B and F : B A be covaiant functos. Futhe suppose that F is ight adjoint to G and that G is exact. Then F sends injectives to injectives. Poof. Let I be an injective object of B. We must show that Hom(F I, ) is exact. Let 0 A A A 0 be an exact sequence in A. The the by the exactness of G, the sequence 0 GA GA GA 0 is exact. By the injectivity of I, the sequence 0 Hom(GA, I) Hom(GA, I) Hom(GA, I) 0 is exact. The adjointness popety of F and G completes the agument.

9 SPECTRAL SEQUENCES 9 If is a fact that f is ight adjoint to the sheaf invese image functo f an exact functo (see [, Ch., Execise.8]). Theefoe, by the lemma, f sends injectives to injectives, and by Theoem 5, fo any sheaf F on X, thee exists a spectal sequence = H p (Y, R q f F) H p+q (X, F) with exact sequence of tems of low degee 0 H (Y, f F) H (X, F) Γ(Y, R f F) H (Y, f F) H (X, F). The above spectal sequence is called the Leay spectal sequence. Fo a nice application of this spectal sequence to geomety, see [], whee it is used to analyze cetain biational isomophisms between sufaces. 8. Goup cohomology and the Hochschild-See spectal sequence Let G be a finite goup and let A be a G-module (equivalently, a Z[G]-module). Let G-mod and Ab denot the categoies of G-modules and abelian goups, espectively. We conside the G-invaiants functo inv G : G-mod Ab, A A G which to a G-module A associates its subgoup of elements invaiant invaiant unde the action of G. It is easy to see that inv G is a left exact, covaiant functo, so we may take its ight deived functos. We define H n (G, A) = R n inv G (A) and call this the n-th cohomology goup of G with coefficients in A. If H is a nomal subgoup of G and A is a G-module (and thus, also an H-module, then A H natually has the stuctue of a G/H-module. In fact, the diagam G-mod inv H G/H-mod inv G inv G/H Ab is easily seen to be commutative. In ode to apply Theoem 5 to this situation, we shall veify that inv H sends injectives to injectives. Evey G/H-module is also a G-module in a natual way. Let ρ : G/H-mod G-mod be this functo, easily checked to be exact. It follows tivially that inv H is ight adjoint to ρ and theefoe, by Lemma 8, the functo inv H peseves injectives. Thus, by Theoem 5, thee exists a spectal sequence = H p (G/H, H q (H, A)) H p+q (G, A) with exact sequence of tems of low degee 0 H (G/H, A H ) inf H (G, A) es H (H, A) G/H t H (G/H, A H ) inf H (G, A) es H (H, A) G/H. The maps inf, es, and t ae called inflation, estiction, and tansgession, espectively. Woking with a moe down-to-eath desciption of these cohomology goups in tems of cetain cocycles and coboundaies, one may explicitly descibe these maps; see [3, Ch. XX, Execise 6].

10 0 MATTHEW GREENBERG Refeences [] A. Achibald, Spectal sequences fo sufaces, Unpublished. [] R. Hatshone, Algebaic Geomety, Gaduate Texts in Mathematics 5, Spinge, 977. [3] S. Lang, Algeba, Addison-Weseley Publishing Company, 993.