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Spectral Sequences: an Introduction Spectral sequences: an Introduction Frederik Caenepeel In these notes we introduce the notion of a spectral sequence and give some basic prop- erties. Even from the definition of a spectral sequence itself, things could appear very technical, so some of the proofs and details will be omitted. For example, the Com- plete Convergence Theorem will not be studied in detail, as it is not directly needed. On the other hand, some known properties like the Universal Coefficient Theorem will be shown using spectral sequences in order to motivate their use and to get more fa- miliar with the subject. We basically follow [2, Ch. 5], but also recommend [1, Ch. XI]. 1 Terminology Definition 1.1 A homology spectral sequence (starting with Ea, a ≥ 0) in an abelian category A consist of the following data: r 1. A family fEpqg of objects of A defined for all integers p; q and r ≥ a; r r r r r 2. Maps dpq : Epq ! Ep−r;q+r−1 that are differentials in the sense that d d = 0, so r that the lines of slope (1 − r)=r in the lattice E∗∗ form chain complexes in A; r+1 r r 3. Isomorphisms between Epq and the homology of E∗∗ at the spot Epq: r+1 ∼ r r Epq = Ker(dpq)=Im(dp+r;q−r+1): r The total degree of the term Epq is n = p+q, so the terms of total degree n lie on a line r of slope −1, and each differential dpq decreases the total degree by one. A morphism f : E ! E0 between two homology spectral sequences is a family of maps r r 0r r r r r r+1 fpq : Epq ! Epq in A (for r suitably large) with d f = f d and such that each fpq is r the map induced by fpq on homology. It follows easily that there exists a category of homology spectral sequences in any abelian category A. r+1 ∼ r r r Epq = Ker(dpq)=Im(dp+r;q−r+1) is a subquotient of Epq, so we obtain a nested family a of subobjects of Epq: a r r+1 r+1 r a a 0 = Bpq ⊆ ··· ⊆ Bpq ⊆ Bpq ⊆ ··· ⊆ Zpq ⊆ Zpq ⊆ ··· ⊆ Zpq = Epq r ∼ r r such that Epq = Zpq=Bpq. Indeed, for r > a consider dr dr r r p+r;q−r+1 r r pq r r Zp+r;q−r+1=Bp+r;q−r+1 −! Zpq=Bpq−!Zp−r;q+r−1=Bp−r;q+r−1: r+1 ∼ r+1 r+1 r+1 r r r+1 r r Then Epq = Zpq =Bpq , where Zpq =Bpq = Ker(dpq), Bpq =Bpq = Im(dp+r;q−r+1) and r+1 r+1 Bpq ⊆ Zpq , by which r r+1 r+1 r Bpq ⊆ Bpq ⊆ Zpq ⊆ Zpq: 1 Appendix D. Spectral Sequences 417 Therefore dh (y) is a vertical cycle in Cp_ 2 q+l and determines a class z = [dh(y)] E Cp-2 q+l +-- Cp-1 q+l 1 Cp-1q +-- Cpq Let us compute d1 (z), Therefore z is a cycle (for d1 ) and determines an element d2 [x] := [z] E FJ,- 2 q+l· We leave to the reader the task of proving that the resulting map d"' : E;q E;_ 2 q+l is well-defined (independent of the choice of x for in- stance) and that it is a differential. Note that in the notation the figure 2 is an index and does not mean do d here. D.3 The Spectral Sequence. The first part of the theory of spectral se- quences consists in proving that the construction of (E2 , can be pushed further to get an infinite sequence of modules E;q and differentials d: : E;q E;_r q+r-1 related to one another by the condition q p Note that if, for someFigure fixed 1: The (p, differentials q), it ever of a homology happens spectral that sequence. E;q = O for some r, then = O for any r' :2: r. In particular, since Cpq = O for p < O or q < O, it follows that E;q = O for p < O or q < O, any r. As a consequence, If our abelian category A is complete and cocomplete (e.g. A = RM ), we introduce for fixed (p, q),the intermediatethe differentials objects ending at and starting from E;q are O if r is large enough (say r = r(p,q)) and¥ therefore ¥ ¥ [ r ¥ \ r Bpq = Bpq and Zpq = Zpq Epq-r _ Er+1r=pqa -_ ... -_ E""pqr= a ¥ ¥ ¥ ¥ and define Epq = Zpq=Bpq. We say that the spectral sequence abuts to Epq. We consider r the terms Epq of the spectral sequence as successive approximations (via successive for r :2: r(p, q). ¥ formation of subquotients) to Epq. r Example 1.2 A first quadrant (homology) spectral sequence is one with Epq = 0 unless p ≥ 0 and q ≥ 0, that is, the point (p;q) belongs to the first quadrant of the plane. At the r r-stage, the differentials d∗∗ go r columns to the left and r − 1 rows up, so for suitably s large r all differentials d∗∗ for s ≥ r leaving and ending at the (p;q)-spot are zero (take r for example r > (p + 1) _ (q + 2)). This means that Epq is ultimately constant in r and ¥ for this stabilized value we clearly have that it equals Epq. We also have a cohomological analogue: Definition 1.3 A cohomology spectral sequence (starting with Ea, a ≥ 0) in an abelian pq pq category A is a family fEr g of objects (r ≥ a; p;q 2 Z), together with maps dr : pq pq p+r;q−r+1 dr : Er ! Er ; which are differentials and give isomorphisms between Er+1 and the homology of Er. ∗∗ pq For such a spectral sequence, the differential dr increases the total degree p+q of Er pq pq pq by one and one defines the objects B¥ ;Z¥ and E¥ analogously. Let’s consider a special type of spectral sequences, namely the bounded ones. Definition 1.4 A homology spectral sequence is said to be bounded if for each n there a are only finitely many nonzero terms of total degree n in E∗∗. 2 418 Appendix D. Spectral Sequences q o p o r Let us now consider the homologyFigure 2: d pqofultimately the total becomes complex zero. Tot C**. There is a canonica! increasing filtration on this complex obtained by taking the first columns: r Because d∗∗ decreases the total degree by one this means that for all p and q there is r r+1 an r0 such that (FiTotEpq = Epq C**)nfor all r ≥= r0.EB As in theCpq example . of a first quadrant spectral ¥ sequence this stable value equals Epq.p+q=n For such spectral sequences we introduce the notion of convergence p<i This induces an increasing filtration (stiH denotedr Fi) onAn= Hn(Tot C**), Definition 1.5 A bounded spectral sequence fEpqg converges to H∗ if there exists a family of objects Hn of A , each having a finite filtration 0 = FsHn ⊆ ··· ⊆ Fp−1Hn ⊆ FpHn ⊆ Fp+1Hn ⊆ ··· ⊆ Ft Hn = Hn; ¥ ∼ D.4 Theorem.such The that Emodulespq = FpHp+ q=Fp−1andHp+q. the We writefiltered down thismodule bounded An= convergence Hn(Tot by C**) a are related by Epq ) Hp+q: Dually, a cohomology spectral sequence is called bounded if there are only finitely o ∗∗ many nonzero terms in each total degree in Ea and such a spectral sequence converges In the literatureto H∗ if this there istheorem a finite filtration is often for each writtenn under the following form: there exists a spectral sequence 0 = Ft Hn ⊆ ··· ⊆ F p+1Hn ⊆ F pHn ⊆ ··· ⊆ FsHn = Hn; E;qwith = H;(H:(c**))::::} Ap+q = Hp+q(TotC**). pq ∼ p p+q p+1 p+q E¥ = F H =F H : The symbol ::::} is to be read "converges to". Most of the time the filtration of r the abutment AnExample is not 1.6 specified,Let fEpqg be abut first quadrantin many homology applications spectral sequence, it is whichnot con-needed verges to H∗. Because for any n we have (see for instance D.6 to D.9). ¥ Note that one could take on TotEp;n −C**p = 0the for filtrationp ≤ 0 or p > byn; columns. This would give a second spectralHn has a finite sequence filtration of of length then +form1: 0 = F−1Hn ⊆ F0Hn ⊆ ··· ⊆ Fn−1Hn ⊆ FnHn = Hn: = H;(H;(C**))::::} Ap+q = Hp+q(Tot C**) . Though the abutment An is the same, the filtration3 is, in general, different. D.5 Leray-Serre Spectral Sequence. There are more general data giving rise to spectral sequences. In particular, instead of starting with a bicomplex, one can start with a filtered complex. Hence one can define E 1 , E 2 , etc and Theorem D.4 expresses the relation between the homology of the graded complex versus the graded module of the homology of the complex. In particular let ¥ ∼ The bottom piece F0Hn = E0n is located on the y-axis, and the top quotient Hn=Fn−1Hn = ¥ En0 is located on the x-axis. Since each arrow landing on the x-axis is zero, and each ¥ a ¥ arrow leaving the y-axis is zero, each En0 is a subobject of En0 and each E0n is a quo- a r r tient of E0n. The terms En0 on the x-axis are called the base terms and the terms E0n on the y-axis are called the fiber terms.
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