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Examples of Spectral Sequences Examples of spectral sequences Man Cheung, Yao-Rui April 16, 2019 Spectral sequences are a collection of useful tools that helps compute the cohomology of various algebraic or topological objects. In this note we will only give examples of first-quadrant cohomological spectral sequences. Dual notions exists for homology and for other quadrants, and one can see [7, Chapter 5] and [3] for definitions and examples. 1 Introduction We briefly introduce the notion of a spectral sequence in this section. Since the construction of spectral sequences is not very illuminating, we have chosen to spend more time on examples; see the next few sections for some of them. Here is an abstract definition of a spectral sequence. Definition 1. Let R be a ring, and let r0 be a fixed nonnegative integer. A (first-quadrant cohomological) spectral sequence E consists of the following data: p;q • a collection of R-modules Er , with integers p; q ≥ 0 and r ≥ r0, p;q p;q p+r;q−r+1 • and a collection of differentials dr = dr : Er −! Er such that p;q 2 p;q p;q (dr ) = 0 and Er+1 is the homology at Er , i.e. ker d : Ep;q −! Ep+r;q−r+1 Ep;q = r r r : r+1 p−r;q+r−1 p;q im dr : Er −! Er p;q th The collection Er = f(Er ; dr): p; q ≥ 0g for a fixed r is called the r -page. The spectral sequence converges if Es = Es+1 = Es+2 = ··· for a large enough s, and we denote this page by E1. Remark. We may sometimes assign an algebra structure to a page Er of a spectral sequence. In this case, the algebra structure ^ needs to respect the p;q p0;q0 p+p0;q+q0 grading, i.e. ^: Er × Er −! Er . The reason why spectral sequences are useful is because of the following definition, which characterizes results that are related to spectral sequences. The examples in this note will make clear of this definition. 1 Definition 2. A filtered chain complex is a chain complex C = (Ci; δi) of R- modules together with a differential-preserving filtration on each Ci, i.e. for each Ci, there is an increasing sequence of R-modules F0Ci ⊂ F1Ci ⊂ · · · ⊂ Fp−1Ci ⊂ FpCi ⊂ Fp+1Ci ⊂ · · · such that [ \ FpCi = Ci; FpCi = 0; δi(FpCi) ⊂ FpCi+1: p p ∗ A filtration on the chain complex induces a filtration FpH (C) on the cohomol- ogy groups H∗(C). The associated graded pieces are defined by n n n GpH = FpH (C)=Fp+1H (C): A spectral sequence E converges to H∗(C), written Ep;q ) Hp+q(C), if E con- r0 verges and p;q ∼ p+q E1 = GpH : As the definition implies, a convergent spectral sequence does not compute the cohomology groups we want, but rather a filtration of it. Hence we have the following question, called the lifting problem. n M p;q How is H (C) related to E1 ? p+q=n There are many instances where the two terms are equal. An instance is when the diagonal p + q = n contains exactly one nontrivial term, which we will see a lot of in the next section. If a nontrivial algebra structure is involved, the following easy proposition will guarantee equality. Proposition 3. If the E1-page is a free graded commutative algebra, then n T ot(E1) equals H (C). We end the introduction by stating a consequence of the definition of spectral sequences, which follows by diagram chasing. Proposition 4. Given a spectral sequence starting from the E2-page and con- p;q p+q verging to the cohomology of C (so E2 ) H (C) in the notation of Definition 2), there is an associated five-term exact sequence 1;0 1 0;1 d2 2;0 2 0 −! E2 −! H (C) −! E2 −−! E2 −! H (C): 2 Three examples of spectral sequences In this section we state three examples of spectral sequences, and give examples for each of them. One can refer to the references for proofs of these spectral sequences. 2 Double complexes Theorem 5 (Double complex spectral sequence [6, Section 1.7]). Let C = Cp;q be a double complex over an abelian category. Then there exists spectral se- quences p;q p;q p;q q;p E0 = C ;E0 = C ; p+q p;q both converging to H (T ot(C)). The differentials in the respective E0 -pages are induced from the maps in the double complex C. Example 6. Let us use the above theorem to prove the Snake Lemma, which says that the following commutative diagram with exact rows 0 / A0 / B0 / C0 / 0 O O O f g h 0 / A / B / C / 0 gives rise to the exact sequence δ 0 / ker f / ker g / ker h / coker f / coker g / coker h / 0: To do this, we first use the second spectral sequence, with E0-page as follows. 0 0 0 AA0 0 BB0 0 CC0 0 0 0 0 By exactness, this spectral sequence converges to zero. Let us now look at the first spectral sequence, with the following E0-page. 0 0 0 0 0 0 A0 B0 C0 0 0 ABC 0 The E1-page will look like 0 0 0 0 0 0 coker f coker g coker h 0 0 ker f ker g ker h 0 and the E2-page is as follow. 0 0 0 0 0 0 ker(coker f −! coker g) 0 0 0 0 0 0 (ker h)=(ker g) 0 3 Since the spectral sequence converges to zero, we have an isomorphism d : ker(coker f −! coker g) −!(ker h)=(ker g): Therefore δ := d−1 induces the connecting homomorphism in the Snake Lemma, while the other maps are the natural ones (which also corresponds to the differ- entials in the E1-page above). Fibrations The next spectral sequence we will look at is associated to a fibration, and is usually given as the first example of a spectral sequence. Theorem 7 (Serre spectral sequence, [3, Theorem 5.1]). Let F −! E −! B be a fibration with B simply connected, and let R be a ring. Then there is a first quadrant spectral sequence of algebras p;q p q p+q E2 = H (B; H (F ; R)) ) H (E; R): The left hand side is the cohomology of B with local coefficients in the cohomol- ogy of F with respect to R. Furthermore, the multiplication structure on E2 is compatible with the cup product structure on cohomology. Example 8. Let K(Z; n) be the Eilenberg-Maclane space with πn(K(Z; n)) = Z and πi(K(Z; n)) = 0 otherwise. We now show by induction that the cohomology ring ( ∗ Q[z]; if n is even; H (K(Z; n); Q) = Q[z]=z2 if n is odd; where jzj = n. (This is still true if we replace Q by a field of characteristic 0.) For n = 1 one has K(Z; 1) = S1, so this is clearly true. For n ≥ 2 we consider ΩK(Z; n) −! PK(Z; n) −! K(Z; n); where ΩK(Z; n) ' K(Z; n − 1) is the loop space and PK(Z; n) ' ∗ is the path space of K(Z; n). Note that π1(K(Z; n)) = 1, so by the Serre spectral sequence p;q p q p+q E2 = H (K(Z; n); Q) ⊗Q H (K(Z; n − 1); Q) ) H (∗; Q): This implies that the E1 page of the spectral sequence is Z at coordinate (0; 0) and zero everywhere. 0;n−1 Let us first consider the case where n ≥ 2 is even. Then E2 is generated by some fixed 2-torsion element z 2 Hn−1(K(Z; n − 1)), and for the page to converge to 0 along this diagonal we require dn(z) = x with dn an isomorphism. n;n−1 Thus E2 is generated by xz, and by the Leibniz rule n 2 dn(xz) = dn(x)z + (−1) xdn(z) = x ; 4 since dn(x) = 0. Inductively, we get the following diagram on the E2-page. 0 0 0 0 0 0 0 0 0 0 z 0 ::: 0 xz 0 ::: 0 x2z 0 0 0 ::: 0 0 0 ::: 0 0 0 . dn. dn. 0 0 ::: 0 0 0 ::: 0 0 0 1 0 ::: 0 x 0 ::: 0 x2 0 This gives us what we want for n ≥ 2 even. Next we consider the case where 0;k(n−1) n;(k−1)(n−1) n ≥ 2 is odd. Then E2 must be isomorphic to E2 for any positive integer n, and the rest of the E2 page is zero except at (0; 0). Similar 0;n−1 n−1 to before, E2 is generated by z 2 H (K(Z; n − 1)), and dn(z) = x with 0;n−1 n;0 dn : E2 −! E2 an isomorphism. Now 2 n−1 dn(z ) = dn(z)z + (−1) zdn(z) = 2xz; n;n−1 and so E2 is generated by xz (since Q has characteristic zero). 0 0 ::: 0 0 0 ::: 1 2 2 2 z 0 ::: 0 xz 0 ::: 0 0 ::: 0 0 0 ::: . dn. 0 0 0 0 0 0 ::: z 0 ::: 0 xz 0 ::: 0 0 ::: 0 0 0 ::: . d . n . 0 0 ::: 0 0 0 ::: 1 0 ::: 0 x 0 ::: th Since the 2n column of the E2 page is zero, we conclude that x must be 2-torsion, as desired. Example 9.
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