What is a spectral sequence?

David Hansen

June 5, 2012

Very short answer: A spectral sequence is a piece of terminology used by algebraists to intimidate other mathematicians. Short answer / slogan: A spectral sequence is a machine which breaks difficult homolog- ical calculations apart into many easier (?) ones, and then glues their results together. Long answer: A (first quadrant cohomology) Er-page spectral sequence consists of the following data:

i,j i,j • AcollectionofabeliangroupsEs for integers i ≥ 0,j≥ 0, s ≥ r.ThedataofEs for s fixed and varying i, j is referred to as the Es-page. i,j i,j i+s,j+1−s • Homomorphisms of abelian groups ds : Es → Es ,thedifferentials,such i−s,j−1+s i,j i,j that imds ⊂ ker ds .ThedifferentialsintoEs are trivial for s>i,andthe i,j differentials out of Es are trivial if s>j+1.

i,j i,j ker ds • Canonical isomorphisms Es+1 $ i−s,j−1+s . imds i,j You can and should think of the Es ’s for fixed s as indexed by integer points in the first quadrant of the plane; the condition on the differentials saysthatthegroupslyingalongeach s−1 line of slope − s form a complex, and the Es+1-page formed by taking the cohomology i,j i,j of the Es-page. Note also that each Es+1 is a quotient of a subgroup of Es ,andthat i,j eventually every differential entering or leaving a fixed entry Es is identically zero, because either the i-coordinate of the source or the j-coordinate of the target is negative. In other i,j words, the entries Es evolve as you “turn the pages”, but eventually they stop changing: i,j i,j they stabilize, and we write E∞ for the stable value of Es .Thefollowingpicturemustbe

1 regarded: 0,3 E•

0,2 d1 / E0,2 / E1,2 E2,2 E3,2 • MRMRR • • • MMRRR d0,2 MMMRRR2 MM RRR 0 1 MM RR d , MM RR( 0,1 1 / 1,1 MMM 2,1 3,1 E• RR E• 0,2 MME• E• R 0,1 d MM RRR d 3 MM RRR2 MM RRR MMM 0 0 R1R0 M d , d ,RR( M& 0,0 1 / 1,0 1 / 2,0 3,0 E• E• E• E•

Note that no entry on the E1-page has stabilized, since a differential leaves every entry 0,0 1,0 on the page. However, on the E2-page, the entries E2 and E2 have stabilized, so e.g. 1,0 1,0 E∞ = E2 . ∗ n This mass of data abuts to an abelian group H = ⊕n≥0H if there exists a descending chain of subgroups

Hn =Fil0Hn ⊃ Fil1Hn ⊃···⊃FiliHn ⊃···⊃FilnHn ⊃ Filn+1Hn =0

i+1 n i n i,n−i n together with canonical isomorphisms Fil H \Fil H $ E∞ ;morecolloquially,H is i,j determined up to extension data by the groups E∞ with i + j = n.Allllllthisinformation is traditionally abbreviated into the notation

i,j i+j Er ⇒ H .

The vast majority of spectral sequences are E2-page spectral sequences, which is to say you begin with the data of the entries on the E2-page. Examples. (Serre) Let f : E → B be a continuous map of topological spaces (say of CW com- −1 plexes). This map f is a Serre fibration if the fibers F = Fb = f (b),b∈ B “satisfying the homotopy lifting property for CW complexes” - fiber bundles are a basic example. Given i aSerrefibration,thecohomologygroupsH (Fb, Q) naturally form the stalks of a locally constant sheaf Hi(F ) over B,andSerreconstructedaspectralsequence

i,j i j i+j E2 = H (B,H (F )) ⇒ H (E,Q). The Serre spectral sequence has some very important extra structure, namely maps

i,j k,l i+k,j+l Er × Er → Er jl which agree on the E2-page with (−1) times the map induced by the cup product maps on the cohomologies of the base and fiber.

2 Exercise: If F → E → B is a fiber bundle, with B path-connected and E orientable, show the equality χ(E)=χ(B)χ(F ) where χ denotes Euler characteristic and χ(F ) is the Euler characteristic of “the” fiber. Exercise: Using the family of fiber bundles SO(n − 1) → SO(n) → Sn−1,inductively calculate the cohomology ring of SO(n). (Grothendieck) Let X be a topological space, with F→X asheafofabeliangroups. Let U be a covering of X by open sets. Given any open set U ⊂ X,therule

Hj(−, F):opensinX → abelian groups j U +→ H (U, F|U ) defines a presheaf on X.Cechcohomologymakessensewithcoefficientsinanypresheaf, and we have the Cech-to-sheaf spectral sequence

i,j ˇ i j i+j E2 = H (U, H (−, F)) ⇒ H (X, F).

Exercise:IfU = {U1,U2} consists of two open sets, prove that the Cech-to-sheaf spectral sequence yields the Mayer-Vietoras long exact sequence

i i i i i+1 ···→H (X, F) → H (U1, F) ⊕ H (U2, F) → H (U1 ∩ U2, F) → H (X, F) → ....

Can you show anything when U consists of three open sets? j Exercise:ThecoveringU is a Leray covering relative to F if H (U, F|U )=0for all j>0 and all U ∈U.SupposingU is a Leray covering, show the spectral sequence degenerates to an isomorphism Hi(X, F) $ Hˇ i(U, F). (Grothendieck) Let X be a topological space, with F→X asheafofabeliangroups. Recall that Hi(X, F) is defined as follows: one chooses a long exact sequence 0 →F→ I0 →I1 →I2 → ... with each Ii an injective sheaf, and Hi(X, F) is the ith cohomology group of the complex

0 → Γ(X, I0) → Γ(X, I1) → Γ(X, I2) → ...

Given a complex G• =0→G0 →G1 → ... of sheaves, the ith cohomology sheaf of G• is the sheaf Hi(G•) defined as the sheafification of the presheaf

ker Γ(U, Gi) → Γ(U, Gi+1) U +→ . imΓ(U, Gi−1) → Γ(U, Gi)

3 In the setup above, note that the complexes F • :0→F→0 and I• :0→I0 → I1 →I2 → ... have the same cohomology sheaves, namely H0(F•) $H0(I•) $Fand Hi(F•) $Hi(I•) $ 0 ∀i ≥ 1 -inotherwords,resolvingF by injective sheaves is equivalent to finding a complex I• of injective sheaves with H0(I•) $F, Hi(I•) $ 0 ∀i ≥ 1. Definition / Theorem: Given any complex of sheaves F•,thereexistsacomplexI• of injective sheaves with Hi(I•) $Hi(F •) ∀i ≥ 0;callanysuchcomplexaninjective resolution of F •.ChooseaninjectiveresolutionI• of F • and define Hi(X, F •) as the ith cohomology group of the complex

0 → Γ(X, I0) → Γ(X, I1) → Γ(X, I2) → ...; then Hi(X, F •) awell-definedabeliangroup(i.e.itdoesn’tdependonthespecific injective resolution I• we chose). The group Hi(X, F •) is the ith hypercohomology group of the complex F •.Thereare two spectral sequences which abut to H∗(X, F •),namely

i,j i j • i+j • E2 = H (X, H (F )) ⇒ H (X, F ) and i,j j i i+j • E1 = H (X, F ) ⇒ H (X, F ). Example: Let X be a smooth complex manifold of complex dimension n.Theholomor- phic de Rham complex of X is the complex

• d 1 d 2 d n ΩX :0→OX → ΩX → ΩX →···→ΩX → 0,

i where ΩX is the sheaf of holomorphic i-forms. The holomorphic functions with zero deriva- 0 • tive are the locally constant functions, so H (ΩX ) $ C;furthermore,aholomorphicanalogue i • of the usual Poincare lemma yields H (ΩX )=0∀i ≥ 1.Hencethefirst hypercohomology spectral sequence degenerates to an isomorphism

i i i • Hsing(X, C) $ H (X, C) $ H (X, ΩX ), so the second hypercohomology spectral sequence now reads

i,j j i i+j E1 = H (X, ΩX ) ⇒ Hsing(X, C).

i In other words, sheaf cohomology of the holomorphic objects ΩX suffices to compute the singular cohomology of X!This,tomeatleast,ismuchmoresurprisingthanthedeRham isomorphism - arbitrary smooth differential forms are very mushy objects, but holomorphic differential forms are extremely rigid.

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