NOTES ON SPECTRAL SEQUENCES - Unicampgtag/calibre/spectral_sequences.pdf · NOTES ON SPECTRAL SEQUENCES ... graded module associated to the ltration of the module Kis de ned by G(K)

NOTES ON SPECTRAL SEQUENCES

AMAR ABDELMOUBINE HENNI1

Abstract. Spectral sequences are a powerful technical tool in algebraic topol-

ogy and algebraic geometry to compute homology/cocomology, and homotopy.

Introduced by J. Leray in order to compute sheaf cohomology, they were usedlater by J. P. Serre to compute cohomology of fibrations, while A. Grothendieck

used them to compute the derived functors of a composed functor. Since then,

they turned out to be a very important research tool. These are introductorynotes to the subject.

Contents

1. Filtrations 22. Inductive limit 33. The spectral sequence associated to a filtred differential module 43.1. The first approximation 83.2. The E∞−term: 93.3. The graded case 104. More on the approximation of E∞ by Er and degeneration of a spectral

sequence 134.1. Degeneration of a spectral sequence 164.2. The five-terme sequence 175. The spectral sequence associated to a double complex 196. The spectral sequence associated to the composition of functors 217. Aplications 217.1. The Cech-de Rahm theorem 217.2. The Leray spectral sequence 217.3. The Local-to-global spectral sequence 21References 22

[email protected]

1

2 A. A. HENNI

1. Filtrations

Let A be a ring (with unity) and K be an A−module;

Definition 1.1.

• A decreasing filtration is a sequence of submodules Kp, p ∈ I ⊆ Z, of K,satisfying

0 ⊆ · · · ⊆ Kp+1 ⊆ Kp ⊆ · · · ⊆ K;⋃p

Kp = K

• An increasing filtration is a sequence of submodules Kp, p ∈ I ⊆ Z, of K,satisfying

K ⊇ · · · ⊇ Kp+1 ⊇ Kp ⊇ · · · ⊇ 0;⋃p

Kp = K

We say that the module K is filtred if it is endowed with a filtration Kp. Thegraded module associated to the filtration of the module K is defined by

G(K) = ⊕pKp/Kp+1

and it is given the obvious grading.

Convention 1.2. From now on, the term filtration will mean decreasing filtration.We will denote the filtration by the lower index, Kp, and the grading by the upperindex Kp.

Let K = ⊕pKp be a graded module, then one can define a decreasing filtrationby putting Kp = ⊕i≥pKi. Indeed, since Kp+1 = ⊕i≥p+1 ⊆ ⊕i≥pKi = Kp, one hasKp+1 ⊆ Kp.

Moreover, one has

G(K) = ⊕p (Kp/Kp+1) = ⊕p(⊕i≥pKi/⊕i≥p+1 K

i)︸︷︷︸

Kp

Hence

G(K) ∼= ⊕pKp ≡ K.

Definition 1.3. We say that a filtration {Kp}p∈I , of a graded module K, is com-patible with the grading if

Kp = ⊕qKp ∩Kq.

Example. Consider a bigraded module K = ⊕i,jKi,j . Define the first (stupid) fil-tration by ′Kp = ⊕i ≥ pKi,j , and the second (stupid) filtration by ′′Kp = ⊕j ≥ pKi,j .The first filtration is compatible with the grading since

⊕′i,jKp ∩Ki,j = ⊕i,j(⊕l≥p,mKl,m ∩Ki,j

)= ⊕l≥p,mKl,m =′ Kp.

Convention 1.4. A compatible filtration of K is sometimes called a graded modulefiltration and the module K is called a filtred graded module.

Definition 1.5. Let K be a filtred graded module. The filtration is called regularif there is an integer ni such that Kp ∩Ki = 0 pour p ≥ ni.

NOTES ON SPECTRAL SEQUENCES 3

Example. Let K be a bigraded module, and let TK = ⊕n(⊕i+j=nKi,j

)be its total

associated module . Then the first filtration {′TKp}p∈I , of TK, is regular if the firstgrading is bounded above or the second grading is bounded below:

Indeed, if the first grading is bounded above then there is an integer s such thatKi,j = 0, for all i ≥ s. Hence

′TKp ∩ TKq =(⊕i≥p,jKi,j

)∩(⊕k+l=qKk,l

),

which is zero for p ≥ s, since(⊕i≥p>sKi,j

)= 0.

Let K and L be two filtred graded A−modules and f : K → L a morphism ofA−modules, that is, f(Kp) ⊆ Lp.Definition 1.6. f is a morphism of filtres graded A−modules if it is compatiblewith the filtrations {Kp}p∈I and {Lq}q∈J , that is, f(Kp) ⊆ Lp.

Consider the associated graded modules G(K) = ⊕p (Kp/Kp+1) and G(L) =⊕p (Lp/Lp+1) , of the graded modules K and L respectively. Since both modules arefiltred graded (i.e., the filtration is compatible with the grading), then we have alsoan induced map: fp,p+1 : Kp/Kp+1 → Lp/Lp+1 such that f(Kp/Kp+1) ⊆ Lp/Lp+1.Moreover, this this defines a (0, 0)−bidegree morphism fG : G(K) → G(L), of theasssociated graded modules.

Lemma 1.7. Let f : K → L be a morphism of filtred graded A−modules K and L,and suppose that the filtration {Kp}p∈I of K is regular. Then

(i) If fG : G(K)→ G(L) is a monomorphism, then f is a monomorphism,(ii) If fG : G(K)→ G(L) is an epimorphism, then f is an epimorphism.

Proof. (i) Let x ∈ Kp and x its class in Kp/Kp+1. Suppose that fG(x) = 0 ∈Lp/Lp+1. Then there is a large enough integer s ≥ p such x ∈ Ks (Ks ⊆Kp+1 ⊆ Kp for s� p ). Since the filtration is regular, then x ∈ Ks∩Ki = 0for some i. Thus x = 0.

(ii) Left for the reader to check.�

2. Inductive limit

In this section we consider an Abelian Category A.Definition 2.1. An ordered set I is called a decreasing filtred set for the familyof objects {Ei}i∈I , in A, if for every (i, j) ∈ I × I such that i � j, there exists amorphism f ij : Ei → Ej , in Mor(A), such that

f ii = IdEi, ∀i ∈ I,

andf ik = f jk ◦ f

ij ,

for every (i, j, k) such that i � j � k.

4 A. A. HENNI

Suppose that the sum object E = (· · · + Ei + Ei+1 + · · · ), associated withthe family decreasingly filtred by I, exists as an object in A. One can define anequivalence relation on E, namely, for x ∈ Ei and y ∈ Ej ; we have x ∼ y if and

only if there exists k ∈ I such that i � k and j � k for which f ik(x) = f jk(y).

Definition 2.2. The quotient object E :=(E/ ∼

)∈ Obj(A) is called the inductive

limit of the family {Ei}i∈I , and it is denoted by E = lim−−→i∈I

.indEi.

Moreover, there exist maps fi : Ei → E, for all i ∈ I, obtained by the obvious

composition Eiai↪→ E

q� E/ ∼, satisfying:

(i) fi = fj ◦ f ij for all i � j;(ii) for xi ∈ Ei, xj ∈ Ej ; fi(xi) = fj(xj) if and only if ∃k ∈ I, with i � k and

j � k, such that fki (xi) = fkj (xj);(iii)

⋃i fi(Ei) = E.

Universal Property. If there exists an object H ∈ Obj(A) and a family of mor-phisms hi : Ei → H in Mor(A) satisfying

(i) hi = hj ◦ f ij for all i � j;(ii) for xi ∈ Ei, xj ∈ Ej ; hi(xi) = hj(xj) if and only if ∃k ∈ I, with i � k and

j � k, such that fki (xi) = fkj (xj);(iii)

⋃i hi(Ei) = H.

Then, there is a unique morphism h : E → H such that hi = h ◦ fi, for all i ∈ I.

Remark 2.3. The inductive limit is also called the direct limit. The dual notionof inductive limit, is called the projective limit or the inverse limit. The reader mayconsult [2]

Example. (i) Let F be a sheaf on a scheme X. The stalk Fx, of F , at a pointx of X, is defined to be the inductive limit of of the system {Ui → F(Ui)/r

ij :

Ui � Uj are restriction maps for i ≥ j}i∈I ([3, Ch.II,§1]).

(ii) The completion of a ring A with respect to the ideal I, also called the I−adiccompletion of A, is obtained as the inverse limit with respect to the system[3, Ch.II,§9]

· · · → A/I3 → A/I2 → A/I.

3. The spectral sequence associated to a filtred differential module

We briefly recall the following:

Definition 3.1. (i) A differential module (K, d) over a ring A is an A−moduleK provided with a an endomorphism d : K → K such that d2 = d ◦ d = 0.

(ii) A morphisme f : (K, d)→ (′K,′ d) is a morphism of A−modules

f : K →′ K


suche that f ◦ d =′ d ◦ f, i.e., the following diagram commutes:

Kf //

d

��

′K

′d

��K

f // ′K

(iii) A filtred differential module (K, d) is a differetial module provided with afiltration {Kp}p∈I such that d(Kp) ⊆ Kp, for all p ∈ I.

In the following, we want to compute the cohomology of the differential moduleM (we omit the diferential d from now on) using a process of successive aproxima-tions that is given by the filtration. Recall that for an injective map of differentialmodules M ↪→ K induces long exact sequence in cohomology :

0→ Z(M)→ Z(K)→ Z(K/M)δ→ H(M)→ H(K)

δ→ H(K/M).

Where Z(M) = {x ∈ M/dx = 0} (respect. Z(K), Z(K/M) ) is the group ofcocycles in M (respect. K, K/M ). We denote the set of coboudaries in M (respect.K, K/M ) by B(M) (respect. B(K), B(K/M) ).

So in general H(M) → H(K) is not injective. For instance, for every termKp of the filtration, we have the injective morphism ip : Kp ↪→ K which inducesa morphism of cohomologies i]p : H(Kp) → H(K). The non-zero element [x] =

{x + dy ∈ Z(Kp)} ∈ H(Kp) might have zero image i]p([x]) if there is an elementz ∈ K such that ip(x) = dz, even if x /∈ B(Kp); that is, one can have:

0 6= [x] = {x = a+ dy ∈ Z(Kp)/a /∈ B(Kp)} ∈ H(Kp) but

i]p([x]) = {i(x) = dz ∈ Z(K)} ∈ H(Kp).

We will use H(K)p for the image of H(Kp) under i]p. We will also denote byil,m : Kl → Km, for l ≥ m, the obvious morphisms between the terms Kl and Km of

the filtration. Their induced map in cohomology will be denoted by i]l,m. Moreover,

one can easily verify that the abelian groups H(K)p satisfy H(K)p+1 ⊆ H(K)p andform a filtration

0 ⊆ · · ·H(K)p+1 ⊆ H(K)p ⊆ · · ·H(K).

of the module H(K). Indeed a commutative diagram

Kp� � ip // K

Kp+1� �

ip+1

//?�

ip+1,p

OO

K

induces the cohomology diagrams

H(Kp)ip // H(K)

H(Kp+1)ip+1

//

ip+1,p

OO

H(K)

and H(K)p� � // H(K)

H(K)p+1� � //

?�

OO

H(K)

where on the left, the induced maps in cohomology are not injective in general,while on the right, the induced maps are injective.

6 A. A. HENNI

In th following we will also use the following results:

Lemma 3.2. Let G be an abelian group and let A1, A2 and B be subgroups of G.Then

A1 +B

A2 +B' A1

A2 +A1 ∩B.

Lemma 3.3. Let L, M, ′M and ′′M be A−modules such that

L

φ

��

ψ

!!DDDD

DDDD

′M

=={{{{{{{{

φ′// M η

// ′′M

is a commutative diagram in which the lower row is exact. Then

Im ψ ' Im φ/Im φ′.

Lemma 3.4. Let G be an abelian group and A, B subgroups sush that A ⊂ B ⊂ G.Then one have the short exact sequence

B/A ↪→ G/A� G/B

The proof is an application of the Snake Lemma.

Now, we fix a filtred differential A−module K (d(Kp) ⊆ Kp for all p), and wewill construct the approximation process to get the derived module H(K).

Fix an integer r ≥ 0; we define the module Zpr := Zd(Kp mod Kp+r), in otherwords, it is the set {x ∈ Kp/dx ∈ Kp+r}. Clearly, Zpr = Kp for r < 0. We also

define the abelian group dZp−r+1r−1 = d({x ∈ Kp−r+1/dx ∈ Kp}), which is the set

of coboudaries in Kp which comes from r − 1 steps higher. Finaly we form the

generalized coboudary group (dZp−r+1r−1 + Zp+1

r−1 ), in which Zp+1r−1 = {x ∈ Kp+1/dx ∈

Kp+r} ⊂ Zpr and dZp−r+1r−1 . Then we define the module

Er = ⊕pEpr where Epr = Zpr /(dZp−r+1r−1 + Zp+1

r−1 ).

One way to represent the abelian groups that we defined above in the following


a ∈ Zp−r+1r−1 ⊆ Kp−r+1_

r−1

��

?�

OO

Kp−r+1

?�

OO

Kp−r+1

?�

OO

......

...

da ∈ dZp−r+1r−1 Kp

?�

OO

Kp

?�

OO

Kp ⊇ Zpr 3 x_

r

��

?�

OO

Kp+1

?�

OO

b ∈ Zp+1r+1 ⊂ Kp+1_

r−1

��

?�

OO

Kp+1

?�

OO

......

...

Kp+r

?�

OO

db(= [0]) ∈ Kp+r

?�

OO

Kp+r 3 ([0] =)dx?�

OO

where in column on the right we have x ∈ Zpr ([dx] = 0 ∈ Kp/Kp+r), in the

column on the left we have da ∈ dZp−r+1r−1 and the middle column we have b ∈ Zp+1

−1r([db] = 0 ∈ Kp/Kp+r). Then Epr is defined by the elements x ∈ Kp that are sentto r steps below by the differential d, modulo the elements b ∈ Kp+1 that are sentr − 1 steps below and the elements da ∈ Kp+1 that comes from r − 1 stepd higherunder d.

Definition 3.5. The family of A−modules {Epr}p∈Z,rN is called the spectral se-quence associated to the filtred differential module (K, d).

Remark 3.6.

(i) In the definition of the generalized coboudaries we used dZp−r+1r−1 instead of

dZp−rr . The meaning of the +1 will be clear when we relate Er with Er+1.

(ii) Remark that, for a fixed p ∈ Z, we have

0 ⊆ · · · dZp−r+1r−1 ⊆ dZp−rr ⊆ · · · ⊆ dK ⊆ Zp∞ ⊆ · · ·Z

pr+1 ⊆ Zpr ⊆ · · · ⊆ Kp.

Hence, as r ∈ N, is increasing, we are taking smaller cocycle groups andbigger coboundaries groups.

(iii) For the decreasing filtration we will also denote K−∞ and K∞ = 0.

8 A. A. HENNI

3.1. The first approximation. The zero’th step of the approximation E0 is givenby the terms

Ep0 = Zp0/(dZp+1−1 + Zp+1

−1 ),

where dZp+1−1 = {x ∈ Kp+1/dx ∈ K} = Kp+1 ∩ K = Kp+1, dZ

p+1−1 = {dx/x ∈

Kp+1} = Kp+1∪dK ⊆ dZp+1−1 and Zp0 = Kp. Hence Ep0 = Kp/Kp+1 and E0 = G(K).

To compute the modules Er for r ≥ 1 we will use the following

Theorem 3.7. There exists a differential dr, of degree r on the module Er, suchthat Hdr (E) is canonically isomorphic to Er+1.

Proof. First, the existence of dr is induced by the differential d; indeed we have

d(Zpr ) ⊆ Zp+rr , and d(dZp−r+1r−1 + Zp+1

r−1 ) ⊆ dZp+rr−1 ,

and descend to the quotient Epr to give a map dr : Epr → Ep+rr of degree r. It isobvious that dr ◦ dr = 0. Thus (Er, dr) is differential module.

The cocycles of dr: A degree p element x ∈ Kp is a cocycle in Er, for the

differential dr, if dx ∈ (dZp+1r−1 + Zp+r+1

r−1 ), that is,

dx = dy + z where y ∈ Zp+1r−1 , z ∈ Zp+r+1

r−1 .

Equivalently Kp 3 x = y + u such that

y ∈ Zp+1r−1 , du = z ∈ Zp+r+1

r−1 ,

i.e., u ∈ Kp such that du ∈ Kp+r+1. Hence u ∈ Zpr+1.

Cocycles dr of degree p : Zdr = (Zp+1r−1 + Zp

r+1)/(dZp−r+1r−1 + Zp+1

r−1 ).

The coboundaries of dr: The degree p elements (in Kp) which are cobound-aries for the differential dr are given by the abelian group dZp−rr which also contains

dZp−r+1r−1 . Hence the coboundaries of dr of degree p :

Bdr = dZp−rr /(dZp−r+1

r + Zp+1r−1 ).

Finally, the cohomology of degree p, with respect to dr, is

Hpdr

(Er) = Zdr/Bdr .

By using Lemma 3.2 one has:

Zdr = (Zp+1r−1 + Zpr+1)/(dZp−r+1

r−1 + Zp+1r−1 )

' Zpr+1/(dZp−r+1r−1 + Zpr+1 ∩ Z

p+1r−1 )

' Zpr+1/(dZp−r+1r + Zp+1

r )

and

dZp−rr /(dZp−r+1r + Zp+1

r−1 ) ' (dZp−rr + Zp+1r−1 )/(dZp−r+1

r + Zp+1r−1 )(1)

Hence


Hpdr

(Er) =Zpr+1/(dZ

p−r+1r + Zp+1

r )

(dZp−rr + Zp+1r−1 )/(dZp−r+1

r + Zp+1r−1 )

' Zpr+1/dZp−rr + Zp+1

r−1 ' Epr+1.

�

Definition 3.8. The family of differential A−modules {(Er, dr)}r∈N subject toEr+1 = H(Er) is called the spectral sequence associated to the filtred differentialmodule (K, d).

Now one has the tool to compute the first approximation E1 by taking the derivedmodule H(E0) with respect to the differential d0 on Ep0 = Kp/Kp+1, namely

E1 = H(G(K)).

3.2. The E∞−term: By taking successive cohomologies, we theoretically end upconstructing the E∞−term of the spectral sequence. Also, we can just apply thedefinition and use remak 3.6; then Ep∞ is by definition2:

Ep∞ = Zp∞/(dZp−∞∞ + Zp+1

∞ ).

Where Zp∞ = {x ∈ Kp/dx = 0} are cocycles in Kp, dZp−∞∞ = Kp ∩ dK are the

coboundries of Kp in K.More precisely, we have

Theorem 3.9. The filtration {H(K)p}p∈Z of the module H(K) provides a canonicalisomorphism E∞ ' H(G(K)).

Proof. For H(K) and its filtration (3), the associated graded module is G(H(K)) =⊕p(H(K)p/H(K)p+1). Let [z] ∈ H(K)p/H(K)p+1 be a class of cocycles in the levelp. Then, [z] = 0 if z is homologous to a + db, where a is a cocycle in Kp+1, thatis a ∈ Zp+1

∞ , and db is a boundary in the level p, that is, db ∈ dZp−∞∞ a cycle andb ∈ K = K−∞. In other words, [z] ∈ Zp∞/(dZp−∞∞ + Zp+1

∞ ), which is by definitionEp∞. Thus

E∞ ' ⊕p(H(K)p/H(K)p+1) ' G(H(K)).

�

The above theorem makes precise the object we are approximating by the spec-tral sequence, namely, the graded module G(H(K)), of the derived module H(K),associated to the filtration {H(K)p}p∈Z.

2The notation dZp−∞∞ emphasises that we are working on the level p in the filtration. It is

also denoted by Bp∞, where Bp

r−1 = dZp−r+1r−1 in our notation.

10 A. A. HENNI

3.3. The graded case. Now we consider the case in which the differential module(K, d) is graded, that is K = ⊕pKp, and that the filtration is compatible with thegrading, i.e., Kp = ⊕qKp ∩Kq. We also suppose that the differential d has degree1, that is, d : Kp → Kp+1. We want to define new terms Er which are bigraded.For this, we look at the modules Zpr dZ

pr when restricted to a specific grading; we

define

Zp,qr = Zpr ∩Kp+q = {x ∈ Kp ∩Kp+q/dx ∈ Kp+r ∩Kp+q+1};

Bp,qr = dZp−r,q+r−1r = dZp−rr ∩Kp+q−1;

Ep,qr = Zp,qr /(Bp,qr−1 + Zp+1,q−1r−1 ), Er = ⊕p,qEp,qr

We call p the filtring grading, p + q the total grding and q the complementarygrading. This ristriction to the terms Epr , to the total grading p + q yields a newdifferential dp,qr : Ep,qr → Ep+r,q−r+1

r . Moreover, we have

Theorem 3.10. The differential dp,qr : Ep,qr → Ep+r,q−r+1r . on the modules Ep,qr

yields a canonical isomorphism Er+1 ' Hdr (Er).

Proof. Let us put Kp,q = Kp ∩Kp+q;

The cocycles of dr: An element of bidegree (p, q) x ∈ Kp,q is a cocycle in Er,

for the differential dr, if dx ∈ (dZp+1,q−1r−1 + Zp+r+1,q−r

r−1 ), that is,

x = dy + z where y ∈ Zp+1,q−1r−1 , z ∈ Zp+r+1,q−r

r−1 .

Equivalently Kp,q 3 x = y + u such that

y ∈ Zp+1,p−1r−1 , du = z ∈ Zp+r+1,q−r

r−1 ,

i.e., u ∈ Kp,q such that du ∈ Kp+r+1,q−r. Hence u ∈ Zp,qr+1.

Cocycles dr of degree p :

Zdr = (Zp+1,q−1r−1 + Zp,q

r+1)/(Bp,qr−1 + Zp+1,q−1

r−1 ).

The coboundaries of dr: The bidegree (p, q) elements (in Kp,q) which arecoboundaries for the differential dr are given by the abelian groupBp,qr = dZp−r,q+r−1r

which also contains Bp,qr−1 = dZp−r+1,q+r−2r−1 . Hence the coboundaries of dr of degree

p :

Bdr = dZp−r,q+r−1r /(dZp−r+1,q+r−2

r + Zp+1,q−1r−1 ).

Finally, the cohomology of degree p, with respect to dr, is

Hpdr

(Er) = Zdr/Bdr .


By using Lemma 3.2 one has:

Zdr = (Zp+1,q−1r−1 + Zp,qr+1)/(dZp−r+1,q+r−2

r−1 + Zp+1,q−1r−1 )

' Zp,qr+1/(dZp−r+1,q+r−2r−1 + Zp,qr+1 ∩ Z

p+1,q−1r−1 )

' Zpr+1/(dZp−r+,q+r−2r + Zp+1,q−1

r )

and

dZp−r,q+r−1r

(dZp−r+1,q+r−2r + Zp+1,q−1

r−1 )'

(dZp−r,q+r−1r + Zp+1,q−1r−1 )

(dZp−r+1,p+r−2r + Zp+1,q−1

r−1 ).

Hence

Hpdr

(Er) =Zp,qr+1/(dZ

p−r+1,q+r−2r + Zp+1,q−1

r )

(dZp−r,q+r−1r + Zp+1,q−1r−1 )/(dZp−r+1,q+r−2

r + Zp+1,q−1r−1 )

' Zp,qr+1/(dZp−r,q+r−1r + Zp+1,q−1

r−1 ) = (Zp,qr+1/Bp,qr + Zp+1,q−1

r−1 )

' Ep,qr+1.

�

Since the filtration {Kp}p∈Z is compatible with graduation of Kq of K, then wehave a filtration {H(K)p}p∈Z, of the derived module H(K), which is compatible with

the graduation Hp(K) = ker(Kp d→Kp+1)

Im (Kp−1 d→Kp), that is, H(K)p = ⊕qH(K)p ∩Hq(K). This

follows from the induced maps in cohomology associated to Kp,q−p = Kp ∩Kq ↪→Kq. Putting Hq(K)p := H(K)p ∩Hq(K), we will have as previousely

Theorem 3.11. The module Ep,q∞ is canonically isomorphic to Hp+q(K)p/Hp+q(K)p+1.

The proof is very similar to Theorem 3.9 and it is left to the reader as an excercise.

Definition 3.12. The family of differential bigraded A−modules

{(Er = ⊕p,qEp,qr , dp,qr : Ep,qr → Ep+r,q−r+1r )}r∈N

subject to Er+1 = H(Er) is called the spectral sequence associated to the filtredgraded differential module (K = ⊕iKi, d : Ki → Ki+1).

With the definition above, one can think of the spectral sequence associatedwith a filtred graded differential A−module (K, d) as book of infinit pages Erof A−modules {Ep,qr , dp,qr }(p,q)∈Z×I , for r ≥ 1, equiped with a differential mapdr, induced by d, and such that the page Er+1 is the cohomology H(Er), withrespect to dr of the previous page Er. Moreover, the page at infinity is canonicallyisomorphic to the graded module G(H(K)), associated to the induced filtration{Hp+q(K)p}p∈Z,q∈I, of the derived module H(K);

12 A. A. HENNI

The first step

E0 : ...

d0

��

...

d0

��

...

d0

��· · · K−1∩K1

K0∩K1

d0

��

⊇ K0∩K1

K1∩K1

d0

��

⊇ K1∩K1

K2∩K1

d0

��

· · ·

· · · K−1∩K0

K0∩K0

d0

��

⊇ K0∩K0

K1∩K0

d0

��

⊇ K1∩K0

K2∩K0

d0

��

· · ·

· · · K−1∩K−1

K0∩K−1

d0

��

⊇ K0∩K−1

K1∩K−1

d0

��

⊇ K1∩K−1

K2∩K−1

d0

��

· · ·

......

...

The second step

E1 : ......

...

· · · d1 // E−1,11

d1 // E0,11

d1 // E1,11

d1 // · · ·

· · · d1 // E−1,01

d1 // E0,01

d1 // E1,01

d1 // · · ·

· · · d1// E−1,−11

d1 // E0,−11

d1 // E1,−11

d1 // · · ·

......

...


The r′th step

Er : ......

...

· · · Er−2,−r+3r Er−1,−r+3

r Er,−r+3r

· · · Er−2,−r+2r Er−1,−r+2

r Er,r+2r

· · · Er−2,r+1r Er−1,−r+1

r Er,−r+1r

· · ·

E−1,2r

dr

::uuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuE0,2r E1,2

r

E−1,1r E0,1r

dr

99ssssssssssssssssssssssssssssssssssssssssssssE1,1r

E−1,0r E0,0r

dr

99ssssssssssssssssssssssssssssssssssssssssssssE1,0r

......

...

4. More on the approximation of E∞ by Er and degeneration of aspectral sequence

Now, we will consider the case in which the filtration is regular, i.e, there existsan integer n(q) such that Kp ∩ Kq = 0 for all p > n(q). The cocycles z ∈ Zp,qrsatisfy drz ∈ Zp+r,q−r+1

r ⊆ Kp+r ∩ Kp+q+1. But, by regularity, there exists aninteger n(p+ q + 1) such that Kp+r ∩Kp+q+1 = 0, for p+ r > n(p+ q + 1). Hencedz = 0 for all r > n(p+ q + 1)− r, and

dr : Ep,qr → Ep+r,p−r+1r is the zero map for all r > n(p+ q + 1)− p.

Moreover, for the above values of r, one has Zp+r,q−r+1r is identically zero. It follows

that for a regular filtration and for large enough values of r, the differential map dris trivial on Er, i.e., Zdr := ker dr is the whole Er. Hence we have an application

14 A. A. HENNI

Er :→ Er+1 = H(Er) = ⊕p,q(Ep,qr

Bp,qr

), which is surjective;

Er � Er+1 = H(Er) for r large enough .

Furthermore, whenever s > r > n(p+q+1)−p we have epimorphisms Ep,qr � Ep,qs .Indeed for the values in question, we also have

0 ⊆· · ·Bp,qr ⊆Bp,qr+1 ⊆· · · ⊂Bp,qs ⊆· · · ⊆Bp,q∞|⋂

Zp,q∞ = · · · = Zp,qs · · · =Zp,qr+1 =Zp,qr ⊆· · · ⊆Kp,q

so that

(Bp,qr−1 +Zp+1,q−1r−1 ) = (Bp,qr−1 +Zp+1,q−1

∞ ) ⊆ (Bp,qs−1 +Zp+1,q−1∞ ) = (Bp,qs−1 +Zp+1,q−1

s−1 ).

By using the fact that Zp,qs = Zp,q∞ = Zp,qs for s > r > n(p+ q+ 1)− p, and Lemma3.4, one has the surjective map:

fsr : Ep,qr // // Ep,qs

Zp,q∞

(Bp,qr−1+Z

p+1,q−1∞ )

Zp,q∞

(Bp,qs−1+Z

p+1,q−1∞ )

Moreover frr : Ep,qr → Ep,qr is the identity, and for t > s > r > n(p+ q + 1)− p,we have

Ep,qrfsr // //

ftr '' ''OOOOOOOOOOOOO Ep,qs

fts��

Ep,qt

that is, f tr = f ts ◦ fsr . Then the E∞−term is given by the inductive limit

Ep,q∞ = lim−→r.indEp,qr for r > n(p+ q + 1)− p.

and there exist maps fr : Ep,qr � Ep,q∞ such that

Ep,qrfr // //

fsr ��

Ep,q∞

Ep,qs

fs

77 77ooooooooooooo

that is, fr = fs ◦ fsr .In this case, one can see the approximation process as an inductive limit associ-

ated to a regular filtration. We say that the spectral sequence converges to or abutsto3 Ep,q∞ .

3From the french verb aboutir.


Let us consider now a morphism f : K −→ L of filtred complexes K and L; thenf(Kp) ⊆ Lp and f(Kq) ⊆ Lq, and

Kf //

Kd

��

L

Ld

��K

f// L

commutes. Hence, for r finite, f(Zp,qr (K)) ⊆ Z(L)p,qr , f(Bp,qr (K)) ⊆ Bp,qr (L) andthere is an induced map fp,qr : Ep,qr (K) −→ Ep,qr (L) such that

Ep,qr (K)fp,qr //

Kdp,qr

��

Ep,qr (L)

Ldp,qr

��Ep+r,q−r+1r (K)

fp+r,q−r+1r

// Ep+r,q−r+1r (L)

These morphisms are compatible with the canonical isomorphisms Er+1 = H(Er)for finite r. On the other hand, we also have a well defined morphism

E∞(K)f∞ // E∞(L)

G(H(K))fG

// G(H(L))

which essentialy fG. S This might be viewed as

fG = lim−→r.ind(fr : Er(K)→ Er(L)).

Suppose now that the filtrations {Kp}p∈Z and {Lq}q∈Z, of K and L respectively,are regular. Then, the induced filtrations {H(K)q}q∈Z and {H(L)q}q∈Z are alsoregualr.

Theorem 4.1. Let f : K −→ L be a morphism of filtred complexes K and L,and assume the filtrations {Kp}p∈Z and {Lq}q∈Z, of K and L respectively, are

regular; If there exists a large enough integer r for which Er(K)fr→ Er(L) is an

isomorphism, then the induced morphism in cohomology f ] : H(K)→ H(L), is alsoan isomorphism.

Proof. If there exists such a large enought integer r such that Er(K)fr→ Er(L), is

an isomorphism then, for every s > r we have also have an isomorphism Es(K)fs→

Es(L), induced by taking (s− r) cohomology steps, since ft is compatible with dtfor all r ≤ t ≤ s. Hence by taking the inductive limit, this induces an isomorphism

between E∞ and E∞, which is just G(H(K))fG→ G(H(L)). It then follows from

Theorem 1.7 that f ] : H(K)→ H(L) is an isomorphism too. �

16 A. A. HENNI

4.1. Degeneration of a spectral sequence.

Definition 4.2. The spectral sequence Ep,qr , associated to a regular filtration of agraded differential A−module (K, d) is said to be degenerate if there is an integerr ≥ 1 for wich it the differential dr is trivial and Er stabilizes, that is

dr = 0, and Er = Er+1 = · · ·E∞.

In most interesting cases for algebraic geometry and algebraic topology, thespectral sequences considered will degenerate ”quickly”, that is the first or thesecond step.

For instance, this might happen if there is an integer r ≥ 1 such that

∀n ∈ Z; ∃q(n) such that En−q,qr = 0, ∀q 6= q(n).

This means that, for a given n ∈ Z, the only non vanishing terms of the bigraded

module Er are En−q(n),q(n)r , for some q(n) depending on n;

♣

♣

NN

NN

N

♣

NNNNNNNNNNN

♣

♣

n = 4

NNNNNNNNNNNNNNNN

♣

n = 3

NNNNNNNNNNNNNNNNNNNNNn = 2

NNNNNNNNNNNNNNNNNNNNNNNNNNn = 1

NNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNN

♣

♣

n = 0♣

NNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNN

n = −1

NNNNNNNNNNNNNNNNNNNNNNNNNN

n = −2

NNNNNNNNNNNNNNNNNNNNN

n = −3

NNNNNNNNNNNNNNN

n = −4

NNNNNNNNNN

NN

NN

N

In the figure above we represent an Er−term of the spectral sequence in whichthe only non vanishing tems, represented by ♣, are given for some q(n).

Since the E∞−term is the associated graded module G(H(K)) associated to theregular4 filtration {H(K)p}p∈Z, then the module Hn(K) is filtred and the gradedmodule G(Hn(K)) associated to its filtration is

G(Hn(K)) = ⊕qEn−q,q∞ .

we have a filtration of the

Theorem 4.3. Let K be a filtred complex such that the filtration is regular, andsuppose that there exists an integer r ≥ 1 such that

∀n ∈ Z; ∃q(n) such that En−q,qr = 0, ∀q 6= q(n).

4Again, the filtration in cohomology induced by the regular filtration {H(K)p}p∈Z, is also

regular; H(K)p ∩ Hq(K)=0 for some q large enough.


Then there is an ijsomorphism

Hn(K) = En−q(n),q(n)r .

Proof. First, we remark that the filtration of the group Hn(K) is given by Hn(K)p :=Hn(K)∩H(K)p, with

⋃p Hn(K)p = Hn(K). Since the filtration is also regular, then

we also have⋂p Hn(K)p = 0.

We have

G(H(K)) ' E∞ = ⊕p,qEp,q∞= ⊕n(⊕nqEn−q,q∞ )

' ⊕nG(Hn(K))

That is G(Hn(K)) ' ⊕nqEn−q,q∞ . By assumption, En−q(n),q(n)∞ = E

n−q(n),q(n)r are

the only non zero terms, hence for each n, there is only one and G(Hn(K)) 'En−q(n),q(n)r . On the other hand, we have

G(Hn(K)) = ⊕q(Hn(K)q/Hn(K)q+1)

= (⋃q

Hn(K)q)�(⋂q

Hn(K)q)

= Hn(K)

�

4.2. The five-terme sequence. In this subsection we will consider positive filtra-tion, that is, Kp = K for all p ≤ 0;

0 ⊆ · · · ⊆ K1 ⊆ K0 = K ⊆ K−1 = K ⊆ · · · ⊆ K

Lemma 4.4. For a positive filtration of the complex K, there exists a morphism

Hn(K)→ E0,nr , for allr ≥ 1.

Proof. Suppose p < 0;

Zp,qr = Zrp ∩Kp+q

= {x ∈ Kp/dx ∈ Kp+r} ∩Kp+q

= {x ∈ K/dx ∈ Kp+r} ∩Kp+q

but we also have Zp+1,q−1r−1 = {x ∈ K/dx ∈ Kp+r} ∩Kp+q = Zp,qr , hence

Ep,qr = Zp,qr /(Bp,qr−1 + Zp+1,q−1r−1 ) ∼= 0 and Ep,q∞ = 0 for all p < 0.

On the other hand, the module Hn(K) is filtred with Hn(K)p, and the gradedmodule associated to this filtration is given by ⊕npEp,n−q∞ . For p ≤ 0;

Hn(K)p = i]p(Hn(Kp))

= Hn(K), since Kp = K, and ip = IdK .

Then, one has Ep,n−p∞ = Hn(K)/Hn(K)p+1, for p ≤ 0. In particular, for p = 0;E0,n∞ = Hn(K)/Hn(K)1, that is, we have a surjective morphism

Hn(K) � E0,n∞ .

18 A. A. HENNI

The elements of E0,nr which are coboundary, with respect to dr, come from

E−r,n+r−1r , which is trivial since −r ≤ 0. In other words, none of the elements ofE0,nr is a coboundary for dr. Moreover, by considering the identification Er+1 =

H(Er) we have

· · · ⊆ E0,nr+1 ⊆ E0,n

r · · · ⊆ E0,n2 ⊆ E0,n

1 .

This is because B0,ndr

= 0 and E0,nr+1 ' Z

0,ndr

(Er) ↪→ E0,nr . In particular we have

Hn(K) //

''PPPPPPP E0,n∞

��E0,nr

∀r ≥ 1.

�

Definition 4.5. A filtration is said to be inferior to the grading if it satisfies:

Kn ∩Kp = 0 p > n.

Inthe following we will consider positive filtrations which are inferior to thegrading. In particular, such a filtrations are regular, and since Zp,qr ⊆ Kp+q ∩Kp,it then follows that Ep,qr = 0 for q < 0.

Lemma 4.6. For a positive filtration which is inferior to the grading, there existsa canonical morphism En,0r → Hn(K).

Proof. First, we have En,0rdr→ En+r,−r+1

r = 0 which applies all the elements of En,0r

to 0, since −r + 1 < 0, for r > 1, i.e., dr = 0 for all r ≥ 2. It follows that we havesurjective morphisms

En,0r � En,0r+1 r ≥ 2.

In particular En,0r � En,0∞ for r ≥ 2, by inductive limit. Hence En,02 � En,0∞ .On the other hand,

En,0∞ = Hn(K)n/Hn(K)n+1,

and since Hn(K)p = 0 for p > n, we thus have

En,0∞ = Hn(K)n

which obviously a subgroup of Hn(K). To resume, we have

En,02// //

++WWWWWWWWWWWWWW En,0r

��En,0∞

� � // Hn(K)

Hence the canonical morphism En,02 � Hn(K).�


Theorem 4.7 (The five-term exact sequence). Let K be a filtred complex,such that the filtration is positive and inferior to the grading;

Kp = K, for p ≤ 0 Kq ∩Kp = 0 for p > q.

Then we have the following canonical exact sequence:

0→ E1,02 → H1(K)→ E0,1

2d2→ E2,0

2 → H2(K).

Proof. The first two maps from the left and the last map on the right exist byLemma4.4 and Lemma4.6 while dr exists by construction of the spectral sequence.The exactness is left as excercise for the reader. �

This sequence can be very usefull in many practicle situations.The following definition and results are inspired by Theorem?? in theparticular

case of Leray spectral sequence for fibred space, when the fibre or the base is ann−sphere.

Definition 4.8. Let K be a filtred complex.

(i) We say that we are in the case of a spherical base if there exists an integern ≥ r such that Ep,qr = 0 for p 6= 0, n.

(ii) We say that we are in the case of a spherical fibre if there exists an integern ≥ r such that Ep,qr = 0 for q 6= 0, n.

Theorem 4.9. Let K be a filtred complex, and the filtration is regular;

(i) If we are in the case of a spherical base for some n, then there exits a longexact sequence:

· · · → En,i−nr → Hi(K)→ E0,ir

dr→ En,i+1−nr → Hi+1(K)→ · · ·

(ii) If we are in the case of a spherical fibre for some n, then there exits a longexact sequence:

· · · → Ei,0r → Hi(K)→ Ei−n,nrdr→ Ei+1,0

r → Hi+1(K) · · ·

5. The spectral sequence associated to a double complex

In this section we construct the spectral sequences associated to a filtred doublecomplex K = ⊕p,qKp,q provided with two differentials: ′d : Kp,q → Kp+1,q and′′d : Kp,q → Kp,q+1, each of degree 1, such that ′d ◦′′ d +′′ d ◦′ d = 0. The totalcomplex TK is defined by TK = ⊕nTnK, where TnK = ⊕p+q=nKp,q, and thetotal differential d =′ d+′′ d.

We consider the first filtration as being the stupid filtration formed by the abeliansub-groups ′Kp = ⊕i≥p,jKi,j . Then there is an associated spectral sequence withterms denoted by ′Ep,qr .

We also define the second filtration as the stupid filtration formed by the abeliansub-groups ′′Kq = ⊕i,j≥qKi,j , and we denote by ′′Ep,qr the terms of the spectralsequence corresponding to it.

We now introduce the following notations: ′Kqp :=′ Kp∩TKp+q = ⊕p+qi=pK

i,p+q−i

for the p−th term of the first filtration of the degree n submodule of the total

20 A. A. HENNI

module and ′′Kqp :=′′ Kp ∩ TKp+q = ⊕p+qj=pK

p+q−j,j for the p−th term of thesecond filtration of the total degree n submodule of the total module.

The zero’th term of the first spectral sequence is then given by

′Ep,q0 =(′Kq

p/′Kq

p+1

)' Kp,q,

hence the differential

′d0 :′ Ep,q0 ' Kp,q →′ Ep,q+10 ' Kp,q+1,

induced by the total differential d, is nothing but ′′d. Thus the first term of the firstspectral sequence is

′Ep,q1 = Hq′′d(K

p,∗); ′d0 ∼=′′ d.

That is, the cohomology of the complex Kp,∗ with respect to the differential ′′d :Kp,q → Kp,q+1.

On the other hand, the zero’th term of the second spectral sequence is

′′Ep,q0 =(′′Kq

p/′′Kq

p+1

)' Kq,p.

The induced differential

′′d0 :′′ Ep,q0 ' Kq,p → Ep,q+10 ' Kq+1,p,

is in this case identifiable with ′d, and the first term of the second spectral sequenceis

′′Ep,q1 = Hq′d(K

∗,p); ′′d0 ∼=′ d.i.e., the cohomology of the complex K∗,p with respect to the differential ′d : Kq,p →Kq+1,p.

Let us compute the second term ′Ep,q2 of the first spectral sequence; to do thiswe will use the exact sequence of complexes

0→ K∗p+1/K∗p+2 → K∗p/K

∗p+2 → K∗p/K

∗p+1 → 0

where

K∗p+1/K∗p+2∼= Kp+1,∗−1, K∗p/K

∗p+2∼= Kp+1,∗−1⊕Kp,∗, K∗p+1/K

∗p+2∼= Kp,∗.

The differential of the first and the last term is ′′d, while the one in the middleterm is given by ( ′′d 0

′d ′′d

).

Taking the long exact sequence in cohomology, one has the connecting morphism

Hq′′d(K

p,∗)′δ→ Hq+1

′′d (Kp+1,∗),

which is obviously induced by ′d. On the other hand we have

Hq′′d(K

p,∗) =′ Ep,q1 and Hq+1′′d (Kp+1,∗) =′ Ep+1,q

1

Thus, we have′Ep,q2

∼= Hp′d(H

q′′d(K)).

Similarly, we obtain′′Ep,q2

∼= Hp′′d(H

q′d(K)).


Remark 5.1. Both spectral sequences ′Er and ′′Er converge to the same derivedmodule G(H(TK)), i.e., the graded module associated to the cohomology of thetotal module TK, while the filtrations associated to them are different. In practicalsituations, this provides many interesting information (see applications).

6. The spectral sequence associated to the composition of functors

7. Aplications

7.1. The Cech-de Rahm theorem.

7.2. The Leray spectral sequence.

7.3. The Local-to-global spectral sequence.

22 A. A. HENNI

References

[1] R. Godement,”Topologie Algebrique et Theorie des faisceaux”, Hermann, Paris, 1958.[2] P. J. Hilton, U. Stammbach, ”A Course in Homological Algebra”, Springer-Verlag, New York,

1971.

[3] R. Hartshorne, ”Algebraic geometry”, Springer-Verlag, GTM, 52.

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NOTES ON SPECTRAL SEQUENCES - Unicampgtag/calibre/spectral_sequences.pdf · NOTES ON SPECTRAL SEQUENCES ... graded module associated to the ltration of the module Kis de ned by G(K)