l.A. ADDITIVE STRUCTURE

3

graded. An example of a filtration is the following. Let X be a CW complex, and

let X^ denote its fc-skeleton. Then

FzW{X;R)=kerlHJ{X(i-l);R) —HJ(X;R)}

defines a filtration of the cohomology module of X.

Associated to a (bounded and convergent) filtration is the bigraded module

G(M) which is defined by

G ( M ) ^ =

In general, it can be difficult to recover the giaded module M from G(M) (this is

a problem about module extensions). However, if M* is of finite type, and if R is

a field, then

•pij njri

T

[ r t

( G ( M ) ) = ©

i 9

^ - M .

yez

1.4. DIFFERENTIAL GRADED MODULES

A map / : M* — Nm of degree r between graded modules is an ^-linear map which

increases degrees by r, i.e. f{Mt) C JVz+r. A differential is a map d : M* — M*

of degree 1, with d2 = 0. The cohomology of (M, d) is the graded module

ff(M)=kei[Jtf^Mi+1'

An ^-linear map / : M " —• J V has bidegree (r, s) if /(M^' ) C

7V2+r^+s.

A

differential d of bidegree (r, 1 — r) is a map d : M — M of bidegree (r, 1 — r) with

d2

= 0. The cohomology of (M, d) is the bigraded module

H ^ ( M ) = ^ ^ ^ — i.

im[A^-

r

'i-

1 + r

— M^]

Note that d is a differential on Tot(M), and that Tot(H**(M)) = H*(Tot(M)).

The pair (M, d) is called a differential (bijgraded module.

1.5. SPECTRAL SEQUENCES

An E2-spectral sequence is a collection of differential bigraded modules E

r

, indexed

b y r = 2,3,4,---, endowed with differentials dr of bidegree (r, 1 — r), such that

E

r +

i = H(E

r

).

The spectral sequence converges if for every pair (i,j) there exists a number n = riij

such the two maps

Ei'

are trivial for all r n. The resulting module E£ ^

E^J+1

^ E ^

2

^ E ^

3

^ • • • is

denoted by E ^ , and one says that the spectral sequence converges to the bigraded

module E££. The spectral sequence collapses if all differentials dr vanish, and it